Question

How do you find the absolute value of $10-7i$?

Answer 1

The numbers of the form $a+bi$ are imaginary. Here, a and b are real numbers. The absolute value is defined as the distance in the complex plane from the complex number to the 0. The absolute value of $a+bi$ is found as, $\sqrt{{{a}^{2}}+{{b}^{2}}}$. By substituting the value of a and b for a complex number, we can find the absolute value of the complex number.

Complete step by step solution:
We are asked to find the absolute value of the complex number $10-7i$. We know that the absolute value is defined as the distance in the complex plane from the complex number to 0. The absolute value of $a+bi$ is found as, $\sqrt{{{a}^{2}}+{{b}^{2}}}$. Comparing the given complex number with $a+bi$, we get $a=10\And b=-7$. Substituting these values in the absolute value expression, we get the absolute value of $10-7i$ as $\sqrt{{{\left( 10 \right)}^{2}}+{{\left( -7 \right)}^{2}}}$.
We know that the square of 10 is 100, and the square of $-7$ is 49. Using these values, we get $\sqrt{{{\left( 10 \right)}^{2}}+{{\left( -7 \right)}^{2}}}=\sqrt{100+49}$. Adding 100 and 49, we get 149.
$\Rightarrow \sqrt{149}$
As we know that 149 is a prime number, the above square root expression can not be simplified further.

Note: To solve the problems on complex numbers, one should know the different properties of the complex number. Here, $i$ is an imaginary number which equals $\sqrt{-1}$. Some of the important properties of a complex number are as follows,
$\left( a+bi \right)\left( a-bi \right)={{a}^{2}}+{{b}^{2}}$
$\overline{a+bi}=a-bi$, this is called a conjugate of a complex number.