In this explainer, we will learn how to use de Moivreβs theorem to find the roots of unity and explore their properties.
In complex numbers, the roots of unity are complex numbers satisfying
We know that there is only one real-valued solution, , to this equation if is an odd integer, since we can solve the equation by applying the root on both sides of the equation. If is an even integer, there are two real-valued solutions, and , to this equation.
On the other hand, for , there are other solutions of this equation which are not real numbers. In particular, for , we know that the solutions of these equation are
We can begin to notice a pattern here. Clearly, there is only one solution to the equation , which is . There are two real-valued solutions and for the equation . Moreover, we observed that there are three distinct solutions to . At this point, we can conjecture that there are distinct solutions to the equation .
We can prove this conjecture by applying de Moivreβs theorem for roots.
Theorem: De Moivreβs Theorem for Roots
For a complex number , the roots of are given by for .
In the first example, we will apply de Moivreβs theorem to compute the roots of unity in polar form.
Example 1: The πth Roots of Unity
Write a general form for the roots of , giving your answer in polar form.
Answer
Recall that, by de Moivreβs theorem for roots, the roots of a complex number are given by
Hence, to take the root of the right-hand side of the equation, which is 1, we can begin by expressing 1 in polar form. We know that the modulus of 1 is equal to 1, and since 1 is on the positive real axis of an Argand diagram, we also know that the argument of 1 is equal to 0 radians. We can write the polar form of 1 by using and :
Applying de Moivreβs theorem for roots, the roots of unity are given by for . Hence, the roots of in polar form are
In the previous example, we found the polar form of the roots of unity by applying de Moivreβs theorem. In particular, we have proved our conjecture that states that there are distinct complex solutions to the equation . In other words, there are distinct roots of unity.
Using the polar form of the roots of unity obtained in the previous example, we can also write the exponential form of the roots of unity. Recall that the exponential form of a complex number with modulus and argument is , which means the polar and exponential forms of the complex number are related by the identity
We summarize both the polar and exponential forms of the roots of unity below.
Definition: πth Roots of Unity in Polar and Exponential Form
The roots of unity in polar form are
The roots of unity in exponential form are
In particular, the root 1 is called the trivial root of unity.
The expressions for the roots of unity above give the moduli and arguments of the complex numbers. We can see that the moduli of all roots of unity are equal to 1, which means that they all lie on the unit circle in an Argand diagram. The trivial root of unity 1 lies at the intersection of the unit circle and the positive real line in an Argand diagram. The arguments of roots of unity increase in an arithmetic sequence increasing by radians. In an Argand diagram, this means that we can plot the roots of unity by starting with 1 and rotating counterclockwise on the unit circle by consecutively. If we connect consecutive roots of unity with line segments, we will obtain a regular polygon inscribed in the unit circle. Let us observe this pattern in the diagrams below for several different values of .
As expected, the roots of unity for form vertices of a regular -gon inscribed in the unit circle in an Argand diagram, with a vertex at the trivial root 1.
We note that the arguments of the roots of unity do not all lie in the standard range, which is radians. In particular, we note that the cube roots of unity are labeled in the Argand diagram above by 1, , and . The last cube root of unity has argument , which is outside this range. Since this argument is over the upper bound , we can obtain an equivalent argument by subtracting the full revolution of radians from this value:
We note that this equivalent argument lies in the standard range , so we can use this argument to write the third root of unity as .
In the next example, we will find the quintic (fifth) roots of unity so that the arguments lie in the standard range and compute their sum.
Example 2: The Sum of the πth Roots of Unity
- Find the quintic roots of unity.
- What is the value of their sum?
Answer
Part 1
We recall that the roots of unity are given in polar form as for . In this example, we want to find the quintic roots of unity, which are the same as the fifth roots of unity. Hence, we can write the quintic roots of unity by substituting and into this expression:
We recall that the argument of a complex number, by convention, should lie in the standard range . The last two quintic roots of unity have arguments and , which do not lie in this range. Since these arguments are over the upper bound , we can obtain equivalent arguments by subtracting the full revolution of radians from this value:
Using these arguments in the standard range, the last two quintic roots of unity can be written as and . Hence, the quintic roots of unity are
Part 2
In this part, we need to find the sum of the roots. We will show two different methods for this computation. The first method will require a calculator to evaluate the cosine ratios at nonspecial angles, and the second method will use an algebra trick and will not require a calculator. The second method is much simpler once we know the algebra trick.
