In this explainer, we will learn how to evaluate, simplify, and multiply pure imaginary numbers and solve equations over the set of pure imaginary numbers.
Gaining confidence in working with imaginary numbers will enable us to acquire the necessary skills to work effectively with complex numbers more generally.
Historically, the introduction of complex numbers was primarily associated with the idea of solving equations. In particular, in the 16th century, mathematicians were working to find algebraic solutions to the cubic equation. Intriguingly, the equations mathematicians were trying to solve often had purely real solutions. However, the methods required to solve them ended up with the need to evaluate the square roots of negative numbers. In particular, Tartaglia’s method for solving cubics of the form often led to the need to evaluate the square root of negative numbers even when the solutions were all real. For example, applying his method to the equation results in the following solution:
However, by inspection, we can see the equation has three real solutions: 0, 1, and . At the time, many people would have dismissed an expression like this as nonsensical. However, the mathematician Rafael Bombelli saw the usefulness of working with the square roots of negative numbers and, as a result, today we credit him as the first person to formalize their properties.
Let us recap the definition of imaginary numbers.
Definition: Imaginary Numbers
An imaginary number is a number of the form , where is a real number and is defined by
We note that and are the two solutions to the equation , which has no real solutions. Alternatively, we can also think of them as the square roots of .
It is important to be aware that imaginary numbers are a subset of a larger set known as complex numbers. Let us also recall the definition of complex numbers.
Definition: Complex Numbers
A complex number is a number of the form , where and are real numbers.
The real part of a complex number is defined to be , and the imaginary part of is defined to be . These two parts can be written, respectively, as
In this explainer, we will only consider pure imaginary numbers, which means complex numbers without a real component.
As mentioned, the introduction of imaginary numbers enables us to solve equations which have no real solutions. We begin by looking at a simple example of this.
Example 1: Solving Equations Using Imaginary Numbers
Solve the equation .
Answer
We begin with isolating by dividing both sides of the equation by two:
Taking the square root of both sides, we get remembering that in taking the square root we need to consider both the positive and the negative solutions. We can rewrite
Hence,
Substituting this back into the equation, we can check our answer. Here we check our answer for :
Since , we can rewrite this as as required.
By applying the familiar rules of arithmetic and algebra, we can learn to easily work with imaginary and complex numbers. In the next few examples, we will apply many of the rules we are confident using with real numbers to solve problems involving purely imaginary numbers.
Example 2: Working with Positive Powers of Imaginary Numbers
Simplify .
Answer
In solving problems like this, it can be helpful to consider each part individually. Beginning with the first part, we can apply the properties of indices, or the commutativity of multiplication, to rewrite it as follows:
Recalling that, by definition, , this reduces to
In a similar way, we consider the second term. By applying the rules of indices or the commutativity of multiplication, we can rewrite
We can easily evaluate . However, how do we deal with ? For some students, the first time they see raised to a power other than two, they are unsure how to handle it. However, we already have all the tools we need to deal with this: if we simply rewrite , we can use our knowledge that to discover that . Therefore, the second part reduces to
Finally, we can multiply the two parts together, which gives us our final answer of
In the previous example, we found that . This raises the question of what happens as we raise to higher powers. We already know that , , and . So what is ? We can calculate this in an analogous way to how we calculated by noting that
Raising this equation to the th power, we get
Multiplying this equation by the powers of from one to three, we get the following identities:
We can also express these identities as the following cycle.
Example 3: Powers of 𝑖
Simplify .
Answer
Firstly, we want to simplify . To do this, we express 45 in the form , where is an integer between 0 and 3. This will enable us to apply our knowledge of the powers of to eliminate the exponent from the expression.
Since , we can express . Now we can apply our knowledge of the powers of , in particular that , to simplify the expression to get . Alternatively, we could have approached this as follows:
By applying the rules of exponents, we can express this as
Since , we have
Hence
At this point, a student new to complex numbers might get a little stuck. However, we should not forget the algebraic and arithmetic tools we already know. Recall that when we want to rationalize the denominator in an expression like , we can multiply both the numerator and the denominator by , which gives us . In a similar way, thinking of as , we can apply the same technique, which results in the following calculation:
Since , we have
Hence, we have
In the previous example, we learnt how to deal with . Given that we can express in index form as , and this is equal to , we might start to wonder whether negative powers of also follow a similar cycle and the same rules we found for positive powers. As it turns out, they do and we have the following fact.
Theorem: Integer Powers of the Imaginary Number 𝑖
For all integers , the following rules are true:
We can express this in a cycle as shown.
We can now look at an example of applying these rules.
Example 4: Simplifying Integer Powers of 𝑖
Given that is an integer, simplify .
Answer
To apply the rules of the powers of , we need to first express in the form , where is an integer between 0 and 3. We note that and . Hence, which we can rewrite as
This is nearly in the correct form. We wanted to ensure that was between 0 and positive 3; however, here it is . We can easily solve this by expressing . Substituting this back in gives
Now we can apply the rules for integer powers of , in particular , to get
We finish by looking at one last example of arithmetic with complex numbers where we need to be careful when trying to apply the familiar rule of arithmetic.
Example 5: Arithmetic with Imaginary Numbers
Simplify .
Answer
We need to be careful here not to fall into the trap of assuming that is true for all numbers. It is certainly true for positive real numbers. However, it is not true for negative numbers as we will see. To avoid this trap, we need to first express these square roots in terms of as follows: and
Now we can multiply them together and simplify:
Expressing 60 as a product of prime factors, , we can see that . Substituting this in and using the fact that , we find
Had we tried to use the rule , we would have incorrectly concluded that the answer is .
This last example highlights the fact that although almost all of the rules of algebra and arithmetic can be applied to complex numbers, we need to be careful when we are dealing with fractional powers and roots to first express the square root of a negative number in terms of before trying to manipulate it.
Key Points
- We can solve many problems involving imaginary and complex numbers applying the familiar rules of arithmetic and algebra.
- We need to be careful when working with noninteger powers of any number which is not both positive and purely real; some of the rules we are familiar with do not apply to negative or complex numbers in general. For example, is not true for arbitrary complex numbers. In particular, it is not true for two negative numbers.
- The integer powers of form a cycle:
Using these rules, we can simplify some calculations involving complex numbers.
