In math, the **integers** are numbers that do not contains fractions or decimals. The set includes zero, the natural numbers (counting numbers), and their additive inverses (the negative integers). Examples of integers include -5, 0, and 7. Here is everything you need to know about the integers, including examples of them and their rules for addition, subtraction, multiplication, and division.

### What Are Integers?

Integers are numbers without decimals or fractions. But, unlike the whole numbers, integers include negative numbers. Typical symbols for the set of integers include the capital letter Z, as well as the blackboard bold or boldface versions of the letter. The set is Z = {…, -3, -2, -1, 0, 1, 2, 3, …}. It is a countably infinite set. All of the integers are both rational and real numbers.

### Examples of Integers and Numbers That Are Not Integers

Integers include:

- Negative integers
- Zero
- Positive integers

So, examples of integers are:

- -57
- -13
- -1
- 0
- 5
- 124

Examples of numbers that are not integers are:

- 1/3
- 0.45
- -22.6
- 132.4
- -1/2
**โ**3- Pi
- 3
*i* - 7 โ

### Addition and Subtraction

The rules for adding and subtracting integers are essentially the rules for working with positive and negative numbers.

When adding two integers that have the same sign, the answer has the same sign. For example:

- 1 + 3 = 4
- (-4) + (-3) = -7
- 4 + 0 = 4
- 0 + (-2) = -2

Adding integers with different signs works the same as subtracting integers. The answer has the same sign as the number with the larger value. Note that when you subtract a negative integer, it is the same as adding its positive value. For example:

- (-7) + 2 = -5 which also equals 2 – 7
- 4 + (-8) = 4 โ 8 = -4
- 3 – (-2) = 3 + 2 = 5
- (-5) – (-3) = (-5) + 3 = -2 which also equals 3 – 5
- 8 – 0 = 8

### Multiplication and Division

The rules for multiplication and division are straightforward:

- If both integers are positive, the answer is positive.
- When both integers are negative, the answer is positive. The two negative signs cancel each other out.
- If the numbers have different signs, the answer is negative.
- When adding or dividing several numbers, add up how many are positive and how many are negative. If there are an odd number of negative signs, the answer is also negative. Otherwise, it is positive.
- Multiplying any integer by 0 equals 0.
- Dividing by zero is either infinity or else undefined (depending on the type of math you are doing).
- Multiplying two integers gives another integer, but this is not a rule for division.

Here are examples of multiplication and division using integers.

- 4 x 5 = 20
- (-2) x (-3) = 6
- (-6) x 3 = -18
- 7 x (-2) = -14
- 2 x (-3) x 4 = -24
- (-2) x 2 x (-3) = 12
- 12 / 2 = 6
- (-2) / 6 = -1/3
- (-10) / 5 = -2
- 14 / (-7) = -2
- (-6) / (-2) = 3

### Rules and Properties of Integers

The algebraic properties of integers include:

- Closure
- The identity element
- Associative property
- Commutative property
- Distributive property
- Inverse elements
- No division by zero

Note that the integers are closed under subtraction, but are not closed under division. In other words, if you divide one integer by another, the answer is not necessarily another integer. For example, 2 divided by 4 is 1/2 or 0.5 (not an integer). Division of an integer by zero is undefined.

Here is a summary of these properties.

Addition/Subtraction | Multiplication/Division | |
---|---|---|

Closure | a + b = integer; a – b = integer | a x b = integer; not closed for division |

Identity Element | a + 0 = a; a – 0 = a | a + 1 = a; a รท 1 = a |

Associative Property | a + (b + c) = (a + b) + c; a – (b – c) – (a – b) – c | a x (b x c) = (a x b) x c |

Commutative Property | a + b = b + a | a x b = b x a |

Distributive Property | a x (b + c) = (a x b) + (a x c) and | (a + b) x c = (a x c) + (b x c) |

Inverse Elements | a + (-a) = 0 | -1 and 1 are the only invertible integers. |

No Zero Divisors | If a x b = 0, then a = 0 and/or b = 0 |

### References

- Clapham, Christopher; Nicholson, James (2014).
*The Concise Oxford Dictionary of Mathematics*(5th ed.). Oxford University Press. ISBN 978-0-19-967959-1. - Frobisher, Len (1999).
*Learning to Teach Number: A Handbook for Students and Teachers in the Primary School*. The Stanley Thornes Teaching Primary Maths Series. Nelson Thornes. ISBN 978-0-7487-3515-0. - Goldrei, Derek (1998).
*Classic Set Theory: A Guided Independent Study*(1st ed.). Boca Raton, FL: Chapman & Hall/CRC. ISBN 978-0-412-60610-6. - Mac Lane, Saunders; Birkhoff, Garrett (1999).
*Algebra*(3rd ed.). American Mathematical Society. ISBN 0-8218-1646-2. - Mendelson, Elliott (2008).
*Number Systems and the Foundations of Analysis*. Dover Books on Mathematics. Courier Dover Publications. ISBN 978-0-486-45792-5.