×

# |x|

We say $$\sqrt{4}$$ = $$\pm2$$ but it is also true that $$\sqrt{x^2}$$=|x|. How is it possible that both of the above statements are true simultaneously?

Note by Samarth Agarwal
2 years, 7 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

One must take note that yes, for every positive real number, there are two square roots with the same magnitude but different polarity (same numerical value but one is positive and one is negative).

The square root sign (radical sign) refers to the positive square root only. So actually \sqrt{4} = 2, and in fact, ±\sqrt{4} = ±2.

This also applies to \sqrt{[x]^[2]} . Since \sqrt{[x]^[2]} cannot be negative, it has to be |x|. Even if the value of x is negative, \sqrt{[x]^[2]} = |x|.

This is where people have misconceptions. Since x ≠ |x| for negative x, \sqrt{[x]^[2]} ≠ x.

- 2 years, 7 months ago

Thanks for the solution.......You may check the formatting guide provided by Brilliant or may read about LaTeX on Wikipedia

- 2 years, 7 months ago

Oops, I'm not used to LaTeX in Brilliant. Help pl0x...

- 2 years, 7 months ago