An irrational number cannot be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne0$. Its decimal form is non-terminating and non-repeating.

1.3 Revisiting Irrational Numbers

You have already met $\sqrt2$, $\sqrt3$

Proof by contradiction

The method has five steps:

1. Assume the opposite of what you want to prove.
2. Write the assumed rational number in lowest form.
3. Use algebra and Theorem 1.2.
4. Reach a fact that contradicts "lowest form".
5. Reject the assumption and state the conclusion.

Theorem 1.3: $\sqrt2$ is irrational

Assume $\sqrt2$ is rational. Then

$$\sqrt2=\frac ab,$$

where $a$ and $b$ are coprime positive integers.

Squaring gives $a^2=2b^2$. So $2$ divides $a^2$, and Theorem 1.2 gives $2\mid a$

Substitution gives $4c^2=2b^2$, so $b^2=2c^2$

Both $a$ and $b$ are even. This contradicts the assumption that they are coprime. Therefore, $\sqrt2$ is irrational.

Example 5

Prove that $\sqrt3$ is irrational.

Solution: Assume $\sqrt3=\frac ab$ in lowest form. Then

Rational and irrational combinations

• rational $+$ irrational is irrational
• rational $-$ irrational is irrational
• non-zero rational $\times$ irrational is irrational
• irrational $\div$ non-zero rational is irrational

These rules are used only when the rational multiplier or divisor is non-zero.

Example 6

Show that $5-\sqrt3$ is irrational.

Solution: If $5-\sqrt3$ were rational, subtracting it from the rational number $5$ would make $\sqrt3$

Example 7

Show that $3\sqrt2$ is irrational.

Solution: If $3\sqrt2$ were rational, dividing it by the non-zero rational number $3$ would make $\sqrt2$

Try each printed exercise before opening its answer. A proof earns marks for its reasoned chain, not only its final line.

Exercise 1.1

1. Express each number as a product of its prime factors

Answer:

• $140=2^2\times5\times7$
• $156=2^2\times3\times13$
• $3825=3^2\times5^2\times17$
• $5005=5\times7\times11\times13$
• $7429=17\times19\times23$

2. Find the LCM and HCF of the following pairs and verify that LCM x HCF equals the product

Answer:

• $26,91$: HCF $=13$, LCM $=182$
• $510,92$: HCF $=2$

3. Find HCF and LCM by prime factorisation

Answer:

• $12,15,21$: HCF $=3$, LCM $=420$
• $17,23,29$: HCF $=1$

4. Given HCF$(306,657)=9$, find the LCM

Answer:

$$\operatorname{LCM}=\frac{306\times657}{9}=22338.$$

5. Check whether $6^n$ can end with digit $0$

Answer: No. $6^n=2^n3^n$ has no prime factor $5$.

6. Explain why each expression is composite

Answer:

• $7\times11\times13+13=13(7\times11+1)$
• $7\times6\times5\times4\times3\times2\times1+5=5(7\times6\times4\times3\times2\times1+1)$

Each has a non-trivial factor.

7. Circular path meeting time

Answer: The friends take $18$ and $12$ minutes. They meet together at the starting point after LCM$(18,12)=36$ minutes.

Exercise 1.2

1. Prove that $\sqrt5$ is irrational

Answer: Assume $\sqrt5=a/b$ in lowest form. The equation $a^2=5b^2$

2. Prove that $3+2\sqrt5$ is irrational

Answer: If it were rational, subtracting $3$ and dividing by $2$ would make $\sqrt5$ rational, a contradiction.

3. Prove that the listed numbers are irrational

Answer:

• $1/\sqrt2$: otherwise its reciprocal $\sqrt2$

1.4 Summary

• Every composite number has one prime factorisation apart from order.
• If prime $p$ divides $a^2$, then $p$ divides $a$.
• The contradiction proof shows $\sqrt2$

Note to the Reader

For three positive integers, the textbook gives separate HCF and LCM relations. Do not extend the two-number product rule without checking the stated formula.

Board-answer checklist

• Have you shown prime factors clearly?
• Have you used the correct HCF/LCM powers?
• Have you stated "lowest form" and the final contradiction in every proof?

Scope note: This is an unnumbered Class IX support lesson. It is not a numbered core section of the current NCERT Class 10 Chapter 1 PDF or active 2026-27 board scope.

