Expressing Square Roots of Negative Numbers as Multiples of [latex]\,i[/latex]

We know how to find the square root of any positive real number. Similarly, we can find the square root of any negative number. The difference is that the root is not real. If the number inside the radicand is negative, the root is said to be an imaginary number. The imaginary number[latex]\,i\,[/latex]is defined as the square root of[latex]\,-1.[/latex]

So, using the properties of radicals,

[latex]{i}^{2}={\left(\sqrt{-1}\right)}^{2}=-1[/latex]

We can write the square root of any negative number as a multiple of[latex]\,i.\,[/latex]Consider the square root of [latex]\,-49.[/latex]

[latex]\begin{array}{ccc}\hfill \sqrt{-49}& =& \hfill \sqrt{49\cdot \left(-1\right)}\\ & =& \sqrt{49}\sqrt{-1}\hfill \\ & =& 7i\hfill \end{array}[/latex]

We use[latex]\,7i\,[/latex]and not[latex]\,-7i\,[/latex]because the principal root of[latex]\,49\,[/latex]is the positive root.

A complex number is the sum of real and imaginary numbers. A complex number is expressed in standard form when written[latex]\,a+bi\,[/latex]where[latex]\,a\,[/latex]is the real part and[latex]\,b\,[/latex]is the imaginary part. For example,[latex]\,5+2i\,[/latex]is a complex number. So, too, is[latex]\,3+4i\sqrt{3}.[/latex]

Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number, and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.

Plotting a Complex Number on the Complex Plane

We cannot plot complex numbers on a number line as 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[latex]\,\left(a,b\right),[/latex] where [latex]\,a\,[/latex] represents the coordinate for the horizontal axis and[latex]\,b\,[/latex] represents the coordinate for the vertical axis.

Let’s consider the number[latex]\,-2+3i.\,[/latex]The real part of the complex number is[latex]\,-2\,[/latex]and the imaginary part is 3. We plot the ordered pair[latex]\,\left(-2,3\right)\,[/latex]to represent the complex number[latex]\,-2+3i,[/latex] as shown in Figure[latex]\, 2 \,[/latex].

Complex Plane

In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis, as shown in Figure[latex]\, 3 \,[/latex].

Plotting a Complex Number on the Complex Plane

Plot the complex number[latex]\,3-4i\,[/latex]on the complex plane.

Show Solution

The real part of the complex number is[latex]\,3,[/latex] and the imaginary part is –4. We plot the ordered pair[latex]\,\left(3,-4\right)\,[/latex] as shown in Figure[latex]\, 4 \,[/latex].

Adding and Subtracting Complex Numbers

We can perform arithmetic operations on complex numbers just as with real numbers. To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts.

Access the online resource below for additional instruction and practice with adding and subtracting complex numbers.

Multiplying Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work separately with the real and imaginary parts.

Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or, more specifically, the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term,[latex]\,{i}^{2},[/latex] it equals[latex]\,-1.[/latex]

Access the online resource below for additional instruction and practice with multiplying complex numbers.

Dividing Complex Numbers

Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in the standard form[latex]\,a+bi.\,[/latex]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[latex]\,a+bi\,[/latex] is [latex]\,a-bi.\,[/latex]For example, the product of[latex]\,a+bi\,[/latex] and[latex]\,a-bi\,[/latex] is

The result is a real number.

Note that complex conjugates have an opposite relationship: The complex conjugate of[latex]\,a+bi\,[/latex] is [latex]\,a-bi[/latex], and the complex conjugate of [latex]\,a-bi\,[/latex] is [latex]\,a+bi.\,[/latex]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[latex]\,c+di\,[/latex] by [latex]\,a+bi,[/latex] where neither [latex]\,a\,[/latex] nor [latex]\,b\,[/latex] equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator and multiply.

[latex]\frac{c+di}{a+bi}\text{ where }a\ne 0\text{ and }b\ne 0[/latex]

Multiply the numerator and denominator by the complex conjugate of the denominator.

Apply the distributive property.

Simplify, remembering that[latex]\,{i}^{2}=-1.[/latex]

Access the online resource below for additional instruction and practice with dividing complex numbers.

Simplifying Powers of i

The powers of[latex]\,i\,[/latex] are cyclic. Let’s look at what happens when we raise[latex]\,i\,[/latex] to increasing powers.

We can see that when we get to the fifth power of[latex]\,i,[/latex] it is equal to the first power. As we continue to multiply[latex]\,i\,[/latex] by increasing powers, we will see a cycle of four. Let’s examine the next four powers of[latex]\,i.[/latex]

The cycle is repeated continuously:[latex]\,i,-1,-i,1,[/latex] every four powers.