Integers are an extension of the number system that we use every day. In earlier classes you learnt about natural numbers (1, 2, 3, ...) and whole numbers (0, 1, 2, 3, ...). Integers include all the whole numbers together with their negative counterparts. The set of integers is written as ..., -3, -2, -1, 0, 1, 2, 3, ... and is denoted by the letter Z. Negative integers are used in real life to represent temperatures below zero, debt or money owed, sea level below zero (like the Dead Sea), and floors below ground level in a building.
The study of integers is the foundation for all of algebra and higher mathematics. When we add, subtract, multiply or divide integers, certain rules apply based on the signs of the numbers. Understanding these rules helps us solve problems about profit and loss, temperature change, distances above and below sea level, and scores in games. Once you master the four operations on integers, you will be ready to handle fractions, decimals, rational numbers and algebraic expressions with confidence.
The sum of any two integers is always an integer. For example, (-3) + 7 = 4 and 5 + (-8) = -3. Since 4 and -3 are both integers, integers are closed under addition.
Addition of integers is commutative. If a and b are any two integers, then: a + b = b + a For example, (-6) + 4 = -2 and 4 + (-6) = -2, so (-6) + 4 = 4 + (-6).
Addition of integers is associative. For any integers a, b and c: (a + b) + c = a + (b + c) For example, (2 + 3) + (-4) = 5 + (-4) = 1 and 2 + (3 + (-4)) = 2 + (-1) = 1. Both give the same result, so the way we group the numbers does not matter.
Zero is the additive identity for integers. For every integer a: a + 0 = a and 0 + a = a For example, (-12) + 0 = -12.
For every integer a, there exists an integer -a such that: a + (-a) = 0 The number -a is called the additive inverse of a. For example, the additive inverse of 8 is -8, and 8 + (-8) = 0. The additive inverse of -15 is 15, because (-15) + 15 = 0.
Subtracting an integer is the same as adding its additive inverse. To subtract b from a, we add the additive inverse of b to a: a - b = a + (-b) For example, 7 - (-3) = 7 + 3 = 10 and (-5) - 2 = (-5) + (-2) = -7.
The difference of any two integers is always an integer. For example, 4 - (-6) = 10, which is an integer, so integers are closed under subtraction.
Subtraction of integers is not commutative. In general, a - b is not equal to b - a. For example, 9 - 5 = 4 but 5 - 9 = -4. The two results are different.
Subtraction is not associative either. For example, (8 - 3) - 2 = 5 - 2 = 3 but 8 - (3 - 2) = 8 - 1 = 7. The two results differ, so we must be careful with the order of operations.
Multiplication of two integers is governed by the rules of signs: - Positive x Positive = Positive (for example, 3 x 4 = 12) - Positive x Negative = Negative (for example, 3 x (-4) = -12) - Negative x Positive = Negative (for example, (-3) x 4 = -12) - Negative x Negative = Positive (for example, (-3) x (-4) = 12)
The product of two integers of the same sign is positive, while the product of two integers of different signs is negative.
Any integer multiplied by zero gives zero: a x 0 = 0. Any integer multiplied by 1 gives the same integer: a x 1 = a, so 1 is the multiplicative identity. Multiplication by -1 gives the additive inverse: a x (-1) = -a.
The distributive property is very useful for mental calculation. For example, 15 x 102 = 15 x (100 + 2) = 1500 + 30 = 1530.
Division is the inverse operation of multiplication. The sign rules for division are the same as those for multiplication: - Positive divided by Positive = Positive (for example, 20 / 5 = 4) - Positive divided by Negative = Negative (for example, 20 / (-5) = -4) - Negative divided by Positive = Negative (for example, (-20) / 5 = -4) - Negative divided by Negative = Positive (for example, (-20) / (-5) = 4)
For any non-zero integer a, a / 1 = a and a / a = 1. Division by zero is not defined; we can never divide any number by 0. Also, integers are not closed under division, because 5 / 2 = 2.5, which is not an integer.
Integers appear in everyday situations involving opposites: temperatures above and below zero, profits and losses, heights above and below sea level, deposits and withdrawals from a bank account, and scores in games. To solve such problems, identify the positive and negative directions first, then apply the correct operation. For example, if the temperature at 6 am is -3 degree Celsius and it rises by 7 degrees by noon, the new temperature is -3 + 7 = 4 degree Celsius. If a diver is at 12 m below sea level and descends another 8 m, the new position is -12 + (-8) = -20 m, that is, 20 m below sea level.
| Operation | Sign Combination | Result Sign | Example |
|---|---|---|---|
| Multiplication | (+) x (+) | Positive | 4 x 3 = 12 |
| Multiplication | (+) x (-) | Negative | 4 x (-3) = -12 |
| Multiplication | (-) x (+) | Negative | (-4) x 3 = -12 |
| Multiplication | (-) x (-) | Positive | (-4) x (-3) = 12 |
| Division | (+) / (+) | Positive | 12 / 4 = 3 |
| Division | (+) / (-) | Negative | 12 / (-4) = -3 |
| Division | (-) / (+) | Negative | (-12) / 4 = -3 |
| Division | (-) / (-) | Positive | (-12) / (-4) = 3 |
| Property | Addition | Multiplication |
|---|---|---|
| Closure | a + b is always an integer | a x b is always an integer |
| Commutative | a + b = b + a | a x b = b x a |
| Associative | (a + b) + c = a + (b + c) | (a x b) x c = a x (b x c) |
| Identity | a + 0 = a (0 is identity) | a x 1 = a (1 is identity) |
| Distributive | Not applicable | a x (b + c) = a x b + a x c |
| Subtraction/Division | Not commutative, not associative | Not commutative, not associative |
Integers extend the number system to include negative numbers, allowing us to describe opposites such as temperature, sea level, profit and loss, and bank transactions. The rules of addition, subtraction, multiplication and division with signs, together with the commutative, associative and distributive properties, form the backbone of arithmetic and algebra. Practice with both positive and negative numbers builds the confidence needed for fractions, rational numbers, and algebra in higher classes. Mastery of integers is the first big step towards fluent, accurate mathematics.