This lesson is a support lesson for the Real NumbersReal numbersAll numbers that fit somewhere on the number line.Example: -3, 0, 5/8, √2Close chapter. It helps us understand division properly first, so HCFHCFHighest Common Factor: the biggest common factor of the given numbers.Example: HCF of 12 and 18 is 6.Close questions later do not feel scary.

Also, one small English help before we start: lemmaLemmaA small helpful rule used to prove or solve a bigger result.Example: Here it is the rule a = bq + r.Close means a small helpful rule. So do not panic. Big word, simple idea.

chapter start: what is a real numberReal numberAny number that can be shown on the number line.Example: -3, 0, 5/8, √2Close?

A real numberReal numberAny number that can be shown on the number line.Example: -3, 0, 5/8, √2Close is any number that can be shown on the number line.

Examples:

• 5
• 0
• -3
• 3/4
• 0.25
• √2

So yes, this chapter is about many kinds of numbers. Nice crowd.

quick warm-up: first see it the school way

When we divide 17 by 5, most Indian students think in long-division form first. That is the right starting point.

From that long division, we get:

• dividendDividendThe number that is being divided.Example: In 17 ÷ 5, the dividend is 17.Close = 17
• divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close = 5
• quotientQuotientThe answer we get before the remainder.Example: In 17 ÷ 5, the quotient is 3.Close = 3
• remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close = 2

Now write the same idea in one line:

17 = 5 × 3 + 2

So the long-division picture and the equation are saying the same thing.

the rule in simple English

When a positive integer a is divided by another positive integer b, we can always write:

a = bq + r

where:

• a is the dividendDividendThe number that is being divided.Example: In 17 ÷ 5, the dividend is 17.Close
• b is the divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close
• q is the quotientQuotientThe answer we get before the remainder.Example: In 17 ÷ 5, the quotient is 3.Close
• r is the remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close
• 0 ≤ r < b

what this really means

It simply means:

dividend = divisor × quotient + remainder

So this is not a new monster. It is just your old long division written in math language.

important condition

The remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close must always be smaller than the divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close.

Correct:

23 = 5 × 4 + 3

Wrong:

23 = 5 × 3 + 8

Why is the second one wrong? Because the remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close 8 is bigger than the divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close 5.

easy examples

example 1

Divide 20 by 6.

20 = 6 × 3 + 2

So:

• quotientQuotientThe answer we get before the remainder.Example: In 17 ÷ 5, the quotient is 3.Close = 3
• remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close = 2

example 2

Divide 35 by 7.

35 = 7 × 5 + 0

So:

• quotientQuotientThe answer we get before the remainder.Example: In 17 ÷ 5, the quotient is 3.Close = 5
• remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close = 0

When remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close is 0, the division is exact.

why this lesson matters in this chapter

This idea is used in the Euclid method for finding HCFHCFHighest Common Factor: the biggest common factor of the given numbers.Example: HCF of 12 and 18 is 6.Close (Highest Common Factor).

Instead of doing factorisationFactorisationBreaking a number into smaller multiplying parts.Example: 12 = 2 × 2 × 3Close for big numbers every time, we can keep dividing. That is often faster.

finding HCFHCFHighest Common Factor: the biggest common factor of the given numbers.Example: HCF of 12 and 18 is 6.Close using Euclid's method

rule

1. Divide the bigger number by the smaller number.
2. Look at the remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close.
3. If the remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close is not 0, divide again.
4. In the next step, use the old divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close and the old remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close.
5. When the remainder

solved example 1

Find HCF(455, 42).

First write the steps:

Step 1:

455 = 42 × 10 + 35

Step 2:

42 = 35 × 1 + 7

Step 3:

35 = 7 × 5 + 0

Now see the same idea in long-division style.

455 = 42 × 10 + 35

Now the remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close is 0. So the last divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close is 7.

Answer:

HCF(455, 42) = 7

solved example 2

Find HCF(135, 225).

Take the bigger number first.

225 = 135 × 1 + 90

135 = 90 × 1 + 45

90 = 45 × 2 + 0

Now the remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close is 0, so stop. The last divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close is 45.