Method 1
We need to compute
We can rearrange the sum to write
Recall the property of the exponential form of a complex number with respect to the complex conjugate: , for any real number . This tells us that the pair of complex numbers inside each of the parentheses above are complex conjugates. We know that, for any complex number , , where is the real part of . Hence, the sum of the roots can be written as
It now remains to find the real parts of the complex numbers and . We recall that a complex number in the exponential form can be expressed in the polar form by writing where is the modulus and is the argument of the complex number. For and , the moduli of both complex numbers are 1, which means for both cases. Also, we can see that for , and for . Hence,
This tells us that
Substituting these values, the sum of the roots is written as
Since neither nor is an angle in the unit circle, we cannot evaluate the cosine function at these angles without using a calculator. Using a calculator, we can compute
Substituting these values into the sum, we obtain
Method 2
Next, we will use an algebra trick to compute the sum. We begin by setting the sum equal to an unknown constant :
We multiply both sides of the equation above by . Then, using the property of the exponential form , we can write
We know that by subtracting from the argument of the first complex number. This means that
We can see that the right-hand side of the equation above is the same as the right side of equation (1). Hence, equating the left sides of these equations leads to
Rearranging this equation, we obtain
Since the complex number inside the parentheses on the left-hand side of the equation above is not equal to zero, we can divide both sides of this equation by this number to obtain
Hence, the sum of the quintic roots of unity is 0.
In the previous example, we found that the sum of the quintic roots of unity is equal to zero. In fact, this is a general property of the roots of unity, as we will see later.
In the next example, we will consider the reciprocal of an root of unity.
Example 3: Reciprocals of the πth Roots of Unity
Let be an root of unity.
- Which of the following is the correct relationship between
and ?
- Express in terms of positive powers of .
Answer
Part 1
In this example, we find where is an root of unity. Recall that de Moivreβs theorem for integer powers tells us that, given a complex number ,
Here, the integer is . Also, we know that the modulus of an root of unity is equal to 1. In other words, for some argument . Hence, applying de Moivreβs theorem for integer powers,
This is the complex number with modulus 1 and argument . Since the argument of this complex number has the opposite sign to the argument of , this means that they are located on opposite sides of the real axis in an Argand diagram.
As we can observe in the Argand diagram above, this means that the real parts of and are equal, while the imaginary parts of these two complex numbers have the opposite signs. In other words, they are complex conjugates of each other. Hence,
This is option D.
Part 2
In this part, we want to express for some positive integer . Since is an root of unity, we know that
We can multiply both sides of this equation by and use the rule of exponents to write
Since is a positive integer, we have
In the previous example, we noted that the reciprocal of an root of unity is equal to its complex conjugate. This means that the product of a root of unity with its conjugate is equal to 1.
Property: Conjugate of Roots of Unity
Let be an root of unity. Then
So far, we have considered properties of the roots of unity. Some roots of unity are shared for different values of . These are called common roots of unity. In the next example, we will discuss the roots of unity which are shared between different values of .
Example 4: Relationship between the πth Roots of Unity for Different Values of π
- Find the cube roots of unity.
- Find the solutions to .
- What is the relationship between the cube roots of unity and the roots of unity?
Answer
Part 1
Recall that the roots of unity in polar form are given as
We can find the cubit roots of unity by substituting , which means that we need to substitute . This leads to
Finding an equivalent expression for the last cube root of unity so that its argument lies in the standard range , we find that the three cube roots of unity are
Part 2
In a similar way, we can find the roots of unity by substituting and , which leads to
Hence, expressing them with arguments in the principal range, we have
Part 3
Comparing the cube roots of unity to the roots of unity, we find that all the cube roots of unity are also roots of unity.
In the previous example, we found that all cube roots of unity are also sixth roots of unity. We can also observe this fact from a different perspective. A cube root of unity is a complex number satisfying , while a sixth root of unity is a complex solution to . If a complex number satisfies , then we can square both sides of the equation to write
Hence, if a complex number satisfies , then it also satisfies . In fact, we can make this statement more general. Say that we have two positive integers and such that divides ; that is, for some positive integer . If is an root of unity, it must satisfy . Applying the power to both sides of this equation,
This means that is also an root of unity. This tells us that, if and are positive integers such that divides , any root of unity is automatically an root of unity.
