Name: Surds: Introduction
Uploaded: 2019-07-19
Duration: 4 min 21 s
Description: We know that irrational numbers are written in the form of neither terminating nor recurring decimals. Let's learn to write another form of irrational number...

Question

If

a

and

b

are two unequal rational numbers, show that

\frac{a + b}{2}

is a rational number lying between

a

and

b

.

Accepted Answer

Given that $a$ and $b$ are two unequal rational numbers.

Let $a$ is less than $b$ .

So, we get, $a < b$

$\Rightarrow a + b < b + b$ [Add $b$ to both sides]

$\Rightarrow a + b < 2 b$

$\Rightarrow \frac{a + b}{2} < \frac{2 b}{2}$ [Divide both sides by $2$ ]

$\Rightarrow \frac{a + b}{2} < b$ -----(i)

Again, $a < b$

$\Rightarrow a + a < a + b$ [Add $a$ to both sides]

$\Rightarrow 2 a < a + b$

$\Rightarrow \frac{2 a}{2} < \frac{a + b}{2}$ [Divide both sides by $2$ ]

$\Rightarrow a < \frac{a + b}{2}$ -----(ii)

Combining (i) and (ii), we get, $a < \frac{a + b}{2} < b$ .

Now, since $a$ and $b$ are rational numbers, $a + b$ is also a rational number [The sum of two rational numbers is always a rational number.] Also, $2$ is an integer, which is also a rational number. Hence, $\frac{a + b}{2}$ is a rational number [The quotient of two rational numbers is a rational number]

Therefore, $\frac{a + b}{2}$ is a rational number lying between $a$ and $b$ . [Proved]

If $a$ and $b$ are two unequal rational numbers, show that $\frac{a + b}{2}$ is a rational number lying between $a$ and $b$ .

Important Questions on Number System

Learn and Prepare for any exam you want !

If a and b are two unequal rational numbers, show that a+b2 is a rational number lying between a and b.

Important Questions on Number System

Learn and Prepare for any exam you want !

If $a$ and $b$ are two unequal rational numbers, show that $\frac{a + b}{2}$ is a rational number lying between $a$ and $b$ .