Class 9 Maths · Original EZT practice
The World of Numbers (Class 9 Chapter 3): summary and practice
The number line you use today is the result of thousands of years of discovery, much of it in India. This chapter traces how counting began, how zero transformed mathematics, how integers and rational numbers filled the gaps, and why irrational numbers complete the real line. All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 22 October 2026:
1. The dawn of mathematics
Counting began with one-to-one correspondence - a pebble for each cow. The Lebombo Bone, about 35000 years old with 29 notches, is early evidence of humans recording natural numbers.
2. The revolution of shunya
India turned 'nothing' into a number. Pingala's Chandahshastra used zero-like placeholders around the 3rd-2nd century BCE, the Bakhshali manuscript shows an early zero symbol, and Brahmagupta gave rules for computing with zero in 628 CE.
3. Integers and rational numbers
Negative numbers went from debts in the Arthashastra to Brahmagupta's full sign rules. Fractions and their ratios then filled the spaces between integers - that is the rational numbers Q.
4. Irrational and real numbers
The diagonal of a unit square, the square root of 2, cannot be written as any fraction - an irrational number. Rationals and irrationals together form the real numbers R, where decimals either terminate, repeat (like 1/7 and the cyclic number 142857), or never repeat (like pi).
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Count before numbers
Choose the method: Match objects one-to-one - no number words needed.
EZT practice question
A herder puts one pebble in a pot for each cow that leaves. In the evening, what does it mean if pebbles remain after all cows are matched?
Show the worked solution
- Each cow out removed one pebble's 'match'.
- Each cow back removes one pebble.
- Pebbles remaining means fewer cows returned than left.
Answer: Some cows are still missing.
Check: If the pot is empty, every pebble found its cow - the herd is complete. This matching is one-to-one correspondence, the birth of the natural numbers.
Watch for this: Do not count the pebbles as the answer - the point is the leftover, not the total.
Now try it yourself
What does an empty pot at dusk tell the herder?
Check your answer
Every cow has returned.
Every pebble found its matching cow.
Compute with zero
Choose the method: Brahmagupta's rules: a + 0 = a, a - 0 = a, a x 0 = 0.
EZT practice question
Using Brahmagupta's rules from the Brahmasphutasiddhanta (628 CE), evaluate 7 + 0, 7 - 0 and 7 x 0.
Show the worked solution
- Adding zero changes nothing: 7 + 0 = 7.
- Subtracting zero changes nothing: 7 - 0 = 7.
- Multiplying by zero gives zero: 7 x 0 = 0.
Answer: 7, 7 and 0.
Check: These match the rules in the Brahmasphutasiddhanta - the first known text to treat zero as a full number with its own arithmetic.
Watch for this: Division by zero has NO answer - Brahmagupta wrestled with it, and it remains undefined today.
Now try it yourself
0 - 5?
Check your answer
-5.
Taking 5 from nothing leaves a debt of 5.
Multiply with signs
Choose the method: Brahmagupta's sign rules: same signs give positive, different signs give negative.
EZT practice question
Evaluate (-3) x (-4) and (-3) x 4.
Show the worked solution
- Negative times negative: Brahmagupta wrote 'a debt multiplied by a debt is a fortune' - so (-3) x (-4) = 12.
- Negative times positive: a debt multiplied by a fortune is a debt - so (-3) x 4 = -12.
- These sign rules are over 1300 years old and still exact.
Answer: 12 and -12.
Check: Consistency check: (-3) x (4 - 4) = 0, and expanding gives (-3) x 4 + (-3) x (-4) = -12 + 12 = 0 - the rules cannot be otherwise.
Watch for this: A negative times a negative is POSITIVE - same signs always give a positive product.
Now try it yourself
(-5) x (-2)?
Check your answer
10.
Two debts multiplied give a fortune.
Find a rational between two rationals
Choose the method: Average them - the result stays rational.
EZT practice question
Find a rational number between 1/2 and 3/4.
Show the worked solution
- The average (1/2 + 3/4) / 2 lies between them.
- 1/2 + 3/4 = 2/4 + 3/4 = 5/4.