Answer:

HCF(135, 225) = 45

how to think without fear

Many students get scared when they see many steps. Do not worry. Just remember this:

• non-zero remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close → divide again
• remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close 0 → stop
• last divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close → answer

That is the whole game. No drama needed.

memory trick

• Non-zero remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close? Divide again.
• Old divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close and old remainderRemainderThe part left after division.Example: In 17 ÷ 5, the remainder is 2.Close make the next division.
• Remainder = 0? Stop.
• Last divisorDivisorThe number by which we divide.Example: In 17 ÷ 5, the divisor is 5.Close gives the answer.

board tip

Write each division on a new line. Neat steps save marks. Messy steps donate marks to the examiner.

solved mini check

Find HCF(84, 30).

84 = 30 × 2 + 24

30 = 24 × 1 + 6

24 = 6 × 4 + 0

So,

HCF(84, 30) = 6

remember this line

Big number = divisor × quotient + what is left.

Scope note: This is an unnumbered support lesson. Decimal expansion is not a numbered core section of the current NCERT Class 10 Chapter 1 PDF or active 2026-27 board scope.

This lesson helps students connect fractions and decimals. It is one of the easiest parts of the chapter once the rule becomes clear.

start with a simple truth

A rational numberRational numberA number that can be written in the form p/q where q is not 0.Example: 3/4, -7/5, 2, 0.125Close can have:

• a terminating decimalTerminating decimalA decimal that ends after a finite number of digits.Example: 0.325Close
• or a repeating decimalRepeating decimalA decimal whose digits repeat in a fixed pattern forever.Example: 0.3333...Close

It will never become a non-repeating endless decimal. That kind of decimal is irrational.

examples first

terminating decimalTerminating decimalA decimal that ends after a finite number of digits.Example: 0.325Close

13/40 = 0.325

another terminating decimalTerminating decimalA decimal that ends after a finite number of digits.Example: 0.325Close

29/125 = 0.232

non-terminating repeating decimalRepeating decimalA decimal whose digits repeat in a fixed pattern forever.Example: 0.3333...Close

7/12 = 0.58333...

the main rule

First, write the fraction in lowest formLowest formA fraction whose numerator and denominator have no common factor except 1.Example: 2/5 is the lowest form of 6/15.Close. Then look only at the denominator.

• if the denominator has only prime factorsPrime factorsFactors that are themselves prime numbers.Example: Prime factors of 12 are 2, 2, 3.Close 2 and/or 5, the decimal terminates
• if the denominator has any prime factorPrime factorA factor that is also a prime number.Example: 2 and 3 are prime factors of 12.Close other than 2 or 5, the decimal repeats

why 2 and 5 matter

Our decimal system is based on 10. And:

10 = 2 × 5

So denominators built from 2 and 5 fit nicely into decimal form.

do not skip lowest formLowest formA fraction whose numerator and denominator have no common factor except 1.Example: 2/5 is the lowest form of 6/15.Close

This is very important. Always reduce the fraction first.

Example:

6/15

If you look directly at denominator 15, you may get confused. First reduce:

6/15 = 2/5

Now denominator is only 5. So the decimal terminates. In fact,

2/5 = 0.4

solved example 1

Check whether 13/40 terminates.

Factorise the denominator:

40 = 2³ × 5

Only 2 and 5 are present. So the decimal terminates.

Answer:

13/40 = 0.325

solved example 2

Check whether 7/12 terminates.

Factorise the denominator:

12 = 2² × 3

A prime factorPrime factorA factor that is also a prime number.Example: 2 and 3 are prime factors of 12.Close 3 is present. So the decimal does not terminate. It repeats.

Answer:

7/12 = 0.58333...

solved example 3

Check whether 11/24 terminates.

Factorise the denominator:

24 = 2³ × 3

Since 3 is present, the decimal will not terminate. It will repeat.

solved example 4

Check whether 9/20 terminates.

Factorise denominator:

20 = 2² × 5

Only 2 and 5 are present. So the decimal terminates.

In fact,

9/20 = 0.45

memory trick

2 and 5 are friends of base 10.

board tip

If the question asks about decimal type, do not start long division first. Prime factorise the denominator. That is cleaner and faster.

compare these four cases

• 3/8: denominator 8 = 2³, so terminating
• 7/25: denominator 25 = 5², so terminating
• 5/6: denominator 6 = 2 × 3, so repeating
• 11/15: denominator 15 = 3 × 5, so repeating

remember this line

If the denominator has only 2 and 5, the decimal ends. If not, it repeats.