Integers are a type of number that includes all whole numbers (positive, negative, or zero) and their opposites. They are often represented using the symbol “Z” (for the German word “Zahlen,” which means “numbers”).
What are Integers?
Integers are whole numbers, including positive and negative numbers, as well as zero. They are different from fractions or decimal numbers. Integers can be added, subtracted, multiplied, and divided just like any other numbers.
The set of integers is represented using the symbol “Z”, includes:
- Positive Numbers: Positive numbers are numbers greater than zero. Example: 1, 2, 3, . . .
- Negative Numbers: Negative numbers are numbers less than zero. Example: -1, -2, -3, . . .
- Zero is a number that represents “nothing” or “none”.
For example, the set of integers includes the following numbers:
Z ={ …, -3, -2, -1, 0, 1, 2, 3, … }and so on, where the dots indicate that the set goes on infinitely in both directions.
Integer Operations
Integer operations refer to the basic arithmetic operations that can be performed on integers. These operations include
- Addition of Integers
- Subtraction of Integers
- Multiplication of Integers
- Division of Integers
Addition of Integers
Addition is the process of combining two or more integers to find their sum. When adding two integers of the same sign, we simply add their absolute values and use the same sign as the integers being added.
For example, 3 + 5 = 8 and -3 + (-5) = -8. When adding integers of opposite signs, we subtract their absolute values and use the sign of the larger absolute value. For example, 3 + (-5) = -2.
Subtraction of Integers
Subtraction is the process of finding the difference between two integers. When subtracting an integer from another integer of the same sign, we simply subtract their absolute values and use the same sign as the integers being subtracted.
For example, 8 – 3 = 5 and -8 – (-3) = -5. When subtracting integers of opposite signs, we add their absolute values and use the sign of the integer with the larger absolute value. For example, 3 – (-5) = 8.
Multiplication of Integers
Multiplication is the process of finding the product of two or more integers. When multiplying two integers with the same sign, the product is positive.
For example, 3 x 5 = 15 and (-3) x (-5) = 15. When multiplying integers with opposite signs, the product is negative. For example, (-3) x 5 = -15.
Rules of Integers in Multiplication
| Product of Signs | Result | Example |
|---|---|---|
| (+) × (+) | + | 3 × 5 = 15 |
| (+) × (-) | – | 3 × (-5) = -15 |
| (-) × (+) | – | (-3) × 5 = -15 |
| (-) × (-) | + | (-3) × (-5) = 15 |
Division of Integers
Division is the process of finding how many times one integer can be divided by another integer. When dividing two integers with the same sign, the quotient is positive.
For example, 10 ÷ 2 = 5 and (-10) ÷ (-2) = 5. When dividing integers with opposite signs, the quotient is negative. For example, (-10) ÷ 2 = -5.
Properties of Integers
Integers have various unique properties that apply only to them. Some of the important properties of integers include:
Closure Property
The closure property of integers states that the result of adding, subtracting, multiplying, or dividing two integers is always another integer. For example, when you add or subtract two integers, the result is always an integer.
Associative Property
The associative property of integers states that the way in which you group integers in an expression does not affect the result. For example, (2+3)+4 = 2+(3+4) = 9.
Commutative Property
The commutative property of integers states that the order in which you add or multiply two integers does not affect the result. For example, 2+3 = 3+2 and 2×3 = 3×2.
Distributive Property
The distributive property of integers states that the product of an integer and the sum or difference of two integers is equal to the sum or difference of the products of the integer and each of the integers. For example, 2x(3+4) = 2×3 + 2×4.
Identity Property
The identity property of integers states that the sum of any integer and zero is equal to the integer itself. For example, 3+0 = 3.
Inverse Property
The inverse property of integers states that the sum of an integer and its additive inverse (opposite) is equal to zero. For example, 3+(-3) = 0.
Even and Odd Property
Every integer is either even or odd. An even number is a multiple of 2, while an odd number is not a multiple of 2.
Frequently Asked Questions on Integers
Q: What are integers?
A: Integers are whole numbers that can be positive, negative, or zero. Examples of integers include -3, 0, 5, and 100.
Q: How are integers different from fractions or decimals?
A: Integers are different from fractions or decimals because they are not expressed as a ratio of two numbers (like fractions) or as a number with a decimal point (like decimals). Integers are simply whole numbers, whether positive, negative, or zero.
Q: What is the difference between positive and negative integers?
A: Positive integers are whole numbers greater than zero (1, 2, 3, etc.). Negative integers are whole numbers less than zero (-1, -2, -3, etc.). Zero is neither positive nor negative.
Q: What is the absolute value of an integer?
A: The absolute value of an integer is the distance that integer is from zero on the number line. For example, the absolute value of -3 is 3, and the absolute value of 5 is 5.
Q: What is the opposite of an integer?
A: The opposite of an integer is the integer with the same absolute value but the opposite sign. For example, the opposite of 5 is -5, and the opposite of -3 is 3.
Q: How do you add or subtract integers?
A: To add integers, you simply add the numbers together. To subtract integers, you can either add the opposite (or negative) of the second number, or you can change the subtraction sign to addition and change the sign of the second number. For example, to subtract 5 from 8, you can add the opposite of 5 (-5) to 8: 8 + (-5) = 3. Or, you can change the subtraction sign to addition and change the sign of 5: 8 + (-5) = 3.
Q: What is the order of operations for integers?
A: The order of operations for integers is the same as for any other type of mathematical expression: parentheses, exponents, multiplication and division (performed from left to right), and addition and subtraction (performed from left to right).