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Author: MathKind
Number theory is a branch of mathematics which helps to study the set of positive whole numbers, say
1, 2, 3, 4, 5, 6,. . . ,
which are also called the set of natural numbers and sometimes called “higher arithmetic”. Number theory helps to study the relationships between different sorts of numbers. Natural numbers are separated into a variety of times. Here are some of the familiar and unfamiliar examples with quick number theory introduction.
Odd Numbers – 1, 3, 5, 7, 9, 11, 13, 15, 17, 19…..
Even Numbers – 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 . . .
Square Numbers – 4, 9, 16, 25, 36, 49, 64, 81,100 . . .
Cube Numbers – 8, 27, 64, 125, 216, 343, 512 . . .
Prime Numbers – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,53, 59, 61 . . .
Composite Numbers – 4, 6, 8, 9, 10, 12, 14, 15, 16,18, 20, 21, 22, 24 . . .
1 (modulo 4) Numbers – 1, 5, 9, 13, 17, 21, 25, . . .
3 (modulo 4) Numbers – 3, 7, 11, 15, 19, 23, 27, . . .
Triangular Numbers – 3, 6, 10, 15, 21, 28, 26, 45,. . .
Perfect Numbers – 6, 28, 496, 8128, . . .
Fibonacci Numbers -1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. . .
Many of these types of numbers like odd, even, square, cube prime and composite numbers are already familiar to you. Other cases, such as the “modulo 4” numbers, Triangular numbers, perfect numbers and Fibonacci numbers are not familiar to you.
Here are some of the most important number theory applications. Number theory is used to find some of the important divisibility tests, whether a given integer m divides the integer n. Number theory have countless applications in mathematics as well in practical applications such as
It is also defined in hash functions, linear congruence, Pseudorandom numbers and fast arithmetic operations.
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1, 2, 3, 4, 5, 6,. . . ,
which are also called the set of natural numbers and sometimes called “higher arithmetic”. Number theory helps to study the relationships between different sorts of numbers. Natural numbers are separated into a variety of times. Here are some of the familiar and unfamiliar examples with quick number theory introduction.
Odd Numbers – 1, 3, 5, 7, 9, 11, 13, 15, 17, 19…..
Even Numbers – 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 . . .
Square Numbers – 4, 9, 16, 25, 36, 49, 64, 81,100 . . .
Cube Numbers – 8, 27, 64, 125, 216, 343, 512 . . .
Prime Numbers – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,53, 59, 61 . . .
Composite Numbers – 4, 6, 8, 9, 10, 12, 14, 15, 16,18, 20, 21, 22, 24 . . .
1 (modulo 4) Numbers – 1, 5, 9, 13, 17, 21, 25, . . .
3 (modulo 4) Numbers – 3, 7, 11, 15, 19, 23, 27, . . .
Triangular Numbers – 3, 6, 10, 15, 21, 28, 26, 45,. . .
Perfect Numbers – 6, 28, 496, 8128, . . .
Fibonacci Numbers -1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. . .
Many of these types of numbers like odd, even, square, cube prime and composite numbers are already familiar to you. Other cases, such as the “modulo 4” numbers, Triangular numbers, perfect numbers and Fibonacci numbers are not familiar to you.
Odd Numbers :
The numbers that are not evenly divided by 2 are called odd numbers.Even Numbers :
The numbers that are evenly divided by 2 are called even numbers.Square Numbers :
A number multiplied by itself is called square numbersCube Numbers :
A number multiplied by itself 3 times is called cube numbers.Prime numbers :
If a number has only two factors: 1 and the number is called prime numbersComposite Numbers :
Composite number has more than two factors. The composite numbers are numbers which are not prime numbers. The number 1 is neither prime nor composite.Modulo 4 Numbers :
A number is said to ne 1 (modulo 4 ) number if it leaves a remainder 1 when divided by 4.Similarly, if a number leaves a remainder 3 when divided by 4, it is said to be 3 (modulo 4) number.Triangular Numbers :
A number is said to be triangular number, when that number of pebbles can be arranged in a triangle using one pebble at the top, two pebbles in next row, three pebbles in next row and so on.Fibonacci Numbers :
Fibonacci numbers are created starting with 1 and 1, then get the next number in the list and adds the previous two numbers. Say, 1+1 =2 and then adds 1+2 you get 3, then adds 2+3 gives 5, then 3+5 gives 8 and so on.Applications of Number Theory
Here are some of the most important number theory applications. Number theory is used to find some of the important divisibility tests, whether a given integer m divides the integer n. Number theory have countless applications in mathematics as well in practical applications such as
- Security System like in banking securities
- E – commerce websites
- Coding theory
- Barcodes
- Making of modular designs
- Memory management system
- Authentication system
It is also defined in hash functions, linear congruence, Pseudorandom numbers and fast arithmetic operations.
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Mathematics is the science of numbers as Aristotle defined. Here we have collected all the important mathematics definitions. Browse these definitions or use the Search function for a specific definition.
Math Definitions
There are currently 184 definitions in this directory
Abacus
An abacus has beads that slide on rods. It can be used to count, add, subtract, multiply and more.
Abscissa
The horizontal ("x") value in a pair of coordinates. How far along the point is.
Always written first in an ordered pair of coordinates such as (12,5).
In this example, the value "12" is the abscissa.
(The second value "5" shows how far up or down and is called the Ordinate)
Always written first in an ordered pair of coordinates such as (12,5).
In this example, the value "12" is the abscissa.
(The second value "5" shows how far up or down and is called the Ordinate)
Absolute Error
The difference between the actual and measured value.
Shown as a positive value.
Example: When your instrument measures in "1"s, then any value between 6½ and 7½ is measured as "7", so the absolute error is ½.
Shown as a positive value.
Example: When your instrument measures in "1"s, then any value between 6½ and 7½ is measured as "7", so the absolute error is ½.
Absolute Value
How far a number is from zero.
Examples:6 is 6 away from zero, so the absolute value of 6 is 6−6 is 6 away from zero, so the absolute value of −6 is 6
In other words it is the magnitude of a number, no negatives allowed.
The symbol "|" is placed either side to mean "Absolute Value", so we write: |−6| = 6
Examples:6 is 6 away from zero, so the absolute value of 6 is 6−6 is 6 away from zero, so the absolute value of −6 is 6
In other words it is the magnitude of a number, no negatives allowed.
The symbol "|" is placed either side to mean "Absolute Value", so we write: |−6| = 6
Acceleration
How fast velocity changes.
Usually measured as m/s2 ("meters per second squared").
Example: going from 5 m/s (5 meters per second) to 6 m/s in exactly one second is an acceleration of 1 meter per second per second.
And two lots of "per second" becomes "per second squared".
So the acceleration is 1 m/s2.
Usually measured as m/s2 ("meters per second squared").
Example: going from 5 m/s (5 meters per second) to 6 m/s in exactly one second is an acceleration of 1 meter per second per second.
And two lots of "per second" becomes "per second squared".
So the acceleration is 1 m/s2.
Acre
A US Standard Unit of area, usually used to measure land.
1 acre = 4,840 square yards.
1 acre is about 0.4 hectares in the Metric system, or exactly 4,046.8564224 square meters.
1 acre = 4,840 square yards.
1 acre is about 0.4 hectares in the Metric system, or exactly 4,046.8564224 square meters.
Additive Identity
The "Additive Identity" is 0, because adding 0 to a number does not change it: a + 0 = 0 + a = a
Adjacent Angles
Two angles that have a common side and a common vertex (corner point), and don't overlap.
Algebra
Algebra uses letters (like x or y) or other symbols in place of values, and then plays with them using special rules.
Example: x + 3 = 7"x" is used in place of a value we don't know yet and is called the "unknown" or the "variable".
In this case the value of "x" can be found by subtracting 3 from both sides of the equal sign like this:
Start with: x + 3 = 7Subtract 3 from both sides: x + 3 − 3 = 7 − 3Calculate: x + 0 = 4Answer: x = 4
Example: x + 3 = 7"x" is used in place of a value we don't know yet and is called the "unknown" or the "variable".
In this case the value of "x" can be found by subtracting 3 from both sides of the equal sign like this:
Start with: x + 3 = 7Subtract 3 from both sides: x + 3 − 3 = 7 − 3Calculate: x + 0 = 4Answer: x = 4
Algorithm
A step-by-step solution.
Each step has clear instructions. Like a recipe.
Example: one algorithm for adding two digit numbers is:1. add the tens2. add the ones3. add the numbers from steps 1 and 2
So to add 15 and 32 using that algorithm:1. add 10 and 30 to get 402. add 5 and 2 to get 73. add 40 and 7 to get 47
Long Division is another example of an algorithm: when you follow the steps you get the answer.
