Operations with Complex/Imaginary Numbers
So far in your math education, you have dealt solely with real numbers. But there is a system of numbers that are called imaginary numbers that result when you take the square root of a negative number. We use the letter \(i\) to represent imaginary numbers.
So far in your math education, you have dealt solely with real numbers. But there is a system of numbers that are called imaginary numbers that result when you take the square root of a negative number. We use the letter \(i\) to represent imaginary numbers.
Definition of an imaginary number:
\(i=\sqrt{-1}\) therefore
\(i^2=\left(\sqrt{-1}\right)^2=-1\)
\(i=\sqrt{-1}\) therefore
\(i^2=\left(\sqrt{-1}\right)^2=-1\)
Using this, we can simplify square roots of negative numbers, which is useful when finding the non-real solutions to a quadratic equation (the quadratic functions that have no x-intercepts will still have two solutions but those solutions are not real numbers).
Simplifying the square root of a negative radicand is similar to simplifying the square root of a positive radicand. You will still use the product property, but you will factor out \(-1\) from the radicand before proceeding.
Example 1: Simplify \(\sqrt{-81}\).
\(\begin{align}&\sqrt{-81}\ & \ &\text{1) Rewrite original problem.}\\&\sqrt{-1\cdot81}\ & \ &\text{2) Factor -1 from the radicand.}\\&\sqrt{-1}\cdot\sqrt{81}\ & \ &\text{3) Apply the product property.}\\&i\cdot9\ & \ &\text{4)} \sqrt{-1}=i\ \text{and simplify the}\ \sqrt{81}.\\&9i\ & \ &\text{5) Rewrite.}\end{align}\)
Example 2: Simplify \(3i\sqrt{-27}\).
In addition to imaginary numbers, there is a number system called the complex numbers. Complex numbers are written in standard form as \(a+bi\ \text {or}\ \ a-bi\), where \(a\) is the real part and \(bi\) is the imaginary part. All real numbers are actually complex numbers with an imaginary part of \(0i\).
We can add, subtract, multiply and divide complex numbers.
To add or subtract complex numbers, you are combining like terms. Add/subtract the real parts and add/subtract the imaginary parts.
Example 3: Write the expression \(\left(12-3i\right)+\left(4+7i\right)\) as a complex number in standard form.
\(\begin{align}\left(12-3i\right)&+\left(4+7i\right)\ & \ &\text{1) Rewrite original problem.}\\12+4&-3i+7i\ & \ &\text{2) Write like terms next to each other.}\\16&+4i\ & \ &\text{3) Combine like terms.}\end{align}\)
Example 4: Write the expression \(3\left(15+6i\right)-7\left(2-3i\right)\) as a complex number in standard form.
You can multiply complex numbers by using the distributive property. Just remember to substitute \(-1\) any time you have \(i^2\).
Example 5: Write the expression \(3i\left(5-7i\right)\) as a complex number in standard form.
Example 5: Write the expression \(3i\left(5-7i\right)\) as a complex number in standard form.
\(\begin{align}&3i\left(5-7i\right)\ & \ &\text{1) Rewrite original problem.}\\&15i-21i^2\ & \ &\text{2) Distribute.}\\&15i-21\left(-1\right)\ & \ &\text{3) Substitute}\ i^2\ \text{with} -1.\\&15i+21\ & \ &\text{4) Multiply -21 and -1.}\\&21+15i\ & \ &\text{5) Write in standard form.}\end{align}\)
Example 6: Write the expression \(\left(-2+6i\right)\left(4+5i\right)\) as a complex number in standard form.
To divide complex numbers, multiply the numerator and the denominator of the fraction by the conjugate of the denominator in order to have a rational number in the denominator. (Remember, the conjugate will just change the sign between the real part and the imaginary part). The goal is to have a rational number in the denominator. Always replace \(i^2\) with \(-1\). Also, to write in standard form, split the expression into two fractions, then reduce each fraction.
Example 7: Write the expression \(\large{\frac{7}{\left(4+5i\right)}}\) as a complex number in standard form.
\(\begin{align}&\frac{7}{\left(4+5i\right)}\ & \ &\text{1) Rewrite original problem.}\\&\frac{7\left(4-5i\right)}{\left(4+5i\right)\left(4-5i\right)}\ & \ &\text{2) Multiply numerator and denominator by the conjugate.}\\&\frac{28-35i}{16-20i+20i-25i^2}\ & \ &\text{3) Distribute.}\\&\frac{28-35i}{16-25\left(-1\right)}\ & \ &\text{4) Substitute}\ i^2\ \text{with -1}.\\&\frac{28-35i}{16+25}\ & \ &\text{5) Multiply -25 and -1.}\\&\frac{28-35i}{41}\ & \ &\text{6) Combine like terms in denominator.}\\&\frac{28}{41}-\frac{35}{41}i\ & \ &\text{7) Split into two fractions and write in standard form.}\end{align}\)
Example 8: Write the expression \(\large{\frac{\left(5+8i\right)}{\left(3-2i\right)}}\) as a complex number in standard form.