NCERT Solutions: Real Numbers (Exercise 1.3)

# Real Numbers (Exercise 1.3) NCERT Solutions - Mathematics (Maths) Class 10

Q.1. Prove that √5 is irrational.
Solutions: Let us assume, that 5 is rational number.
i.e. 5 = x/y (where, x and y are co-primes)
y5= x
Squaring both the sides, we get,
(y5)2 = x2
⇒ 5y2 = x2……………………………….. (1)
Thus, x2 is divisible by 5, so x is also divisible by 5.
Let us say, x = 5k, for some value of k and substituting the value of x in equation (1), we get,
5y2 = (5k)2
⇒ y2 = 5k2
is divisible by 5 it means y is divisible by 5.
Clearly, x and y are not co-primes. Thus, our assumption about 5 is rational is incorrect.
Hence, 5 is an irrational number.

Q.2. Prove that 3 + 2√5 + is irrational.
Solutions: Let us assume 3 + 2√5 is rational.
Then we can find co-prime x and y (y ≠ 0) such that 3 + 2√5 = x/y
Rearranging, we get,

Since, x and y are integers, thus,
is a rational number.
Therefore, √5 is also a rational number. But this contradicts the fact that √5 is irrational.
So, we conclude that 3 + 2√5 is irrational.

Q.3. Prove that the following are irrationals:
(i) 1/√2
(ii) 7√5
(iii) 6 + 2
Solutions:
(i) 1/2
Let us assume 1/√2 is rational.
Then we can find co-prime x and y (y ≠ 0) such that 1/√2 = x/y
Rearranging, we get,
√2 = y/x
Since, x and y are integers, thus, √2 is a rational number, which contradicts the fact that √2 is irrational.
Hence, we can conclude that 1/√2 is irrational.

(ii) 75
Let us assume 7√5 is a rational number.
Then we can find co-prime a and b (b ≠ 0) such that 7√5 = x/y
Rearranging, we get,
√5 = x/7y
Since, x and y are integers, thus, √5 is a rational number, which contradicts the fact that √5 is irrational.
Hence, we can conclude that 7√5 is irrational.

(iii) 6 +2
Let us assume 6 +√2 is a rational number.
Then we can find co-primes x and y (y ≠ 0) such that 6 +√2 = x/y⋅
Rearranging, we get,
√2 = (x/y) – 6
Since, x and y are integers, thus (x/y) – 6 is a rational number and therefore, √2 is rational. This contradicts the fact that √2 is an irrational number.
Hence, we can conclude that 6 +√2 is irrational.

The document Real Numbers (Exercise 1.3) NCERT Solutions | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## Mathematics (Maths) Class 10

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## FAQs on Real Numbers (Exercise 1.3) NCERT Solutions - Mathematics (Maths) Class 10

 1. What are real numbers?
Ans. Real numbers are a set of numbers that include all rational and irrational numbers. They can be positive, negative, or zero, and can be represented on the number line.
 2. What is the difference between rational and irrational numbers?
Ans. Rational numbers are those that can be expressed as a fraction, where the numerator and denominator are both integers. Irrational numbers, on the other hand, cannot be expressed as a fraction and have non-repeating, non-terminating decimal representations.
 3. How can we determine if a number is rational or irrational?
Ans. To determine if a number is rational or irrational, we can check its decimal representation. If the decimal representation terminates or repeats, then the number is rational. If the decimal representation neither terminates nor repeats, then the number is irrational.
 4. Can real numbers be negative?
Ans. Yes, real numbers can be negative. Real numbers include both positive and negative numbers, in addition to zero.
 5. Are all whole numbers also real numbers?
Ans. Yes, all whole numbers are real numbers. Real numbers include all rational and irrational numbers, and whole numbers are a subset of rational numbers.

## Mathematics (Maths) Class 10

115 videos|478 docs|129 tests

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