Prove that √2 + √3 is irrational

Solution :

Consider that √2 + √3 is rational.

Assume √2 + √3 = a , where a is rational.

So, √2 = a - √ 3

By squaring on both sides,

2 = a² + 3 - 2a√3

√3 = a² + 1/2a, is a contradiction as the RHS is a rational number while √3 is irrational

Therefore, √2 + √3 is irrational.

✦ Try This: Prove that √2 is irrational

Consider that √2 is a rational number.

It can be expressed in the form p/q where p, q are co-prime integers and q≠0

√2 = p/q.

p and q are coprime numbers and q ≠ 0

Solving,

√2 = p/q

By squaring both the sides,

2 = (p/q)²

2q² = p² --- (1)

p²/2 = q²

Here 2 divides p and

p is a multiple of 2.

p = 2m

p² = 4m² --- (2)

From equations (1) and (2), we get,

2q² = 4m²

q² = 2m²

q² is a multiple of 2

q is a multiple of 2

p, q have a common factor 2.

p/q is not a rational number.

Therefore, √2 is an irrational number

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1

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NCERT Exemplar Class 10 Maths Exercise 1.3 Sample Problem 3

Prove that √2 + √3 is irrational

Summary:

Irrational numbers are those real numbers that cannot be represented in the form of p/q. In other words, those real numbers that are not rational numbers are known as irrational numbers. √2 + √3 is irrational.

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☛ Related Questions:

• Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q
• Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m
• Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer . . . .