A Story of Numbers

Why do we need different kinds of numbers?

Counting books uses positive whole quantities, measuring a debt needs negative values, sharing a pizza needs fractions, and measuring a diagonal can require an irrational number. The number system developed as new questions demanded new representations. Example: A lift moving from floor −2 to floor 5 travels 7 floor intervals, because 5−(−2)=7.

Counting numbers, zero and place value

Natural numbers are 1,2,3,…; whole numbers are 0,1,2,3,… . Zero represents no quantity and also serves as a placeholder in positional notation. In 50,304, the digits represent 5×10⁴+0×10³+3×10²+0×10+4. Changing a digit's position changes its value.

Integers and the number line

Integers are …,−3,−2,−1,0,1,2,3,… . Values increase from left to right on a number line. Opposite numbers a and −a are equally far from zero. Absolute value |a| is distance from zero, so |−7|=7. Example: −8+13=5; −4−6=−10. Subtracting a negative number means adding its opposite.

Fractions and rational numbers

A rational number is p/q where p and q are integers and q≠0. Examples: −3/4, 7, 0, 0.125=1/8 and 0.333…=1/3. Equivalent fractions name the same number: 2/3=4/6. To add fractions, use a common denominator: 3/4+2/5=15/20+8/20=23/20.

Decimal forms and terminating or recurring expansions

A rational number's decimal expansion terminates or eventually repeats. Examples: 3/8=0.375 and 1/6=0.1666… . In lowest terms, p/q has a terminating decimal precisely when the positive denominator has no prime factors other than 2 and 5. For example, 7/40 terminates because 40=2³×5, whereas 1/3 repeats.

Irrational numbers and the real number line

Numbers such as √2, √3 and π are irrational: they cannot be expressed as a ratio of integers. √2≈1.4142 lies between 1 and 2; it is the diagonal of a unit square. The real numbers contain rational and irrational numbers. A decimal that never ends and never repeats is irrational; a decimal merely having many digits does not prove irrationality.

Comparing and ordering numbers

For negative numbers, −8<−3 because −8 is farther left on the number line. For fractions, use a common denominator or cross-multiplication with positive denominators: 3/5=15/25 and 4/7≈0.5714, so 3/5>4/7. To compare √5 and 2, square the nonnegative values: 5>4, so √5>2.

Patterns, sequences and mathematical reasoning

Look for the rule, not just the next term. Arithmetic sequence 4,7,10,13 adds 3 each time. Geometric sequence 3,6,12,24 multiplies by 2 each time. Squares 1,4,9,16 have successive differences 3,5,7. A finite list alone does not establish a unique pattern: explain the rule you choose.

A historical perspective on number representations

Ancient societies developed different numeral systems for recording trade, calendars and measurement. The Hindu–Arabic place-value system uses ten symbols, 0–9, and powers of ten. India's mathematical traditions contributed significantly to positional decimal notation and treatments of zero. Historical dates and individual priority claims require specific source verification; do not infer them from this overview.

Important properties and common mistakes

Zero is whole but is not positive. Division by zero is undefined. A negative number can be rational, for example −2=−2/1. Not every square root is irrational: √49=7. 0.121212… is rational because its digits repeat. Do not compare negative numbers by their absolute values without reversing the direction: −9<−2.

Revision and formula sheet

Natural numbers N={1,2,3,…}; whole numbers W={0,1,2,…}; integers Z={…,−1,0,1,…}; rational numbers Q={p/q:p,q∈Z,q≠0}; real numbers R contain Q and irrational numbers. Distance between a and b on a number line is |a−b|. Add fractions a/b+c/d=(ad+bc)/bd for nonzero b,d. Decimal termination test: reduced denominator has only prime factors 2 and 5.

Self-assessment and flashcards

Flashcard: What does |−12| mean? Distance from zero, 12. Flashcard: Is 0.25 rational? Yes, 1/4. Flashcard: Is √25 irrational? No, it equals 5. Flashcard: Is 1/7 terminating? No, it repeats. Flashcard: What is the difference between a digit and its place value? The digit is a symbol; place value depends on position.

Original practice questions — fully worked solutions

These are original supplementary questions, not quotations or numbered NCERT textbook exercises.

