Advertisements
Advertisements
प्रश्न
Prove that the following number is irrational: √3 + √2
उत्तर
√3 + √2
Let √3 + √2 be a rational number.
⇒ √3 + √2 = x
Squaring on both the sides, we get
( √3 + √2 )2 = x2
⇒ 3 + 2 + 2 x √3 x √2 = x2
⇒ x2 - 5 = 2√6
⇒ √6 = `[ x^2 - 5 ]/2`
Here, x is a rational number.
⇒ x2 is a rational number.
⇒ x2 - 5 is a rational number.
⇒ `[ x^2 - 5]/2` is also a rational number.
But √6 is an irrational number.
⇒ `[ x^2 - 5]/2` is also a irrational number.
⇒ x2 - 5 is an irrational number.
⇒ x2 is an irrational number.
⇒ x is an irrational number.
But we have assume that x is a rational number.
∴ we arrive at a contradiction.
So, our assumption that √3 + √2 is a rational number is wrong.
∴ √3 + √2 is an irrational number.
APPEARS IN
संबंधित प्रश्न
In the following equation, find which variables x, y, z etc. represent rational or irrational number:
y2 = 9
Classify the numbers π as rational or irrational:
Prove that of the numbers `3 sqrt(7)` is irrational:
State, whether the following numbers is rational or not : ( 2 + √2 )2
Check whether the square of the following is rational or irrational:
`(3sqrt(2))/(2)`
Classify the following number as rational or irrational with justification:
`sqrt(12)/sqrt(75)`
Find whether the variable u represents a rational or an irrational number:
`u^2 = 17/4`
Insert a rational number and an irrational number between the following:
0.15 and 0.16
Insert a rational number and an irrational number between the following:
3.623623 and 0.484848
Classroom activity (Constructing the ‘square root spiral’): Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1 P2 perpendicular to OP1 of unit length. Now draw a line segment P2 P3 perpendicular to OP2. Then draw a line segment P3 P4 perpendicular to OP3. Continuing in this manner, you can get the line segment Pn–1Pn by drawing a line segment of unit length perpendicular to OPn–1. In this manner, you will have created the points P2, P3,...., Pn,.... ., and joined them to create a beautiful spiral depicting `sqrt2, sqrt3, sqrt4,` ...