Computers use algorithms all the time.
"Algorithm" is named after the 9th century Persian mathematician Al-Khwarizmi.
Each step has clear instructions. Like a recipe.
Example: one algorithm for adding two digit numbers is:1. add the tens2. add the ones3. add the numbers from steps 1 and 2
So to add 15 and 32 using that algorithm:1. add 10 and 30 to get 402. add 5 and 2 to get 73. add 40 and 7 to get 47
Long Division is another example of an algorithm: when you follow the steps you get the answer.
Computers use algorithms all the time.
"Algorithm" is named after the 9th century Persian mathematician Al-Khwarizmi.
Alternate Exterior Angles
When two lines are crossed by another line (the Transversal), a pair of angles• on the outer side of those two lines• but on opposite sides of the transversalare called Alternate Exterior Angles.
Alternate Interior Angles
When two lines are crossed by another line (the Transversal), a pair of angles• on the inner side of each of those two lines• but on opposite sides of the transversalare called Alternate Interior Angles.
Alternating Series
An infinite series where the terms alternate between positive and negative.
Example: 1/2 − 1/4 + 1/8 − 1/16 + ... = 1/3
Example: 1/2 − 1/4 + 1/8 − 1/16 + ... = 1/3
Altitude (geometry)
Generally: another word for height.
For Triangles: a line segment leaving at right angles from a side and going to the opposite corner.
For Triangles: a line segment leaving at right angles from a side and going to the opposite corner.
Amplitude
The height from the center line to the peak (or trough) of a periodic function.Or we can measure the height from highest to lowest points and divide that by 2.
Angle Bisector
A line that splits an angle into two equal angles.
("Bisect" means to divide into two equal parts.)
("Bisect" means to divide into two equal parts.)
Angle of Elevation
The "upwards" angle from the horizontal to a line of sight from the observer to some point of interest.If the angle goes "downwards" it is called an Angle of Depression.
Annual Percentage Rate (APR)
The percentage cost of borrowing per year, including interest, fees, etc.
Example. A $1000 loan repaid after one year with $80 interest plus a $10 service fee, has a total finance charge of $90, and so has an APR of 9%.
Example. A $1000 loan repaid after one year with $80 interest plus a $10 service fee, has a total finance charge of $90, and so has an APR of 9%.
Annual Percentage Yield (APY)
The annual rate of return on an investment.
Example: A $1,000 investment at 10% per year earns $100 in one year, and has an APY of 10%.
Example: A $1,000 investment at 5% per half-year earns $102.50 in one year, and has an APY of 10.25%.
Example: A $1,000 investment at 10% per year earns $100 in one year, and has an APY of 10%.
Example: A $1,000 investment at 5% per half-year earns $102.50 in one year, and has an APY of 10.25%.
Anticlockwise
Moving in the opposite direction to the hands on a clock.
Also called Counterclockwise (US English).
Angles are usually measured anticlockwise.
Also called Counterclockwise (US English).
Angles are usually measured anticlockwise.
Area
The size of a surface.The amount of space inside the boundary of a flat (2-dimensional) object such as a triangle or circle, or surface of a solid (3-dimensional) object.
Argument
An input to a function: a variable that affects a functions result.
Example: imagine a function that works out the height of a tree:
h(year) = 20 × year,
then "year" is an argument of the function "h".
Example: imagine a function that works out the height of a tree:
h(year) = 20 × year,
then "year" is an argument of the function "h".
Arithmetic
The basic calculations we make in everyday life: addition, subtraction, multiplication and division.The subject also includes fractions and percentages (related to division), and exponents (related to multiplication).
Arithmetic Sequence
A sequence made by adding the same value each time.Example: 1, 4, 7, 10, 13, 16, 19, 22, 25, ...(each number is 3 larger than the number before it)
Ascending Order
Arranged from smallest to largest. Increasing.
Example: 3, 9, 12, 55 are in ascending order.
Example: 3, 9, 12, 55 are in ascending order.
Asset
Something you own that has value.
Specially if it helps you make money, but it doesn't have to.
Examples: personal property, real estate, stocks/shares, bank accounts
Specially if it helps you make money, but it doesn't have to.
Examples: personal property, real estate, stocks/shares, bank accounts
Associative Law
When adding it doesn't matter how we group the numbers (i.e. which we calculate first).
Example addition: (6 + 3) + 4 = 6 + (3 + 4)Because 9 + 4 = 6 + 7 = 13
Also when multiplying it doesn't matter how we group the numbers.
Example multiplication: (2 × 4) × 3 = 2 × (4 × 3)Because 8 × 3 = 2 × 12 = 24
Example addition: (6 + 3) + 4 = 6 + (3 + 4)Because 9 + 4 = 6 + 7 = 13
Also when multiplying it doesn't matter how we group the numbers.
Example multiplication: (2 × 4) × 3 = 2 × (4 × 3)Because 8 × 3 = 2 × 12 = 24
Asymmetry
Asymmetry means "no symmetry". Something without symmetry is asymmetrical. It is also possible to be symmetrical in one way and asymmetrical in another.
Attribute
A property of an object or person etc. Something you can say it has (such as size or color).
Example: The attributes of a dog include height, speed and color.
Example: The attributes of a dog include height, speed and color.
Axes
Plural of Axis.
Pronounced "ak-seez".
Axes often means the "x" and "y" lines that cross at right angles to make a graph.
Pronounced "ak-seez".
Axes often means the "x" and "y" lines that cross at right angles to make a graph.
Bakers Dozen
A "baker's dozen" is 13 (A dozen is 12). In the past a baker could be fined for selling items below weight, so they added one extra "to be sure".
Balance
When both sides have the same quantity or mass.
Here "x" is balanced by 4 "1"s, so x must be 4
Here "x" is balanced by 4 "1"s, so x must be 4
Bank
Banks look after peoples money, give loans and have other financial services. They must be government approved, and must follow special rules set by the government. Banks exist to make a profit for their owners.
Bankrupt
When a person can't pay their debts they can be legally declared bankrupt.
A bankrupt person loses everything except some basic things they own, but all the debt will go away. They receive a bad credit record and may not be able to borrow money again for years.
A bankrupt company gets protection from people it owes money to (so they cannot destroy all of the physical capital and goodwill by breaking it apart and moving it away). This gives more time for the business to work out a good solution.
A bankrupt person loses everything except some basic things they own, but all the debt will go away. They receive a bad credit record and may not be able to borrow money again for years.
A bankrupt company gets protection from people it owes money to (so they cannot destroy all of the physical capital and goodwill by breaking it apart and moving it away). This gives more time for the business to work out a good solution.
Bar Graph
A graph drawn using rectangular bars to show how large each value is.
The bars can be horizontal or vertical.
The bars can be horizontal or vertical.
Base (geometry)
The surface a solid object stands on, or the bottom line of a shape such as a triangle or rectangle. But the top is also called a base when it is parallel to the bottom!
Base (numbers)
Definition 1: The number that gets multiplied when using an exponent.
Examples:
1. in 82, 8 is the base, and the result is 8 × 8 = 64
2. in 53, 5 is the base, and the result is 5 × 5 × 5 = 125
Definition 2: How many digits in a number system.
The decimal number system we use every day has 10 digits {0,1,2,3,4,5,6,7,8,9} and so it is Base 10.
A binary digit can only be 0 or 1, so is Base 2.
A hexadecimal digit can be {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}, so is Base 16.
Examples:
1. in 82, 8 is the base, and the result is 8 × 8 = 64
2. in 53, 5 is the base, and the result is 5 × 5 × 5 = 125
Definition 2: How many digits in a number system.
The decimal number system we use every day has 10 digits {0,1,2,3,4,5,6,7,8,9} and so it is Base 10.
A binary digit can only be 0 or 1, so is Base 2.
A hexadecimal digit can be {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}, so is Base 16.
Billion
A thousand millions.
1,000 x 1,000,000 = 1,000,000,000
Which is a 1 followed by 9 zeros
Using scientific notation: 1 × 109
(In many non-English speaking countries it means a million million, which is a 1 followed by 12 zeros, or 1 × 1012)
Below is a cube made of a million smaller cubes. If each of the small cubes is worth a thousand dollars, then the whole cube is worth a billion dollars.
1,000 x 1,000,000 = 1,000,000,000
Which is a 1 followed by 9 zeros
Using scientific notation: 1 × 109
(In many non-English speaking countries it means a million million, which is a 1 followed by 12 zeros, or 1 × 1012)
Below is a cube made of a million smaller cubes. If each of the small cubes is worth a thousand dollars, then the whole cube is worth a billion dollars.