We can apply this statement to prove an even more general theorem. This time, say that does not divide , but . This tells us that divides both and . Hence, all of roots of unity are also and roots of unity. In fact, these are precisely the common roots of unity.
Theorem: Common Roots of Unity When gcd(π,π) β 1
Say that and are positive integers such that . Then, the common roots of unity shared by the and roots of unity are precisely the roots of unity.
Let us consider an example where we apply this theorem to solve a geometric problem.
Example 5: Number of Common Vertices in Inscribed Polygons
Two regular polygons are inscribed in the same circle where the first has 1βββ731 sides and the second has 4βββ039. If the two polygons have at least one vertex in common, how many vertices in total will coincide?
Answer
Recall that the roots of unity in an Argand diagram lie at the vertices of a regular -gon inscribed in a unit circle, where one of the vertices is at 1. We can place a coordinate system on this plane so that the given circle is the unit circle and the common vertex is the point . Then, we can think of the vertices as complex numbers in an Argand diagram.
In this setting, the vertices of the polygon with 1βββ731 vertices are the roots of unity, while the vertices with 4βββ039 sides are the roots of unity.
We recall that, for any positive integers and , the common roots of unity shared by the and roots of unity are precisely the roots of unity where . In this example, and , so we need to find . We note that the first number is divisible by 3 and the second number is divisible by 7. Using these prime factors, we can write the two numbers as
We see here that 577 is a common factor between these numbers, and it is the greatest factor since the other prime factors 3 and 7 are not shared. This gives us
Hence, the common vertices of these two polygons are represented in the Argand diagram as the roots of unity. We know that there are exactly 577 complex numbers that are the roots of unity.
Therefore, if the two polygons inscribed in the same circle have at least one vertex in common, they will have a total of 577 vertices in common.
In previous examples, we identified common roots of unity. We now turn our attention to another important definition for the roots of unity. Previously, we have listed the roots of unity for , which are labeled in Argand diagrams. Let us list them here, so that the arguments of the roots of unity lie in the standard range .
| The roots of unity | |
|---|---|
| 1 | |
| 2 | , |
| 3 | , , |
| 4 | , , , |
| 5 | , , , , |
| 6 | , , , , , |
| 7 | , , , , , , |
The red numbers in the table above are the roots of unity that have appeared as an root of unity for some . For instance, the root for is red because this root has appeared as a root of unity when and . This means that the green numbers are the roots of unity that appear for the first time in the list. Such roots of unity are called the primitive roots of unity.
Definition: Primitive πth Roots of Unity
A primitive root of unity is a complex number for which is the smallest positive integer satisfying . In other words, a primitive root of unity is an root of unity that is also not an root of unity for any .
From the table above, we have identified all the primitive roots of unity for :
| Primitive roots of unity | |
|---|---|
| 1 | 1 |
| 2 | |
| 3 | , |
| 4 | , |
| 5 | , , , |
| 6 | , |
| 7 | , , , , , |
We can observe that the first nontrivial root of unity for is always a primitive root of unity.
Property: Primitive πth Roots of Unity
Any complex number of the form for positive integer is a primitive root of unity.
This property also tells us that there is always a complex number corresponding to a primitive root of unity. However, we should keep in mind that a primitive root of unity for is not unique, and this property provides a way to obtain one particular primitive root of unity rather than all of them.
Let us prove this property by showing that there is no positive integer , less than , such that . Thus, consider an arbitrary integer satisfying . Then, using de Moivreβs theorem for integer powers, we can write
We also note that the condition implies when we divide each part of the inequality by . Then, the argument of the complex number above satisfies
Since the argument of 1 is equal to 0 or an integer multiple of , this tells us that for any integer satisfying . In other words, is the smallest positive integer for which , which proves the property.
One important property of a primitive root of unity is that it generates all roots of unity by taking consecutive powers.
Property: Primitive πth Roots of Unity
If is a primitive root of unity, then all roots of unity are given by
We can prove this property by showing the following two facts. Let be a primitive root of unity. Then,
- any integer power of is also an root of unity,
- the complex numbers are distinct.
