Integers: Rules, Operations, Properties and Integer Questions
Integers are whole numbers that include negative numbers, zero and positive numbers. This topic covers integer basics, number-line representation, signs, arithmetic operations, properties, absolute value and commonly used formulas. Worked examples explain how to add, subtract, multiply and divide integers accurately.
What are Integers?
The set of integers is represented by ℤ. On a number line, positive integers lie to the right of zero and negative integers lie to the left. The opposite of an integer has the same magnitude but the opposite sign; for example, the opposite of 7 is -7, while the opposite of -4 is 4. Zero is neither positive nor negative.
Integers Formula & Tricks
Important Formulas
An integer and its opposite always have a sum of zero. For example, 9 + (-9) = 0.
Absolute value is always non-negative. Thus, |−8| = 8 and |6| = 6.
To subtract an integer, add its additive inverse. For example, 7 - (-3) = 7 + 3 = 10.
The product of two consecutive integers n and n + 1 is n(n + 1). For example, 6 × 7 = 42.
Quick Tricks
Change the subtraction sign to addition and reverse the sign of the second integer.
Like signs give a positive result, while unlike signs give a negative result.
On a number line, the integer farther to the right is greater. Therefore, every positive integer is greater than zero, and zero is greater than every negative integer.
Integers Concepts
Addition and subtraction of integers
Subtraction is changed into addition of the opposite: a - b = a + (-b). Examples: (-6) + (-4) = -10, (-9) + 5 = -4, and 8 - (-3) = 11.
Multiplication and division sign rules
Multiply or divide the absolute values first, then apply the sign. The rules are (+) × (+) = +, (-) × (-) = +, (+) × (-) = -, and (-) × (+) = -. The same sign rules apply to division. Division by zero is undefined.
Properties of integer operations
Closure means that the result remains an integer. Addition and multiplication are commutative: a + b = b + a and ab = ba. Subtraction and division are not commutative. Addition and multiplication are associative, but subtraction and division are not. Multiplication is distributive over addition and subtraction: a(b + c) = ab + ac and a(b - c) = ab - ac.
Absolute value and opposites
For any integer a, |a| = |-a| and |a| ≥ 0. The integers a and -a are opposites. If a is positive, -a is negative; the opposite of zero is zero.
Ordering and comparing integers
Every positive integer is greater than zero, and zero is greater than every negative integer. Among negative integers, the number with the smaller absolute value is greater. For example, -3 > -8 because 3 is less than 8.
Integers Video Lessons
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Prime Numbers: Test and Find Them Fast
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Integer Rules: Quick Revision
Recall these definitions, sign rules and properties while solving integer questions.
- Integers include negative whole numbers, zero and positive whole numbers; fractions and decimals are not integers.
- For addition with unlike signs, subtract absolute values and use the sign of the number with greater absolute value.
- Subtraction follows a - b = a + (-b).
- Like signs give a positive product or quotient; unlike signs give a negative product or quotient.
- Division by zero is undefined.
- Integers are closed under addition, subtraction and multiplication, but not under division.
- Addition and multiplication are commutative and associative; subtraction and division are neither commutative nor associative.
- The absolute value of an integer is its non-negative distance from zero.
Integers FAQs
Is zero an integer?
Yes. Zero is an integer, but it is neither positive nor negative.
What is the result of subtracting a negative integer?
Subtracting a negative integer becomes addition: a - (-b) = a + b. For example, 6 - (-4) = 10.
Why is -5 greater than -9?
On the number line, -5 lies to the right of -9. Therefore, -5 > -9, even though its absolute value is smaller.
Are integers closed under division?
No. Dividing two integers does not always produce an integer. For example, 7 ÷ 2 = 3.5.
What is the value of |−13|?
|−13| = 13 because absolute value gives the distance from zero.
What is the sign of the product of three negative integers?
The product is negative because an odd number of negative factors gives a negative result. For example, (-2)(-3)(-4) = -24.