Binary
Binary Numbers use only the digits 0 and 1.
Examples:
• 0 in Binary equals 0 in the Decimal Number System,
• 1 in Binary equals 1 in the Decimal Number System,
• 10 in Binary equals 2 in the Decimal Number System,
• 11 in Binary equals 3 in the Decimal Number System,
• 100 in Binary equals 4 in the Decimal Number System,
• etc.
Also called Base 2
Examples:
• 0 in Binary equals 0 in the Decimal Number System,
• 1 in Binary equals 1 in the Decimal Number System,
• 10 in Binary equals 2 in the Decimal Number System,
• 11 in Binary equals 3 in the Decimal Number System,
• 100 in Binary equals 4 in the Decimal Number System,
• etc.
Also called Base 2
Binary Number
Binary Numbers use only the digits 0 and 1.
Examples:• 0 in Binary equals 0 in the Decimal Number System,• 1 in Binary equals 1 in the Decimal Number System,• 10 in Binary equals 2 in the Decimal Number System,• 11 in Binary equals 3 in the Decimal Number System,• 100 in Binary equals 4 in the Decimal Number System,• etc.
Also called Base 2
Examples:• 0 in Binary equals 0 in the Decimal Number System,• 1 in Binary equals 1 in the Decimal Number System,• 10 in Binary equals 2 in the Decimal Number System,• 11 in Binary equals 3 in the Decimal Number System,• 100 in Binary equals 4 in the Decimal Number System,• etc.
Also called Base 2
Binary Operation
An operation that needs two inputs.
A simple example is the addition operation "+":
In 2 + 3 = 5 the operation is "+", which takes two values (2 and 3) and gives the result 5
Subtraction, multiplication and division are also binary operations, and there are many more.
The two inputs are called "operands".
Also, a binary operation should take and return things of the same type! In other words, the operands and the result must belong to the same Set.
An operation that has only one input is called a "unary operation".
Example: the square root function is a unary operation: √(16) = 4 has just one input "16" to produce an output of 4
A simple example is the addition operation "+":
In 2 + 3 = 5 the operation is "+", which takes two values (2 and 3) and gives the result 5
Subtraction, multiplication and division are also binary operations, and there are many more.
The two inputs are called "operands".
Also, a binary operation should take and return things of the same type! In other words, the operands and the result must belong to the same Set.
An operation that has only one input is called a "unary operation".
Example: the square root function is a unary operation: √(16) = 4 has just one input "16" to produce an output of 4
Bisect
To divide into two equal parts.
We can bisect line segments, angles, and more.
The dividing line is called the "bisector"
We can bisect line segments, angles, and more.
The dividing line is called the "bisector"
Bisector
The line that divides something into two equal parts.
You can bisect line segments, angles, and more.
You can bisect line segments, angles, and more.
Bit
A single binary digit: 0 or 1
Example: 110100 has 6 bits.
Symbol is b
Example: 1Mb means 1 million bits
Example: 110100 has 6 bits.
Symbol is b
Example: 1Mb means 1 million bits
Bounds
Either of these two:
Lower bound: a value that is less than or equal to every element of a set of data.
Upper bound: a value that is greater than or equal to every element of a set of data.
Example: in {3,5,11,20,22} 3 is a lower bound, and 22 is an upper bound
But be careful! 2 is also a lower bound (it is less than any element of that set), in fact any value 3 or less is a lower bound.
Likewise any value 22 or above is also an upper bound, such as 50 or 1000.
Example: how tall is a human? We may not know the exact shortest human, but we can say that 0 is a lower bound (can't be less than zero in height, right?)
Lower bound: a value that is less than or equal to every element of a set of data.
Upper bound: a value that is greater than or equal to every element of a set of data.
Example: in {3,5,11,20,22} 3 is a lower bound, and 22 is an upper bound
But be careful! 2 is also a lower bound (it is less than any element of that set), in fact any value 3 or less is a lower bound.
Likewise any value 22 or above is also an upper bound, such as 50 or 1000.
Example: how tall is a human? We may not know the exact shortest human, but we can say that 0 is a lower bound (can't be less than zero in height, right?)
Budget
An estimate of income and spending for some period of time.
Example: Sam has a weekly budget to make sure there is enough money at the end of the week for a night out.
Example: HappyPup Ltd has just finished their Yearly Budget and has set aside $20,000 for the local animal shelter.
Example: Sam has a weekly budget to make sure there is enough money at the end of the week for a night out.
Example: HappyPup Ltd has just finished their Yearly Budget and has set aside $20,000 for the local animal shelter.
Byte
8 binary digits
A single binary digit (called a "bit") can only be 0 or 1
Example: 1 is a bit
Example: 10110110 is a byte
A bit can have only 2 different values: 0 or 1
A byte can have 2×2×2×2×2×2×2×2 = 256 different values
Symbol is B (and b means bit)
Example: 1MB means 1 million bytes (or 8 million bits)
A single binary digit (called a "bit") can only be 0 or 1
Example: 1 is a bit
Example: 10110110 is a byte
A bit can have only 2 different values: 0 or 1
A byte can have 2×2×2×2×2×2×2×2 = 256 different values
Symbol is B (and b means bit)
Example: 1MB means 1 million bytes (or 8 million bits)
Calculate
To work out an answer, usually by adding, multiplying etc.
Example: Calculate the cost of 10 apples when each apple costs 0.50.
Answer: 10 x 0.50 = 5.00
Example: Calculate the cost of 10 apples when each apple costs 0.50.
Answer: 10 x 0.50 = 5.00
Calculus
A branch of mathematics that looks at how things change, or how things add up, by breaking them into really small pieces.
Differential Calculus cuts things into small pieces to find how they change, so we can work out slopes, speed, etc
Integral Calculus joins (integrates) the small pieces together to find how much there is, so we can work out areas within curves, volumes, and more.
Differential Calculus cuts things into small pieces to find how they change, so we can work out slopes, speed, etc
Integral Calculus joins (integrates) the small pieces together to find how much there is, so we can work out areas within curves, volumes, and more.
Capacity
The amount that something can hold.
Usually it means volume, such as milliliters (ml) or liters (l) in Metric, or pints or gallons in Imperial.
Example: This glass has a capacity of 300 ml (but is actually holding only 160 ml)
Usually it means volume, such as milliliters (ml) or liters (l) in Metric, or pints or gallons in Imperial.
Example: This glass has a capacity of 300 ml (but is actually holding only 160 ml)
Capital gain
Money gained when you sell an asset (such as personal property, real estate, etc) for more than it cost.
Example: Buy a building for $1,000,000, then sell it next year for $1,200,000. The capital gain is $200,000
Example: Buy a building for $1,000,000, then sell it next year for $1,200,000. The capital gain is $200,000
Capital loss
Money lost when you sell an asset (such as personal property, real estate, etc.) for less than it cost.
Example: Buy a building for $1,000,000, then sell it next year for $950,000. The capital loss is $50,000
Example: Buy a building for $1,000,000, then sell it next year for $950,000. The capital loss is $50,000
Celsius
Celsius (or "degrees Celsius", or sometimes "Centigrade") is a temperature scale.
It is used to tell how hot or cold something is
It is often written as °C
Water freezes at 0°C and boils at 100°C
It is used to tell how hot or cold something is
It is often written as °C
Water freezes at 0°C and boils at 100°C
Cent
The smallest money value in many countries.
100 cents equals one dollar in the US, Canada and Australia.
100 cents equals one euro in Europe.
100 cents equals one dollar in the US, Canada and Australia.
100 cents equals one euro in Europe.
Centimeter
A measure of length. It is about the width of a fingernail.
There are 100 centimeters in a meter.
The abbreviation is cm
A typical ruler measures up to 30 cm, like the one shown below. It has cm along the top and inches (1 inch = 2.54 cm) along the bottom.
There are 100 centimeters in a meter.
The abbreviation is cm
A typical ruler measures up to 30 cm, like the one shown below. It has cm along the top and inches (1 inch = 2.54 cm) along the bottom.
Century
One hundred years.
The 1st Century was from the Year 1 to the Year 100
The 2nd Century was from 101 to 200
The 20th Century was from 1901 to 2000
The 21st Century is from 2001 to 2100
We are currently in the 21st century.
This is a 5 Dollar gold coin from 1855, which is in the 19th Century
The 1st Century was from the Year 1 to the Year 100
The 2nd Century was from 101 to 200
The 20th Century was from 1901 to 2000
The 21st Century is from 2001 to 2100
We are currently in the 21st century.
This is a 5 Dollar gold coin from 1855, which is in the 19th Century
Closed Curve
A curve that joins up so there are no end points.