If we know these two facts, then we know that are distinct roots of unity, which means that they are all roots of unity.
To show the first statement, let be an arbitrary integer. Then, using the rules of exponents,
We know that since is an root of unity. Hence, the right-hand side of the equation above is equal to 1. Since , is an root of unity. This proves the first statement needed for our proof.
To prove the second statement, we can begin by assuming that this statement is false. Then there are two distinct nonnegative integers and with and satisfying
Let us assume , since the order of these integers is arbitrary. Then, dividing both sides of the equation by and using the rule of exponents, we can write
This tells us that satisfies the equation , where is a positive integer by the assumption . Hence, is the root of unity where . However, this contradicts the fact that is a primitive root of unity. This means that it is not possible to have for any nonnegative integers and smaller than . In other words, the numbers are distinct numbers. This proves the desired property.
Let us now consider a few properties of primitive roots of unity.
Property: Sum of Powers of Primitive πth Roots of Unity
Let be a primitive root of unity for . Then
Since the terms in the left-hand side of the equation above are the roots of unity, this statement tells us that the sum of the roots of unity is equal to zero for any . Remember that we showed that the sum of the quintic roots of unity is equal to zero in example 2. A similar algebraic trick can be used here to prove this statement.
To prove this property, let us begin by setting the left-hand side of the equation equal to an unknown constant :
Multiplying both sides of this equation by ,
Since is an root of unity, we know that . This gives us
We can see that the right-hand sides of both equations are the same, which tells us
Since is a primitive root of unity, so we know that . This means that we can divide both sides of the equation above by to obtain
This proves the desired statement, which leads directly to the following result.
Corollary: Sum of πth Roots of Unity
The sum of the roots of unity for any is equal to zero.
Since the sum of complex numbers is geometrically equivalent to the addition of vectors, the property above gives us another aspect of the symmetry of the roots of unity. To make a physical analogy, if we think of each root of unity as a vector representing the magnitude and the direction of a force acting at the origin of an Argand diagram, this property tells us that the combined effect of all the forces at the origin is zero. In other words, the forces represented by the roots of unity form a perfect equilibrium.
Let us consider an example where we need to apply this property of a primitive root of unity.
Example 6: Summing Power of Primitive Roots of Unity
If is a primitive root of unity, which of the following expressions is equivalent to ?
- 1
Answer
We recall that a primitive root of unity satisfies
Since we are given that is a primitive root of unity, . Hence,
We can rearrange the equation to find an equivalent expression to the given one:
This is option D.
In our final example, we will apply properties of an root of unity to solve a problem.
Example 7: Applications of the πth Roots of Unity
For how many pairs of real numbers does the relation hold?
Answer
Recall the properties of the modulus of a complex number, which states
If we denote , we have ; hence the given equation is . Taking the modulus of this equation, we have
Then, the properties of the modulus stated above give
Hence, subtracting from both sides of the equation, we have
Therefore, either or . If , and are both zero, so we have one pair of real numbers that satisfy the equation in this case. Now we consider the case where . Multiplying both sides of our original equation by gives
Recall that, for any complex number , . Since we know that , we can rewrite the equation above as
We know that the solutions of these equations are called the roots of unity, and we also know there are exactly 2βββ021 unique roots of unity satisfying this equation. Therefore, in the case that , there are 2βββ021 pairs of which satisfy the equation.
In total, there are 2βββ022 pairs of real numbers for which the given relationship holds.
Let us finish by recapping a few important concepts from this explainer.
Key Points
- For any positive integer , there are exactly distinct complex numbers satisfying the equation . The complex solutions of this equation are called the roots of unity.
- The roots of unity in exponential form are given by
- The roots of unity for form vertices of a regular -gon inscribed in the unit circle in an Argand diagram, with a vertex at the trivial root 1.
- Say that and are positive integers such that . Then the common roots of unity shared by the and roots of unity are precisely the roots of unity.
- A primitive root of unity is a complex number for which is the smallest positive integer satisfying . In other words, a primitive root of unity is an root of unity which is also not an root of unity for any . In particular, any complex number of the form is a primitive root of unity.
- If is a primitive root of unity, then all roots of unity are given by They satisfy
