Integers
Overview
Integers include all positive whole numbers, negative whole numbers, and zero. This chapter extends the number line beyond natural numbers and whole numbers to include negative integers. Students learn to compare, order, and perform all four arithmetic operations on integers. Understanding integers is essential for real-life situations like temperature below zero, depths below sea level, and financial transactions involving debits and credits.
Typed Notes
What Are Integers?
Integers are the collection of whole numbers and their negatives: { …, -4, -3, -2, -1, 0, 1, 2, 3, 4, … }. Positive integers (1, 2, 3, …) lie to the right of zero on the number line, while negative integers (-1, -2, -3, …) lie to the left. Zero is neither positive nor negative. We use integers in daily life to represent temperatures below 0°C, floors below ground level in a building, or money owed. The further left a number is on the number line, the smaller its value.
Addition and Subtraction of Integers
To add two integers with the same sign, add their absolute values and keep the common sign. To add integers with different signs, subtract the smaller absolute value from the larger and keep the sign of the integer with the larger absolute value. For example, (-7) + 3 = -4. Subtraction is converted to addition using the rule a – b = a + (–b). So 5 – (–3) = 5 + 3 = 8. The additive inverse of any integer a is –a, and their sum is always zero.
Multiplication of Integers
Multiplying two integers follows simple sign rules. Positive × Positive = Positive (e.g., 4 × 5 = 20). Negative × Negative = Positive (e.g., (–4) × (–5) = 20). Positive × Negative = Negative (e.g., 4 × (–5) = –20). Negative × Positive = Negative (e.g., (–4) × 5 = –20). Multiplying any integer by 1 gives the same integer, and multiplying by –1 gives its additive inverse. The distributive property holds: a × (b + c) = a × b + a × c, which is very useful for simplifying calculations.
Division of Integers
Division of integers also follows sign rules similar to multiplication. Positive ÷ Positive = Positive (e.g., 20 ÷ 4 = 5). Negative ÷ Negative = Positive (e.g., (–20) ÷ (–4) = 5). Positive ÷ Negative = Negative (e.g., 20 ÷ (–4) = –5). Negative ÷ Positive = Negative (e.g., (–20) ÷ 4 = –5). Division by zero is not defined. Zero divided by any non-zero integer equals zero. Unlike multiplication, division of integers is neither commutative nor associative.
Key Concepts
Revision ke liye ek nazar — exam se pehle yahi padhna hai.
- 01Integers = { …, -3, -2, -1, 0, 1, 2, 3, … }; negative integers are to the left of zero on the number line.
- 02On a number line, moving right means adding a positive integer; moving left means adding a negative integer.
- 03Subtraction of an integer is the same as adding its additive inverse: a – b = a + (–b).
- 04Product or quotient of two integers with the SAME sign is POSITIVE; with DIFFERENT signs is NEGATIVE.
- 05Zero multiplied by any integer is zero, and division of zero by any non-zero integer is zero.
Important Formulas
Questions & Solutions
Pehle khud try karo — phir solution kholo. Yahi tariqa topper banata hai.
Evaluate: (–15) + 28 + (–13)
Subtract (–12) from 9.
Find the product: (–6) × (–7) × (–2).
Divide: (–144) ÷ 12.
Using distributive property, find (–4) × (6 + (–3)).
Integers extend whole numbers to include negatives. Addition and subtraction use the number line concept and additive inverse. Multiplication and division follow the same-sign-positive, different-sign-negative rule. Key properties include closure, commutativity, associativity for addition and multiplication, and distributivity of multiplication over addition. These operations on integers form the foundation for algebra and higher mathematics.
Chapter-end quiz
4 MCQsOption tap karo — turant pata chalega sahi ya galat, explanation ke saath.
Q1.What is the additive inverse of –9?
Q2.Which of the following is correct?
Q3.What is (–56) ÷ (–8)?
Q4.On a number line, which integer is 5 steps to the left of –2?
Mark karo — dashboard par progress dikhega.