- (5/4) / 2 = 5/8.
Answer: 5/8.
Check: 0.5 < 0.625 < 0.75 - the average always lands strictly between, and it is a ratio of integers, so it is rational. Between any two rationals there is another - the rationals are dense.
Watch for this: Average is not the only answer - 9/16 or 11/16 work too; any value strictly between is fine.
Now try it yourself
A rational between 0 and 1/4?
Check your answer
1/8.
It is the average of 0 and 1/4.
Meet a cyclic decimal
Choose the method: Divide and watch the remainder cycle repeat.
EZT practice question
What is special about the decimal expansion of 1/7?
Show the worked solution
- 1/7 = 0.142857142857... - the block 142857 repeats forever.
- 142857 is a cyclic number: 142857 x 2 = 285714, x 3 = 428571 - rotations of the same digits.
- And 142857 x 7 = 999999.
Answer: Its decimal repeats the 6-digit block 142857, a cyclic number.
Check: Every rational number has a decimal that terminates or eventually repeats - 1/7 repeats with period 6 because the remainders must cycle.
Watch for this: A repeating decimal is still RATIONAL - non-repeating and non-terminating is what makes a number irrational.
Now try it yourself
What is 142857 x 3?
Check your answer
428571.
The same digits rotated one more step.
Sort rational from irrational
Choose the method: A rational number is a ratio of two integers; an irrational one is not.
EZT practice question
Sort into rational and irrational: 3.14, the square root of 2, 0.333..., pi, 5/8.
Show the worked solution
- 0.333... = 1/3 - a repeating decimal, so rational.
- 5/8 is a ratio of integers - rational. 3.14 = 314/100 - rational.
- The square root of 2 and pi cannot be written as any fraction - irrational.
Answer: Rational: 3.14, 0.333..., 5/8. Irrational: the square root of 2 and pi.
Check: The classic proof assumes root 2 = a/b in lowest terms and reaches a contradiction - so root 2 must be irrational. Aryabhata's 3.1416 is a rational APPROXIMATION of pi, not pi itself.
Watch for this: 3.14 and 22/7 are rational APPROXIMATIONS of pi - pi itself is irrational.
Now try it yourself
Is 0.1010010001... rational?
Check your answer
No.
It never repeats, so no fraction equals it.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Lebombo Bone?
Show answer and reason
About 35000-year-old tally bone with 29 notches.
Early evidence of humans recording numbers.
2. Brahmagupta, 628 CE?
Show answer and reason
Gave the first known rules for computing with zero and negative numbers.
In the Brahmasphutasiddhanta.
3. Rational number?
Show answer and reason
A ratio of two integers, with non-zero denominator.
Like 5/8 or 0.333... which equals 1/3.
4. Cyclic number?
Show answer and reason
142857 - multiplying it by 1 to 6 rotates its digits.
It is the repeating block of 1/7.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Find a rational number between 1/3 and 1/2.
- Evaluate (-2) x (-2) x (-2).
- Is 0.1010010001... rational or irrational? Why?
Know the rule but lose track of the steps?
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Questions about this practice page
- What is the Lebombo Bone?
- A roughly 35000-year-old bone from the Lebombo Mountains with 29 carved notches - among the earliest physical evidence of humans recording natural numbers, probably as a tally or calendar.
- Who gave the first rules for zero?
- The Indian mathematician Brahmagupta, in his Brahmasphutasiddhanta of 628 CE. He stated rules such as a + 0 = a and a x 0 = 0, and treated zero as a number in its own right.
- What is the difference between rational and irrational numbers?
- A rational number is a ratio of two integers, like 5/8 or 0.333...; its decimal terminates or repeats. An irrational number, like the square root of 2 or pi, cannot be written as any fraction and its decimal never repeats.
- Why is 142857 called a cyclic number?
- Because multiplying it by 1, 2, 3, 4, 5 or 6 only rotates its digits - 142857 x 3 = 428571 - and 142857 x 7 = 999999. It is the repeating block in the decimal expansion of 1/7.
- Are these the NCERT exercise questions for The World of Numbers (Class 9)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.