Example: an ellipse is a closed curve. So is a circle.
Example: an ellipse is a closed curve. So is a circle.
Coefficient
A number used to multiply a variable.
Example: 6z means 6 times z, and "z" is a variable, so 6 is a coefficient.
Variables with no number have a coefficient of 1.
Example: x is really 1x.
Sometimes a letter stands in for the number.
Example: In ax2 + bx + c, "x" is a variable, and "a" and "b" are coefficients.
Example: 6z means 6 times z, and "z" is a variable, so 6 is a coefficient.
Variables with no number have a coefficient of 1.
Example: x is really 1x.
Sometimes a letter stands in for the number.
Example: In ax2 + bx + c, "x" is a variable, and "a" and "b" are coefficients.
Collateral
In Finance, collateral is property that a borrower promises to give to a lender if they can't pay back a loan.
Example: you borrow $2,000 and use your car as collateral. If you don't pay back the loan according to the agreement (maybe you miss a monthly payment) they can take your car.
Example: you borrow $2,000 and use your car as collateral. If you don't pay back the loan according to the agreement (maybe you miss a monthly payment) they can take your car.
Compass
An instrument that shows us direction (such as North, South, East and West) by a small magnetic needle that points North/South.
Composite Function
A function made of other functions, where the output of one is the input to the other.Example: the functions 2x+3 and x2 together make the composite function (2x+3)2
Composite Numbers
Composite number has more than two factors. The composite numbers are numbers which are not prime numbers. The number 1 is neither prime nor composite.
Compound Interest
Where interest is calculated on both the amount borrowed plus previous interest. Usually calculated one or more times per year.
Computer
A machine that can be programmed to do things with data.
Computers can usually:
• "input" data from a mouse, keyboard or touch-screen,
• "process" the data using a CPU and memory, and
• "output" the result onto a screen or save it to disk.
Computers can usually:
• "input" data from a mouse, keyboard or touch-screen,
• "process" the data using a CPU and memory, and
• "output" the result onto a screen or save it to disk.
Cone
A solid (3-dimensional) object that has a circular base joined to point by a curved side.
The point is called a vertex.
The point is called a vertex.
Convex
Curved outwards.
Example: A polygon (which has straight sides) is convex when there are NO "dents" or indentations in it (no internal angle is greater than 180°)
The opposite idea is called "concave".
Example: A polygon (which has straight sides) is convex when there are NO "dents" or indentations in it (no internal angle is greater than 180°)
The opposite idea is called "concave".
Coordinates
A set of values that show an exact position.
On graphs it is usually a pair of numbers: the first number shows the distance along, and the second number shows the distance up or down.
Example: the point (12,5) is 12 units along, and 5 units up.
On graphs it is usually a pair of numbers: the first number shows the distance along, and the second number shows the distance up or down.
Example: the point (12,5) is 12 units along, and 5 units up.
Cotangent
In a right angled triangle, the cotangent of an angle is:
The length of the adjacent side divided by the length of the side opposite the angle.
The abbreviation is cot
cot(θ) = adjacent / opposite
The length of the adjacent side divided by the length of the side opposite the angle.
The abbreviation is cot
cot(θ) = adjacent / opposite
Cube Root
The cube root of a number is a special value that, when used in a multiplication three times, gives that number.
Example: 3 × 3 × 3 = 27, so the cube root of 27 is 3.
Example: 3 × 3 × 3 = 27, so the cube root of 27 is 3.
Cubic Meter
A volume that is made by a cube that is 1 meter on each side.
Its symbol is m3
It is equal to 1000 (one thousand) liters.
Example: A box that is 2 meters wide, 2 meters long and 0.25 meters deep has a volume of 2×2×0.25 = 1 m3
Its symbol is m3
It is equal to 1000 (one thousand) liters.
Example: A box that is 2 meters wide, 2 meters long and 0.25 meters deep has a volume of 2×2×0.25 = 1 m3
Curve
A smoothly-flowing line (no sharp changes).
In normal language a curve must bend (change direction), but in mathematics a straight line is also a curve.
In normal language a curve must bend (change direction), but in mathematics a straight line is also a curve.
Cylinder
A solid object with:
• two identical flat ends that are circular or elliptical
• and one curved side.
It has the same cross-section from one end to the other.
• two identical flat ends that are circular or elliptical
• and one curved side.
It has the same cross-section from one end to the other.
Data
A collection of facts, such as numbers, words, measurements, observations or even just descriptions of things.
Decimal Point
A point (small dot) used to separate the whole number part from the fractional part of a number.
Example: in the number 36.9 the point separates the 36 (the whole number part) from the 9 (the fractional part, which really means 9 tenths).So 36.9 is 36 and nine tenths.
Example: in the number 36.9 the point separates the 36 (the whole number part) from the 9 (the fractional part, which really means 9 tenths).So 36.9 is 36 and nine tenths.
Eccentricity
How much a conic section (a circle, ellipse, parabola or hyperbola) varies from being circular.
A circle has an eccentricity of zero.
A circle has an eccentricity of zero.
Factorial
Factorial says to multiply all whole numbers from the chosen number down to 1.
The symbol is "!"
Examples:
4! = 4 × 3 × 2 × 1 = 24
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
The symbol is "!"
Examples:
4! = 4 × 3 × 2 × 1 = 24
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
Fibonacci Numbers
Fibonacci numbers are created starting with 1 and 1, then get the next number in the list and adds the previous two numbers. Say, 1+1 =2 and then adds 1+2 you get 3, then adds 2+3 gives 5, then 3+5 gives 8 and so on.
Finite
Not infinite. Has an end. Could be measured, or given a value.
Example:
There are a finite number of people at a beach.
There are also a finite number of grains of sand at a beach. Hard to count but still finite!
And the length of a beach is also finite.
For infinite or infinity please use search bar or alphabet index above
Example:
There are a finite number of people at a beach.
There are also a finite number of grains of sand at a beach. Hard to count but still finite!
And the length of a beach is also finite.
For infinite or infinity please use search bar or alphabet index above
Geometry
The branch of mathematics that deals with points, lines, shapes and space.
• Plane Geometry is about flat shapes like lines, circles and triangles.
• Solid Geometry is about solid (3-dimensional) shapes like spheres and cubes.
• Plane Geometry is about flat shapes like lines, circles and triangles.
• Solid Geometry is about solid (3-dimensional) shapes like spheres and cubes.
Googol
Is the number written as 1 followed by 100 zeros:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
In Scientific Notation it is 1 × 10100
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
In Scientific Notation it is 1 × 10100
Greatest Common Factor
The greatest number that is a factor of two (or more) other numbers.
When we find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.
Abbreviated "GCF". Also called "Highest Common Factor"
Example: the GCF of 12 and 16 is 4, because 1, 2 and 4 are common factors of both 12 and 16, and 4 is the greatest of them.
When we find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.
Abbreviated "GCF". Also called "Highest Common Factor"
Example: the GCF of 12 and 16 is 4, because 1, 2 and 4 are common factors of both 12 and 16, and 4 is the greatest of them.
Hexadecimal
A Hexadecimal Number is based on the number 16
As well as the familiar digits 0 to 9, there are also the letters A, B, C, D, E and F in place of the decimal numbers 10 to 15.
Example Hexadecimal Number: 82E6B
More Examples:
• 9 in Hexadecimal equals 9 in the Decimal Number System.
• B in Hexadecimal equals 11 in the Decimal Number System.
• 1E in Hexadecimal equals 30 in the Decimal Number System.
Also called Base 16.
As well as the familiar digits 0 to 9, there are also the letters A, B, C, D, E and F in place of the decimal numbers 10 to 15.
Example Hexadecimal Number: 82E6B
More Examples:
• 9 in Hexadecimal equals 9 in the Decimal Number System.
• B in Hexadecimal equals 11 in the Decimal Number System.
• 1E in Hexadecimal equals 30 in the Decimal Number System.
Also called Base 16.
Highest Common Factor
The greatest number or Highest Common Factor that is a factor of two (or more) other numbers.
When we find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.
Abbreviated "GCF". Also called "Highest Common Factor"
Example: the GCF of 12 and 16 is 4, because 1, 2 and 4 are common factors of both 12 and 16, and 4 is the greatest of them.
When we find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.
Abbreviated "GCF". Also called "Highest Common Factor"
Example: the GCF of 12 and 16 is 4, because 1, 2 and 4 are common factors of both 12 and 16, and 4 is the greatest of them.
