⭐ Sabse Important Points · एक नज़र में
- Whole Numbers: Set of numbers {0, 1, 2, 3, ...}. They do not include fractions or decimals.
- Addition Rule: When adding integers with the same sign, add their absolute values and keep the sign. With different signs, subtract the smaller absolute value from the larger and keep the sign of the larger.
- Integers: Set of numbers {..., -3, -2, -1, 0, 1, 2, 3, ...}. They include positive numbers, negative numbers, and zero.
- Subtraction Rule: Subtracting an integer is the same as adding its additive inverse (a - b = a + (-b)).
- Closure Property: Integers are closed under addition, subtraction, and multiplication (result is always an integer), but not under division.
- Multiplication/Division Rule: The product or quotient of two integers with the same sign is positive; with different signs is negative.
Introduction · परिचय
Integers and whole numbers are fundamental concepts in mathematics, forming the basis for numerical operations. Whole numbers include zero and all positive integers (0, 1, 2, 3, ...), while integers include all positive and negative whole numbers along with zero (..., -3, -2, -1, 0, 1, 2, 3, ...). Understanding their properties, operations, and number line representation is crucial for solving problems in SSC MTS.
Whole Numbers and Integers: Definitions and Properties
Whole numbers are non-negative integers starting from 0. Integers extend whole numbers to include negative numbers. Both sets are infinite and have specific properties under addition, subtraction, multiplication, and division.
⭐ Key Points · याद रखने वाली बातें
- Whole Numbers: Set of numbers {0, 1, 2, 3, ...}. They do not include fractions or decimals.
- Integers: Set of numbers {..., -3, -2, -1, 0, 1, 2, 3, ...}. They include positive numbers, negative numbers, and zero.
- Closure Property: Integers are closed under addition, subtraction, and multiplication (result is always an integer), but not under division.
- Commutative Property: Addition and multiplication are commutative for integers (a+b = b+a, a×b = b×a), but subtraction and division are not.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Classify the following numbers as whole numbers or integers: -5, 0, 7, -2, 3.
✅ Solution: Whole numbers: 0, 7, 3; Integers: -5, 0, 7, -2, 3
💡 Explanation: Whole numbers are non-negative, so -5 and -2 are not whole numbers. All given numbers are integers because they are whole numbers or their negatives.
Operations on Integers and the Number Line
Integers can be represented on a number line, which helps visualize addition and subtraction. Rules for operations with positive and negative numbers are essential.
- Start with expression
- Perform multiplication/division left to right
- Perform addition/subtraction left to right
- Apply integer rules for signs
- Get final result
📐 Formula / Rule Box · सूत्र याद रखो
Addition of opposite signs
a + (-b) = a - b
Subtraction as addition
a - b = a + (-b)
Multiplication of signs
(+) × (+) = (+), (+) × (-) = (-), (-) × (-) = (+)
Addition Rule
When adding integers with the same sign, add their absolute values and keep the sign. With different signs, subtract the smaller absolute value from the larger and keep the sign of the larger.
Subtraction Rule
Subtracting an integer is the same as adding its additive inverse (a - b = a + (-b)).
Multiplication/Division Rule
The product or quotient of two integers with the same sign is positive; with different signs is negative.
⭐ Key Points · याद रखने वाली बातें
- Absolute Value: The distance of a number from zero on the number line, always non-negative. |a| = a if a ≥ 0, |a| = -a if a < 0.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Calculate: (-8) + 5 - (-3) × 2
✅ Solution: -8 + 5 - (-6) = -3 + 6 = 3
💡 Explanation: First, perform multiplication: (-3) × 2 = -6. Then, rewrite subtraction as addition: -8 + 5 + 6. Now add: -8 + 5 = -3; -3 + 6 = 3.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Which of the following is not an integer? (a) -7 (b) 0 (c) 3/4 (d) 12
👀 Answer dekho
✅ Answer: (c) 3/4
💡 Explanation: Integers include whole numbers and their negatives, but not fractions. 3/4 is a fraction, so it is not an integer.
Q2Evaluate: (-15) + 8 - (-4)
👀 Answer dekho
✅ Answer: -3
💡 Explanation: (-15) + 8 = -7; then -7 - (-4) = -7 + 4 = -3.
Q3What is the product of (-6) and (-3)?
👀 Answer dekho
✅ Answer: 18
💡 Explanation: The product of two negative integers is positive: (-6) × (-3) = 18.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Forgetting that zero is a whole number and an integer, not a positive number.
- Misapplying signs in multiplication/division: e.g., (-2) × (-3) = -6 (incorrect; correct is +6).
- Ignoring order of operations (BODMAS) when solving integer expressions.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Whole Numbers: {0, 1, 2, 3, ...} – no negatives, no fractions.
- Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...} – include negatives and zero.
- Sign Rules: Same signs → positive; different signs → negative (for multiplication/division).
- Number Line: Right is positive, left is negative; absolute value is distance from zero.