Question

Express the quantity without using absolute value

$\left|1-\sqrt{2}\right|$

Short answer

The quantity of$\left|1-\sqrt{2}\right|\mathrm{is}\sqrt{2}-1$.

Step-by-step solution

Step 1. Apply the concept of absolute value.

If a is a real number, then the absolute value of a is given by:

$\left|a\right|=\left\{\begin{array}{l}a\text{;if}a\ge 0\\ -a\text{;if}a<0\end{array}\right.$

Step 2. Analyze the given value.

Given$\left|1-\sqrt{2}\right|$

Here the value inside modulus is $\left|1-\sqrt{2}\right|$.

$\begin{array}{l}\sqrt{2}\approx 1.41>1\\ \Rightarrow 1-\sqrt{2}<0\end{array}$

Step 3. Write without using absolute value.

$\begin{array}{l}\left|1-\sqrt{2}\right|=-\left(1-\sqrt{2}\right)\mathrm{.................}[1-\sqrt{2}<0]\\ \Rightarrow \left|1-\sqrt{2}\right|=-\left(-\left[\sqrt{2}-1\right]\right)\\ \Rightarrow \left|1-\sqrt{2}\right|=\sqrt{2}-1\end{array}$