Inequality
An inequality says that two values are not equal.
a ≠ b says that a is not equal to b
There are other special symbols that show in what way things are not equal.
a < b says that a is less than b
a > b says that a is greater than b
(those two are known as strict inequality)
a ≤ b means that a is less than or equal to b
a ≥ b means that a is greater than or equal to b.
a ≠ b says that a is not equal to b
There are other special symbols that show in what way things are not equal.
a < b says that a is less than b
a > b says that a is greater than b
(those two are known as strict inequality)
a ≤ b means that a is less than or equal to b
a ≥ b means that a is greater than or equal to b.
Integer
A number with no fractional part (no decimals).
Includes:
• the counting numbers {1, 2, 3, ...},
• zero {0},
• and the negative of the counting numbers {-1, -2, -3, ...}
We can write them all down like this: {..., -3, -2, -1, 0, 1, 2, 3, ...}
Examples of integers: -16, -3, 0, 1, 198
Includes:
• the counting numbers {1, 2, 3, ...},
• zero {0},
• and the negative of the counting numbers {-1, -2, -3, ...}
We can write them all down like this: {..., -3, -2, -1, 0, 1, 2, 3, ...}
Examples of integers: -16, -3, 0, 1, 198
Joule
The unit of energy, work and heat.
Equal to the work done (or energy transferred) by a force of one newton acting over one meter in the direction of the force.
Its symbol is J, which is also kg m2/s2 (kilogram meter squared per second squared)
Equal to the work done (or energy transferred) by a force of one newton acting over one meter in the direction of the force.
Its symbol is J, which is also kg m2/s2 (kilogram meter squared per second squared)
Kilo-
A prefix meaning one thousand (1,000 or 103)
Examples:
• a kilogram is a thousand grams
• a kilometer is a thousand meters
• a kilobyte is a thousand bytes
Symbol is k
Example: 12 km = 12 kilometers = 12 thousand meters (12,000 m)
Examples:
• a kilogram is a thousand grams
• a kilometer is a thousand meters
• a kilobyte is a thousand bytes
Symbol is k
Example: 12 km = 12 kilometers = 12 thousand meters (12,000 m)
Kilogram
A Metric measure of mass (which we feel as weight).
The abbreviation is kg.
1 kg = 1000 grams.
1 kg = 2.205 pounds (approximately).
The abbreviation is kg.
1 kg = 1000 grams.
1 kg = 2.205 pounds (approximately).
Kilometre | Kilometer
A Metric measure of distance. Equal to 1,000 meters.
The abbreviation is km.
Examples:
• it takes about 12 minutes to walk 1 km
• at highway speed a car goes about 100 km in an hour (100 km/h)
The abbreviation is km.
Examples:
• it takes about 12 minutes to walk 1 km
• at highway speed a car goes about 100 km in an hour (100 km/h)
km/h
An abbreviation of "kilometers per hour", a metric measure of speed.
This is a sign showing a speed limit of 50 km/h
1 km/h = 0.6 miles per hour approximately
This is a sign showing a speed limit of 50 km/h
1 km/h = 0.6 miles per hour approximately
Limit
A value we get closer and closer to, but never quite reach
For example, when we graph y=1/x we see that it gets closer... and closer... to zero but does not ever quite get there.
We then say "As x approaches infinity, then 1/x approaches 0"
For example, when we graph y=1/x we see that it gets closer... and closer... to zero but does not ever quite get there.
We then say "As x approaches infinity, then 1/x approaches 0"
Mass
A measure of how much matter is in an object.
Suppose a gold bar is quite small but has a mass of 1 kilogram (about 2.2 pounds), so it contains a lot of matter.
Mass is commonly measured by how much something weighs. But weight is caused by gravity, so your weight on the Moon is less than here on Earth, while the mass stays the same.
Mass is measured in grams, kilograms and, tonnes (Metric) or ounces and pounds (US units).
Suppose a gold bar is quite small but has a mass of 1 kilogram (about 2.2 pounds), so it contains a lot of matter.
Mass is commonly measured by how much something weighs. But weight is caused by gravity, so your weight on the Moon is less than here on Earth, while the mass stays the same.
Mass is measured in grams, kilograms and, tonnes (Metric) or ounces and pounds (US units).
Matrix
An array of numbers.
They can be added, subtracted, multiplied and more.
There is a whole subject called "Matrix Algebra"
The plural is "matrices".
They can be added, subtracted, multiplied and more.
There is a whole subject called "Matrix Algebra"
The plural is "matrices".
Modulo 4 Numbers
A number is said to ne 1 (modulo 4 ) number if it leaves a remainder 1 when divided by 4.Similarly, if a number leaves a remainder 3 when divided by 4, it is said to be 3 (modulo 4) number.
Nano-
A prefix meaning one-billionth (1/1,000,000,000 or 10-9)Examples:• a nanometer is one-billionth of a meter, about the width of DNA (illustrated here)• a nanosecond is one-billionth of a secondSymbol is nExample: 12 ns = 12 nanoseconds = 12 billionths of a second (0.000000012 s)
Natural Number
The whole numbers from 1 upwards: 1, 2, 3, and so on ...
Or from 0 upwards in some fields of mathematics: 0, 1, 2, 3 and so on ...
No negative numbers and no fractions.
Or from 0 upwards in some fields of mathematics: 0, 1, 2, 3 and so on ...
No negative numbers and no fractions.
Negative
Less than zero.
(Positive means more than zero. Zero is neither negative nor positive.)
A negative number is written with a minus sign in front
Example: −5 is negative five.
The word "negative" can be shortened to "−ve"
(Positive means more than zero. Zero is neither negative nor positive.)
A negative number is written with a minus sign in front
Example: −5 is negative five.
The word "negative" can be shortened to "−ve"
Net Income
Income after tax and expenses.
For a business: total sales minus all expenses including tax.
Example: Bravedog Inc sells $400,000 of dog biscuits, spending $180,000 making them. Marketing and other costs and tax are $100,000. Bravedog's gross income is $220,000 and its net income is $120,000.
For a business: total sales minus all expenses including tax.
Example: Bravedog Inc sells $400,000 of dog biscuits, spending $180,000 making them. Marketing and other costs and tax are $100,000. Bravedog's gross income is $220,000 and its net income is $120,000.
Nominal Number
A number used only as a name, or to identify something (not as an actual value or position)
Examples:
· the number on the back of a footballer: "8"
· a zip code: "91210"
· a model number: "380"
· etc
Examples:
· the number on the back of a footballer: "8"
· a zip code: "91210"
· a model number: "380"
· etc
Normal
At right angles to, going directly away from.
For a curve imagine a line just touching and matching the slope there (called a "tangent") and draw a line at right angles to it.
Example: a circle's diameter is always normal to its circumference
For a curve imagine a line just touching and matching the slope there (called a "tangent") and draw a line at right angles to it.
Example: a circle's diameter is always normal to its circumference
Number
A number is a count or measurement.
They are really an idea in our minds. We write or talk about numbers using numerals such as "5" or "five". We could also hold up 5 fingers, or tap the table 5 times. These are all different ways of referring to the same number.
There are also different types of numbers, such as
• whole numbers {1,2,3,...}
• decimals (like 1.48 or 50.5)
• fractions (like 1/2 or 3/8)
• and more.
They are really an idea in our minds. We write or talk about numbers using numerals such as "5" or "five". We could also hold up 5 fingers, or tap the table 5 times. These are all different ways of referring to the same number.
There are also different types of numbers, such as
• whole numbers {1,2,3,...}
• decimals (like 1.48 or 50.5)
• fractions (like 1/2 or 3/8)
• and more.
Number Sense
A person's ability to use and understand numbers:
· knowing their relative values,
· how to use them to make judgments,
· how to use them in flexible ways when adding, subtracting, multiplying or dividing
· how to develop useful strategies when counting, measuring or estimating.
· knowing their relative values,
· how to use them to make judgments,
· how to use them in flexible ways when adding, subtracting, multiplying or dividing
· how to develop useful strategies when counting, measuring or estimating.
Octal
An Octal Number uses only these 8 digits: 0, 1, 2, 3, 4, 5, 6 and 7
Examples:
• 7 in Octal equals 7 in the Decimal Number System
• 10 in Octal equals 8 in the Decimal Number System
• 11 in Octal equals 9 in the Decimal Number System
• 167 in Octal equals 119 in the Decimal Number System
Also called Base 8
Examples:
• 7 in Octal equals 7 in the Decimal Number System
• 10 in Octal equals 8 in the Decimal Number System
• 11 in Octal equals 9 in the Decimal Number System
• 167 in Octal equals 119 in the Decimal Number System
Also called Base 8
Ordinate
The vertical ("y") value in a pair of coordinates. How far up or down the point is.
Always written second in an ordered pair of coordinates such as (12,5).
