3.1: Complex Numbers
Expressing Square Roots of Negative Numbers as Multiples of i
We know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number i is defined as the square root of negative 1.
\[\sqrt{-1}=i\]
So, using properties of radicals,
\[i^2=(\sqrt{-1})^2=−1\]
We can write the square root of any negative number as a multiple of i. Consider the square root of –25.
\[\begin{align} \sqrt{-25}&=\sqrt{25 {\cdot} (-1)}\\ &=\sqrt{25}\sqrt{-1} \\ &= 5i \end{align}\]
We use 5 i and not −5 i because the principal root of 25 is the positive root.
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written \(a+bi\) where \(a\) is the real part and \(bi\) is the imaginary part. For example, \(5+2i\) is a complex number. So, too, is \(3+4\sqrt{3}i\).
Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.
A complex number is a number of the form \(a+bi\) where
- \(a\) is the real part of the complex number.
- \(bi\) is the imaginary part of the complex number.
If \(b=0\), then \(a+bi\) is a real number. If \(a=0\) and \(b\) is not equal to 0, the complex number is called an imaginary number . An imaginary number is an even root of a negative number.
Plotting a Complex Number on the Complex Plane
We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs \((a,b)\), where \(a\) represents the coordinate for the horizontal axis and \(b\) represents the coordinate for the vertical axis.
Let’s consider the number \(−2+3i\). The real part of the complex number is−2 and the imaginary part is \(3i\). We plot the ordered pair \((−2,3)\) to represent the complex number \(−2+3i\) as shown in Figure \(\PageIndex{2}\)
In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis as shown in Figure \(\PageIndex{3}\).
Adding and Subtracting Complex Numbers
Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.
Adding complex numbers:
\[(a+bi)+(c+di)=(a+c)+(b+d)i\]
Subtracting complex numbers:
\[(a+bi)−(c+di)=(a−c)+(b−d)i\]
Dividing Complex Numbers
Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of \(a+bi\) is \(a−bi\).
Note that complex conjugates have a reciprocal relationship: The complex conjugate of \(a+bi\) is \(a−bi\), and the complex conjugate of \(a−bi\) is \(a+bi\). Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.
Suppose we want to divide \(c+di\) by \(a+bi\), where neither a nor \(b\) equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.
\[\dfrac{c+di}{a+bi} \, \text{ where $a{\neq}0$ and $b{\neq}0$} \nonumber\]
Multiply the numerator and denominator by the complex conjugate of the denominator.
\[\dfrac{(c+di)}{(a+bi)}{\cdot}\dfrac{(a−bi)}{(a−bi)}=\dfrac{(c+di)(a−bi)}{(a+bi)(a−bi)} \nonumber\]
Apply the distributive property.
\[=\dfrac{ca−cbi+adi−bdi^2}{a^2−abi+abi−b^2i^2} \nonumber\]
Simplify, remembering that \(i^2=−1\).
\[=\dfrac{ca−cbi+adi−bd(−1)}{a^2−abi+abi−b^2(−1)} \\ =\dfrac{(ca+bd)+(ad−cb)i}{a^2+b^2} \nonumber\]
The complex conjugate of a complex number \(a+bi\) is \(a−bi\). It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.
- When a complex number is multiplied by its complex conjugate, the result is a real number.
- When a complex number is added to its complex conjugate, the result is a real number.