Question

How do you find the absolute value of \[10-7i\]?

Answer 1

Hint: 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.