In this example, the value "5" is the ordinate
Always written second in an ordered pair of coordinates such as (12,5).
In this example, the value "5" is the ordinate
Pi
The ratio of a circle's circumference to its diameter
In other words: all the way around a circle divided by all the way across it.
The symbol is π
No matter how large or small the circle, its circumference is always π times its diameter.
π = 3.14159265358979323846... (the digits go on forever without repeating)
A rough approximation is 22/7 (=3.1428571...), but that is not accurate.
In other words: all the way around a circle divided by all the way across it.
The symbol is π
No matter how large or small the circle, its circumference is always π times its diameter.
π = 3.14159265358979323846... (the digits go on forever without repeating)
A rough approximation is 22/7 (=3.1428571...), but that is not accurate.
Profit
Income minus all expenses.
Example: Sam's Bakery received $900 yesterday, but expenses such as wages, food and electricity came to $650. So the Profit was $900 − $650 = $250.
But if the income is LESS THAN the expenses it is called a "Loss".
Example: Two days ago Sam's Bakery received $480, but expenses were $520.
$480 − $520 = −$40, which is a $40 Loss
Example: Sam's Bakery received $900 yesterday, but expenses such as wages, food and electricity came to $650. So the Profit was $900 − $650 = $250.
But if the income is LESS THAN the expenses it is called a "Loss".
Example: Two days ago Sam's Bakery received $480, but expenses were $520.
$480 − $520 = −$40, which is a $40 Loss
Quantity
How much there is of something.
Example: What is the quantity of rice?
• We can say "a handful"
• Or use a measuring cup and say "40 milliliters"
• Or we can count them and say "1562"
Example: What is the quantity of rice?
• We can say "a handful"
• Or use a measuring cup and say "40 milliliters"
• Or we can count them and say "1562"
Quotient
The answer after we divide one number by another.
dividend ÷ divisor = quotient.
Example: in 12 ÷ 3 = 4, 4 is the quotient.
dividend ÷ divisor = quotient.
Example: in 12 ÷ 3 = 4, 4 is the quotient.
Radian
The angle made by taking the radius and wrapping it round the circle.
One Radian is (180/π) degrees, which is about 57.2958 degrees.
One Radian is (180/π) degrees, which is about 57.2958 degrees.
Ratio
A ratio shows the relative sizes of two or more values.
Ratios can be shown in different ways:
• using the ":" to separate example values
• using the "/" to separate one value from the total
• as a decimal, after dividing one value by the total
• as a percentage, after dividing one value by the total
Example: if there is 1 boy and 3 girls you could write the ratio as:
1:3 (for every one boy there are 3 girls)
1/4 are boys and 3/4 are girls
0.25 are boys (by dividing 1 by 4)
25% are boys (0.25 as a percentage)
Ratios can be shown in different ways:
• using the ":" to separate example values
• using the "/" to separate one value from the total
• as a decimal, after dividing one value by the total
• as a percentage, after dividing one value by the total
Example: if there is 1 boy and 3 girls you could write the ratio as:
1:3 (for every one boy there are 3 girls)
1/4 are boys and 3/4 are girls
0.25 are boys (by dividing 1 by 4)
25% are boys (0.25 as a percentage)
Sale Price
The price after the original price has been reduced by a discount.
The sale price here is $8.00
If the discount is a percentage we must first work out the discount amount:
• discount amount = original price × discount rate
• then subtract that from the original price
Example: the shirt shop is having a 10% discount sale
The t-shirt's normal price is $23
• discount amount = $23 × 10% = $2.30
• then $23 - $2.30 = $20.70 (the Sale Price)
The sale price here is $8.00
If the discount is a percentage we must first work out the discount amount:
• discount amount = original price × discount rate
• then subtract that from the original price
Example: the shirt shop is having a 10% discount sale
The t-shirt's normal price is $23
• discount amount = $23 × 10% = $2.30
• then $23 - $2.30 = $20.70 (the Sale Price)
Semicircle
Half a circle (made by a diameter and the connecting arc)
Its area is half of a circle's area: πr2/2
Its perimeter is half of a circle's perimeter plus the diameter: πr + 2r
Its area is half of a circle's area: πr2/2
Its perimeter is half of a circle's perimeter plus the diameter: πr + 2r
Set
A collection of "things" (objects or numbers, etc).
Here is a set of clothing items.
Each member is called an element of the set.
A set has only one of each member (all members are unique).
Example: {1,2,3,4} is the set of counting numbers less than 5
Here is a set of clothing items.
Each member is called an element of the set.
A set has only one of each member (all members are unique).
Example: {1,2,3,4} is the set of counting numbers less than 5
Simple Interest
Interest calculated as a percent of the original loan.
Example: a 3-year loan of $1,000 at 10% costs 3 lots of 10%
So the interest is 3 × $1,000 × 10% = $300
(Simple interest is almost never used in the real world, with compound interest being preferred.)
Example: a 3-year loan of $1,000 at 10% costs 3 lots of 10%
So the interest is 3 × $1,000 × 10% = $300
(Simple interest is almost never used in the real world, with compound interest being preferred.)
Sphere
A 3-dimensional object shaped like a ball.
Every point on the surface is the same distance from the center.
Every point on the surface is the same distance from the center.
Square Root
A square root of a number is a value that, when multiplied by itself, gives the number.
Example: 4 × 4 = 16, so a square root of 16 is 4.
Note that (−4) × (−4) = 16 too, so −4 is also a square root of 16.
The symbol is √ which always means the positive square root.
Example: √36 = 6 (because 6 x 6 = 36)
Example: 4 × 4 = 16, so a square root of 16 is 4.
Note that (−4) × (−4) = 16 too, so −4 is also a square root of 16.
The symbol is √ which always means the positive square root.
Example: √36 = 6 (because 6 x 6 = 36)
Standard Deviation
A measure of how spread out numbers are.
It is the square root of the Variance,
and the Variance is the average of the squared differences from the Mean.
It is the square root of the Variance,
and the Variance is the average of the squared differences from the Mean.
Subset
Part of another set.
A is a subset of B when every member of A is a member of B.
Example: B = {1,2,3,4,5}
Then A = {1,2,3} is a subset of B
Other subsets of B include {2,3} or {1,4,5} or {4} etc...
But {1,2,6} is NOT a subset of B as it has 6 (which is not in B)
A is a subset of B when every member of A is a member of B.
Example: B = {1,2,3,4,5}
Then A = {1,2,3} is a subset of B
Other subsets of B include {2,3} or {1,4,5} or {4} etc...
But {1,2,6} is NOT a subset of B as it has 6 (which is not in B)
Triangular Numbers
A number is said to be triangular number, when that number of pebbles can be arranged in a triangle using one pebble at the top, two pebbles in next row, three pebbles in next row and so on.
Trigonometry
Trigonometry is the study of triangles: their angles, lengths and more.
(The name comes from Greek trigonon "triangle" + metron "measure").
(The name comes from Greek trigonon "triangle" + metron "measure").
Unequal
Not equal.
Example: 7 and 5 are unequal.
The symbol is ≠ (the "not equal" symbol).
Example: 7 ≠ 5
Example: 7 and 5 are unequal.
The symbol is ≠ (the "not equal" symbol).
Example: 7 ≠ 5
Union
The set made by combining the elements of two sets.
So the union of sets A and B is the set of elements in A, or B, or both.
The symbol is a special "U" like this: ∪
Example:
Soccer = {alex, hunter, casey, drew}
Tennis = {casey, drew, jade}
Soccer ∪ Tennis = {alex, hunter, casey, drew, jade}
In words: the union of the "Soccer" and "Tennis" sets is alex, hunter, casey, drew and jade
So the union of sets A and B is the set of elements in A, or B, or both.
The symbol is a special "U" like this: ∪
Example:
Soccer = {alex, hunter, casey, drew}
Tennis = {casey, drew, jade}
Soccer ∪ Tennis = {alex, hunter, casey, drew, jade}
In words: the union of the "Soccer" and "Tennis" sets is alex, hunter, casey, drew and jade
Velocity
Velocity is speed (how fast something is moving) with a direction.
Saying that Ariel the Dog is running 3 km/h Westwards is a velocity.
(But saying just 3 km/h is a speed.)
Saying that Ariel the Dog is running 3 km/h Westwards is a velocity.
(But saying just 3 km/h is a speed.)
Vertex
A point where two or more line segments meet. A corner.
Examples:
• any corner of a pentagon (a plane shape)
• any corner of a tetrahedron (a solid)
(The plural of vertex is "vertices".)
Examples:
• any corner of a pentagon (a plane shape)
• any corner of a tetrahedron (a solid)
(The plural of vertex is "vertices".)