1. Question 1. Classify 0, −7, 4/5, √9 and √2 into suitable number sets.

   Step-by-step solution: 0 is whole, integer, rational and real. −7 is integer, rational and real. 4/5 is rational and real. √9=3 is natural, whole, integer, rational and real. √2 is irrational and real.

   Final answer: √2 is the only irrational number in the list.

2. Question 2. Represent −3/2 on a number line.

   Step-by-step solution: Convert −3/2=−1.5. Mark zero and −1, −2 at equal spacing; place −1.5 exactly halfway between −1 and −2.

   Final answer: −1.5, midway between −1 and −2.

3. Question 3. Calculate −8+13−6.

   Step-by-step solution: First −8+13=5. Then 5−6=−1.

   Final answer: −1.

4. Question 4. Add 3/4 and 2/5.

   Step-by-step solution: The least common denominator is 20. Convert 3/4=15/20 and 2/5=8/20. Add: 23/20.

   Final answer: 23/20 or 1 3/20.

5. Question 5. Determine whether 7/40 has a terminating decimal.

   Step-by-step solution: 40=2³×5. The reduced denominator has no prime factors besides 2 and 5. Divide: 7÷40=0.175.

   Final answer: Yes; 0.175.

6. Question 6. Is 0.272727… rational? Express it as a fraction.

   Step-by-step solution: Let x=0.272727… . Then 100x=27.272727… . Subtract: 99x=27. Hence x=27/99=3/11.

   Final answer: 3/11, rational.

7. Question 7. Which is greater: −4 or −9? Explain.

   Step-by-step solution: On the number line, −4 lies to the right of −9. A number farther right is greater.

   Final answer: −4>−9.

8. Question 8. Estimate √2 to two decimal places.

   Step-by-step solution: Since 1.41²=1.9881 and 1.42²=2.0164, √2 lies between 1.41 and 1.42. More precisely √2≈1.4142, which rounds to 1.41.

   Final answer: 1.41.

9. Question 9. Explain why √49 is rational but √50 is irrational.

   Step-by-step solution: √49=7, an integer. √50=5√2. Since √2 is irrational and 5 is nonzero rational, 5√2 is irrational.

   Final answer: √49 is rational; √50 is irrational.

10. Question 10. A lift starts at floor −2 and rises to floor 5. How many floors does it move?

    Step-by-step solution: Compute the change in position: 5−(−2)=5+2=7.

    Final answer: 7 floor intervals.

11. Question 11. A recipe uses 3/4 cup of flour. How much is needed for 6 batches?

    Step-by-step solution: Multiply 6×3/4=18/4=9/2=4.5.

    Final answer: 4½ cups.

12. Question 12. Find the next three terms of 3, 6, 12, 24 under a doubling rule.

    Step-by-step solution: Multiply each term by 2: 24×2=48, 48×2=96, 96×2=192.

    Final answer: 48, 96, 192.

13. Question 13. MCQ: Which number is irrational? (A) 0.75 (B) √16 (C) √3 (D) −5

    Step-by-step solution: 0.75=3/4; √16=4; −5=−5/1 are rational. √3 is irrational.

    Final answer: Option C, √3.

14. Question 14. HOTS: Can a rational number have infinitely many decimal digits?

    Step-by-step solution: Yes. Rational decimal expansions can repeat indefinitely. Example: 1/3=0.333… . Infinite length alone does not imply irrationality.

    Final answer: Yes; 1/3 is an example.

15. Question 15. Case study: A bank balance is −₹350. A deposit of ₹600 is made and ₹175 is withdrawn. Find the new balance.

    Step-by-step solution: Calculate −350+600−175=250−175=75.

    Final answer: ₹75.

16. Question 16. Competency: Two runners stand at positions −2.5 km and 1.75 km on a straight path. Find their separation.

    Step-by-step solution: Distance=|1.75−(−2.5)|=|4.25|=4.25.

    Final answer: 4.25 km.

Textbook exercise verification status

Not yet verified: Exact official NCERT chapter topics, diagrams, in-text activities, Figure it Out exercise numbering and question wording. Do not interpret the original questions above as NCERT exercise-wise solutions. The official source is NCERT Ganita Prakash Chapter 3 PDF; compare the current edition before claiming complete textbook alignment.