Arithmetic

What Is Arithmetic?

Arithmetic is a branch of mathematics that studies numbers and the operations applied to numbers.

Dictionary Definition

The Oxford English Dictionary defines arithmetic as "the science of numbers; the art of computation by figures."
Here are some examples from arithmetic. We might be interested in adding, subtracting, multiplying and dividing numbers. We might be interested in doing several of these and have to do them in the correct order. Arithmetic allows us to do this.

Where Does the Word Arithmetic Come From?

Arithmetic comes from the Greek word "arithmētikḗ" meaning “counting”. 'Arithmos' is the Greek word for "number".

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning. A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Operations and Their Order

Operations take numbers to create a new number. The most common operations in arithmetic are adding, subtracting, multiplying and dividing. In this mini-curriculum, you will learn about operations.

Operation
An operation takes values and calculates a new value from them.
The basic operations in arithmetic are addition, subtraction, multiplication and division.


Inverse Operations
An inverse operation is the opposite of an operation. It reverses an operation.
The inverse operation of addition is subtraction, and vice versa.

The inverse operation of multiplication is division, and vice versa.


Order of Operations
The order of operations tells us what order to perform operations in.


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Read more about operations
Read more about inverse operations
Read more about the order of operations

Advanced Arithmetic

When numbers get larger than a single numeral, we must use place value and 'long' arithmetic. In this mini-curriculum, you will learn about place value, long addition, long subtraction, long multiplication and long division.

Place Value
Place value tells us the value of a digit depending on what place it is in a number.
The same digit can have a different value depending on its place value.
A 1 placed here has a value of one.


A 1 placed a column to the left (before a 0) has a value of ten.


Long Addition
Numbers with more than one digit can be added using long addition.


Long Subtraction
Numbers with more than one digit can be subtracted from each other using long subtraction.


Long Multiplication
Numbers with more than one digit can be multiplied using long multiplication.


Long Multiplication with Decimals
When numbers have a decimal point in them, they can be multiplied using long multiplication with decimals.


Long Division
Numbers with more than one digit can be divided using long division.


Long Division with a Remainder
Sometimes, a number will not divide exactly into another: there will be a remainder.


Long Division with Decimals
When numbers have a decimal point in them, they can be divided using long division with decimals.


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Read more place value
Read more about long addition
Read more about long subtraction
Read more about long multiplication
Read more about long multiplication with decimals
Read more about long division
Read more about long division with a remainder
Read more about long division with decimals

Factors

The factors of a number are those numbers that multiply to make that number. They divide exactly into the number. In this mini-curriculum, you will learn about factors.

Factor
A factor is a number which divides exactly into another number. The numbers that are multiplied together to make another number are factors.
2 and 3 are factors of 6 because they multiply to make 6.


Find Factors
To find the factors of a number, find pairs of numbers that multiply to make that number.


Greatest Common Factor
The greatest common factor is the largest factor that is common to two or more numbers.


Find the Greatest Common Factor
To find the greatest common factor, list the factors of both numbers and find the largest number that appears in both lists.
Here is an example that shows that the greatest common factor of 4 and 6 is 2.


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Read more factors
Read more about how to find factors
Read more about the greatest common factor
Read more about how to find the greatest common factor

Multiples

Multiples are the result of multiplying a number by another whole number. In this mini-curriculum, you will learn about multiples.

Multiples
A multiple is the result of multiplying a number by an integer (a whole number).
The multiples of 3 are found by multiplying 3 by the integers: 1, 2, 3, 4, 5...


Least Common Multiple
The least common multiple is the smallest multiple that is common to two or more numbers.
The multiples of 3 are found by multiplying 3 by the integers: 1, 2, 3, 4, 5...


Find the Least Common Multiple
To find the least common multiple, list the multiples of both numbers and find the least number that appears in both lists.
Here is an example that shows that the least common multiple of 3 and 4 is 12.

Multiples of 3 = 3, 6, 9, 12, 15...
Multiples of 4 = 4, 8, 12, 16, 20...

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Read more multiples
Read more about the least common multiple
Read more about how to find the least common multiple

Powers

Powers are the result of multiplying a number by itself. In this mini-curriculum, you will learn about the parts of powers.

Powers
A power is the product of multiplying a number by itself. Here is an example of a power:

This is 3 squared. 3 is multiplied by itself 2 times.

Base
The base is part of a power. It is the number that is multiplied by itself.
In our example, 3 is the base:


Exponent
The exponent is part of a power. It tells you how many times a number is multiplied by itself.
In our example, 2 is the exponent:


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Read more powers
Read more about bases
Read more about exponents

Laws of Exponents

The laws of exponents lets you do arithmetic with powers. In this mini-curriculum, you will learn about the laws of exponents.

Laws of Exponents
The laws of exponents are rules for using exponents. They tell us how to do arithmetic with powers.


Multiplying Powers
When the same numbers with different exponents are multiplied together, the exponents can be added.


Dividing Powers
When the same numbers with different exponents are divided, the exponents can be subtracted.


Powers of Powers
A power of a power is where a number with an exponent is itself raised to an exponent. When a number with an exponent is raised to another exponent, the exponents can be multiplied.


Negative Exponent
A number with a negative exponent is equal to 1 divided by the number with the exponent made positive.


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Read more the laws of exponents
Read more about how to multiply powers
Read more about how to divide powers
Read more about how to find a power of a power
Read more about how to find negative exponents

Reciprocals



Reciprocals of a number are 1 divided by the number. They can be denoted by an exponent of −1. In this mini-curriculum, you will learn about reciprocals.

Reciprocals
The reciprocal of a number is the result of dividing 1 by that number.


Exponent of −1
A number with an exponent of −1 is equal to 1 divided by the number.


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Read more reciprocals
Read more about an exponent of −1