Wage
Payment for work based on hours worked (or sometimes per day or per amount of work done).
Can be paid weekly, two-weekly, or monthly.
Example: Alex earns $20 per hour, and worked 40 hours last week so earned an $800 wage.
You often get paid extra for overtime work.
Can be paid weekly, two-weekly, or monthly.
Example: Alex earns $20 per hour, and worked 40 hours last week so earned an $800 wage.
You often get paid extra for overtime work.
Weight
How heavy something is. The downward force caused by gravity on an object.
Weight and Mass are different things, but weighing scales are designed to estimate the mass sitting on them and so (instead of units of force) it is common to use units of mass like these:
• grams, kilograms and tonnes (in Metric)
• ounces, pounds and tons (in US units)
Weight and Mass are different things, but weighing scales are designed to estimate the mass sitting on them and so (instead of units of force) it is common to use units of mass like these:
• grams, kilograms and tonnes (in Metric)
• ounces, pounds and tons (in US units)
Whole Number
Any of the numbers {0, 1, 2, 3, ...} etc.
There is no fractional or decimal part. And no negatives.
Example: 5, 49 and 980 are all whole numbers.
There is no fractional or decimal part. And no negatives.
Example: 5, 49 and 980 are all whole numbers.
X
The letter "x" is often used in algebra to mean a value that is not yet known.
It is called a "variable" or sometimes an "unknown".
In x + 2 = 7, x is a variable, but we can work out its value if we try!
A variable doesn't have to be "x", it could be "y", "w" or any letter, name or symbol.
It is called a "variable" or sometimes an "unknown".
In x + 2 = 7, x is a variable, but we can work out its value if we try!
A variable doesn't have to be "x", it could be "y", "w" or any letter, name or symbol.
X Axis
The line on a graph that runs horizontally (left-right) through zero.
It is used as a reference line so you can measure from it.
It is used as a reference line so you can measure from it.
X Coordinate
The horizontal value in a pair of coordinates: how far along the point is.
The X Coordinate is always written first in an ordered pair of coordinates (x,y), such as (12,5).
In this example, the value "12" is the X Coordinate.
Also called "Abscissa"
The X Coordinate is always written first in an ordered pair of coordinates (x,y), such as (12,5).
In this example, the value "12" is the X Coordinate.
Also called "Abscissa"
Y Axis
The line on a graph that runs vertically (up-down) through zero.
It is used as a reference line so you can measure from it.
It is used as a reference line so you can measure from it.
Y Coordinate
The vertical value in a pair of coordinates. How far up or down the point is.
The Y Coordinate is always written second in an ordered pair of coordinates (x,y) such as (12,5).
In this example, the value "5" is the Y Coordinate.
Also called "Ordinate"
The Y Coordinate is always written second in an ordered pair of coordinates (x,y) such as (12,5).
In this example, the value "5" is the Y Coordinate.
Also called "Ordinate"
Z-score
How many standard deviations a value is from the mean.
In this example, the value 1.7 is 2 standard deviations away from the mean of 1.4, so 1.7 has a z-score of 2.
Similarly 1.85 has a z-score of 3.
So to convert a value to a Standard Score ("z-score"):
· first subtract the mean,
· then divide by the standard deviation
In this example, the value 1.7 is 2 standard deviations away from the mean of 1.4, so 1.7 has a z-score of 2.
Similarly 1.85 has a z-score of 3.
So to convert a value to a Standard Score ("z-score"):
· first subtract the mean,
· then divide by the standard deviation
Zepto-
A prefix meaning one-sextillionth (1/1,000,000,000,000,000,000,000 or 10-21)
The same as multiplying by 10-21 (which is 0.000 000 000 000 000 000 001)
Prefix is z
Example: 12 zg = 12 zeptograms = 0.000 000 000 000 000 000 012 g
The mass of 1 single molecule of glucose (a simple sugar found in many foods) is 0.3 zg
The same as multiplying by 10-21 (which is 0.000 000 000 000 000 000 001)
Prefix is z
Example: 12 zg = 12 zeptograms = 0.000 000 000 000 000 000 012 g
The mass of 1 single molecule of glucose (a simple sugar found in many foods) is 0.3 zg
Zero
The whole number between −1 and 1, with the symbol 0
Shows that there is no amount.
Example: 6 − 6 = 0 (the difference between six and six is zero)
Zero is not positive and is also not negative.
When we add zero to a number the result is just the number, unchanged.
When we multiply a number by zero we get zero.
Zero is also used as a "place-holder" so that you can write a numeral properly.
Example: 502 (five hundred and two) could be mistaken for 52 (fifty two) without the zero in the tens place.
Shows that there is no amount.
Example: 6 − 6 = 0 (the difference between six and six is zero)
Zero is not positive and is also not negative.
When we add zero to a number the result is just the number, unchanged.
When we multiply a number by zero we get zero.
Zero is also used as a "place-holder" so that you can write a numeral properly.
Example: 502 (five hundred and two) could be mistaken for 52 (fifty two) without the zero in the tens place.
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Algebra is a branch of Mathematics that substitutes letters for numbers. An algebraic equation depicts a scale, what is done on one side of the scale with a number is also done to either side of the scale.
In almost every field of study, from computer science to engineering, algebra is immensely important. So, learners must grasp every algebra formula to prepare themselves for calculations beyond basic math. Not only for learners in school but even candidates appearing for competitive exams should have a firm grasp of algebraic formulae to excel.
Important Formulas in Algebra
Here is a list of Algebraic formulas –
- (a+b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- a2 – b2 = (a – b)(a + b)
- a2 + b2 = (a – b)2 + 2ab
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
- (a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
- (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
- (a – b)3 = a3 – 3a2b + 3ab2 – b3
- a3 – b3 = (a – b)(a2 + ab + b2)
- a3 + b3 = (a + b)(a2 – ab + b2)
- (a + b)3 = a3 + 3a2b + 3ab2 + b3
- (a – b)3 = a3 – 3a2b + 3ab2 – b3
- (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4)
- (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4)
- a4 – b4 = (a – b)(a + b)(a2 + b2)
- a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
- If n is a natural number, an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
- If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
- If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)
- (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….
- Laws of Exponents (am)(an) = am+n (ab)m = ambm (am)n = amn
- Fractional Exponents a0 = 1
Do you know?
- Algebra was introduced by the Greeks back in the 3rd century.
- It was the Babylonians who created the algebraic equation and formulae we still use in the 21st century to solve diverse problems.
- Modern algebra was brought in by Rene Descartes in the 16th century.
Here’s what is involved in the study of algebra
The study of algebra revolves around in-depth learning of terms, concepts and formulae. The idea is simple. Here, mathematical symbols, known as variables, represent quantity without having any fixed value and these are manipulated to derive solutions.
A basic example:
Algebra asks questions like what is the value of x if x + 15 = 20? To get the result, you need to do another calculation, i.e. 20 – 15 = 5. So, the value of x is 5.
Once you learn the basics (elementary algebra), advanced levels gradually become easier.
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Isn’t it amazing, if you can calculate faster in competitive exams or in daily life? In competitive exams you need to solve more number of problems in less time. So, people, nowadays running behind the coaching centers for shortcuts. But it is absolutely useless if you don’t have conceptual clarity. Here is the speed math rule for adding large numbers. Check other quicker math rules here.
Adding large numbers just in your head can be difficult. This method shows how to simplify this process by making all the numbers a multiple of 10. Here is an example:
644 + 238 = ?
While these numbers are hard to contend with, rounding them up will make them more manageable. So, 644 becomes 650 and 238 becomes 240.
Now, add 650 and 240 together. The total is 890. To find the answer to the original equation, it must be determined how much we added to the numbers to round them up.
650 – 644 = 6 and 240 – 238 = 2
Now, add 6 and 2 together for a total of 8
To find the answer to the original equation, 8 must be subtracted from the 890.
890 – 8 = 882
So the answer to 644 +238 is 882.
This is speed math trick 8. Please check other speed math tricks here.
Are you ready to play a brain game? Ok, that’s Equi Math for you. Click here to get the Android Game. This game will help in decision making in quick time. It is very easy at beginning and gets difficult on the higher levels. Try to reach as much higher level as possible. On game over, try again to improve your best. That way you will have quick solving skills.
Write down your feedback on comments section below.
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Isn’t it fantastic to be able to compute faster in competitive tests or in everyday life? You have to answer more questions in less time on competitive exams. People are increasingly scurrying behind coaching centers in search of shortcuts. But without conceptual clarity, it is all completely pointless. The quick math formula to multiply any two-digit integer by 11 is as follows. Check other quicker math rules here.
Rule: Multiply any two-digit number by 11
There is an easy trick for multiplying any two-digit number by 11. Here it is:
11 x 25
Take the original two-digit number and put a space between the digits. In this example, that number is 25.
2_5
Now add those two numbers together and put the result in the center:
2_(2 + 5)_5
2_7_5
The answer to 11 x 25 is 275.
This is quicker math trick 7. Please check other speed math tricks here.
If the numbers in the center add up to a number with two digits, insert the second number and add 1 to the first one. Here is an example for the equation 11 x 88
8_(8 +8)_8
(8 + 1)_6_8
9_6_8
There is the answer to 11 x 88 = 968
Are you ready for a brain game? Ok, that’s Equi Math for you. Click here to get the Android Game. This game will help in decision making in quick time. It is very easy at beginning and gets difficult on the higher levels. Try to reach as much higher level as possible. On game over, try again to improve your best. That way you will have quicker solving skills.
So, this was the rule to multiply any number by 11. Write down your feedback on comments section below. Also check the other speed math rules for faster calculations.
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When multiplying the number 5 by an even number, there is a quick way to find the answer. This is quicker math trick 6. Please check other quicker math tricks here.
For example, 5 x 4 = ?
Step 1: Take the number being multiplied by 5 and cut it in half, this makes the number 4 become the number 2.
Step 2: Add a zero to the number to find the answer. In this case, the answer is 20.so, 5 x 4 = 20
Another example, 5 x 824 = ?
Step 1: Take the number being multiplied by 5 and cut it in half, this makes the number 824 become the number 412.
Step 2: Add a zero to the number to find the answer. In this case, the answer is 4120.so, 5 x 824 = 4120
When multiplying an odd number times 5, the formula is a bit different.
For instance, consider 5 x 3 = ?
Step 1: Subtract one from the number being multiplied by 5, in this instance the number 3 becomes the number 2.
Step 2: Now halve the number 2, which makes it the number 1. Make 5 the last digit. The number produced is 15, which is the answer.5 x 3 = 15
Another example 5 x 713 = ?
Step 1: Subtract one from the number being multiplied by 5, in this instance the number 713 becomes the number 712.
Step 2: Now halve the number 712, which makes it the number 356. Make 5 the last digit. The number produced is 3565, which is the answer.5 x 713 = 3565
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Isn’t it great, if you can calculate faster in competitive exams or in daily life? In competitive exams you need to solve more number of problems in less time. So, people , nowadays running behind the coaching centers for shortcuts. But it is absolutely useless if you don’t have conceptual clarity. Here is the speed math rule for multiplying a Number ending with 9. Check other quicker math rules here.
Rule for Multiplying a Number ending with 9
Multiply by 10 (append a ‘0’ after the multiplicand) and then multiply by one more than the tens digit of the multiplier (number ending in 9). After that, subtract the given number from the result.
For example, multiply 713 by 39.
First multiply 713 by 10 to get 7,130.
One more than the tens digit (3) of the multiplier is 4. Multiply 7,130 by 4 (using Short Cut doubling twice).
7,130 x 4 = 28,520
Now, subtract the given number from this finding to get the final answer.
28,520 – 713 = 27807 (Answer)
This is quicker math trick 5. Please check other quicker math tricks here.
Another Example: 24,653 × 79 =?
Multiply 24653 by 10 to get 246530.
Multiply 246530 by 8 to get (246530 x 8 =) 1972240.
Subtract the given number to get the answer (1972240 – 24653 =) 1947587.
This short cut can be applied to any number, no matter how many digits it has, so long as the unit’s digit is 9. Of course, as the number gets larger, multiplying the two numbers of the first step will become cumbersome unless a short cut can be used. However, most two- and three digit numbers ending in 9 can be readily squared, once a facility with the other short-cut methods has been achieved.
Are you ready to play a brain game? Ok, that’s Fire Ball Sort for you. Click here to get the Android Game.So, this was the rule for multiplying a Number ending with 9. Write down your feedback on comments section below. Also check the other quicker math rules for faster calculations.
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Isn’t it fantastic to be able to compute more quickly in everyday situations or on competitive exams? You have to answer more questions in less time on competitive exams. People are increasingly scurrying behind coaching centers in search of shortcuts. But without conceptual clarity, it is all completely pointless. This is the quick math formula for calculating the square of numbers consisting 1 only. Check other quicker math rules here.
To find square of numbers consisting 1’s only such as 11,111,1111 or 11111 and so on is very easy. You can calculate the correct result in a fraction of seconds. You just need to know the quicker math method for this calculation. Once you will know and understand the method you will be amazed. You can use this speed math technique in competitive exams and other necessary scenarios.
Rule
Like most of the other vedic math techniques, in this too we will calculate the result in two parts. And then we will concatenate the parts to get ultimate result.
To find square of numbers consisting 1’s only such as 11,111,1111,11111 and so on is very easy. You can calculate the correct result in a fraction of seconds. You just need to know the quicker math method for this calculation. Once you will know and understand the method you will be amazed. You can use this speed math technique in competitive exams and other necessary scenarios.
Like most of the other vedic math techniques, in this too we will calculate the result in two parts. And then we will concatenate the parts to get ultimate result.
For example to calculate square of 111, there are 3 ones in the given number. So the first part will be “123”. And the second part will start with (3-1)=2 and will reach 1 in reverse order i.e the second part is “21”. So the result is 12321.
Now we will take another example, (1111)2=?
There are 4 ones in the given number. So the first part will be “1234” and the second part will start from 3 and will reach one in reverse order i.e “321”. So the result is 1234321.
Now you can see how easily we can calculate the square of numbers consisting ones only. For other quicker math techniques, please check our “Quicker Math” section.
Exercise for square of numbers consisting 1 only
As exercise calculate the result of (111111)2. You just need to count the number of 1’s in it and you will have the correct result in a fraction of seconds.
See as we promised, now you can find result of such complicated calculation very quickly. Best of luck for your exams.
Are you ready to play a brain game? Ok, that’s Equi Math for you. Click here to get the Android Game. This game will help in decision making in quick time. It is very easy at beginning and gets difficult on the higher levels. Try to reach as much higher level as possible. On game over, try again to improve your best. That way you will have quicker solving skills.
So, this was the rule for calculating Square of numbers consisting 1 only. Write down your feedback on comments section below. Also check the other quicker math rules for faster calculations.
Don’t forget to share. Please subscribe to Mathkind here for math tricks and more puzzles.
Please share your thoughts and feedback on this, in comments section below.
Isn’t it great if you are able to calculate faster in a competitive exam or in your everyday life? In a competitive exam, you have to solve a large number of questions in a short period of time. Nowadays, people are rushing to the coaching centers for shortcuts. However, it is all for nothing if you do not have a good conceptual understanding. Here is a quicker math rule for calculating the Square of a given number near 10^x.
Check other quicker math rules here.
To square numbers close to the bases of powers of 10 i.e. 10, 100, 1000 and so on easily at extremely fast speed, just follow the steps below:
Step-1: Find the surplus or deficit from the base 10, 100, 1000 & so onStep-2: Add the surplus (if it is more than base) with or subtract the deficit (if it is less than base) from the whole number given and put the result.
Step-3: Find the square of surplus or deficit and write the result in the last places. Be sure that, since 10 has 1 zero so, 1 more digit to go for numbers near 10, since 100 has 2 zeros so, 2 more digits to go for numbers near 100, accordingly since 1000 has 3 zeros, 3 more digits to go for numbers near 1000 and so on. So, carry forward or put extra zero(s) if necessary to place the digits accurate.
We use the algebraic formula
x2 = (x2 – y2) + y2 = (x + y)(x -y) + y2
Ex 1: (98)2 = (98 – 2) (98 + 2) + 22 = 9600 + 4 = 9604
Ex 2: (103)2 = (103 + 3)(103 – 3) + 32 = 10600 + 9 = 10609
Ex 3: (993)2 = (993 – 7)(993 + 7) + 72 = 986000 + 49 = 986049
Ex 4: (1008)2 = (1008 – 8)(1008 + 8) + 82 = 1016000 + 64 = 1016064
This is quicker math trick 3 on square of a number near to 10^x. Please check other speed math tricks here.
Are you ready to play a brain game? Ok, that’s Equi Math for you. Click here to get the Android Game. This game will help in decision making in quick time. It is very easy at beginning and gets difficult on the higher levels. Try to reach as much higher level as possible. On game over, try again to improve your best. That way you will have quick solving skills.
Don’t forget to share. Please subscribe to Mathkind here for math tricks and more puzzles.
Please share your thoughts and feedback on this, in comments section below.