Complex numbers form the basis in mathematics, primarily in algebra and calculus, and are of tremendous importance in several U.S. competitive tests like the SAT, ACT, GRE, and AP Calculus. Complex numbers continue the real number system by using imaginary numbers for the solution of equations with no real solutions like  𝑥2 +1=0. A complex number is a number written in the form z=a+bi, in which “a” is the real component and “bi” is the imaginary component. Complex numbers are a necessary foundation for understanding advanced mathematical concepts, such as polynomials, electrical engineering, and quantum mechanics. This tutorial will examine the definition, properties, operations, and graphical representation of complex numbers, giving students a good foundation for exam preparation.

What are Complex Numbers?

The sum of a real and an imaginary number is a complex number. Complex numbers are represented by the sign z and has the formula an ib. The actual numbers in this instance are a and b. The real component, represented by Re(z), is the value 'a', while the imaginary element, represented by Im(z), is the value 'b'.  An imaginary number is another name for ib. 

History

 
A Greek mathematician, Hero of Alexandria, first mentioned complex numbers in the first century when attempting to figure out the square root of a negative integer. However, he just turned the negative into a positive and obtained the numeric root value. Additionally, Italian mathematician Gerolamo Cardano found the true identity of a complex number in the 16th century while attempting to find the negative roots of formulae using cubic and quadratic polynomials.

If x and y are any two real numbers then any number of the form x + iy is termed a complex number. Here x and y are real numbers and i in maths denotes the imaginary number also read as “iota”. Also, the value of \(i = \sqrt{-1}\) and the complex no. is denoted by z.

As per the complex number meaning, it is the addition of two numbers, one real and one imaginary. Let us understand these two types of numbers.

Real Numbers: All sorts of positive, zero, negative, integer, fractions, rational, irrational, etc. are counted under real numbers. For instance: 23, -35, 0, 1/8, 7.6, √5, etc., are listed under real numbers.

Imaginary Numbers: In the previous heading, we read about real numbers, let us now understand imaginary numbers. A number which is of form ix denotes the imaginary number. Here, the value of \(i = \sqrt{-1}\) and is expressed as Im().

In simple terms, you can understand that numbers that are not real are imaginary. When an imaginary number is squared, it gives a negative outcome. For example, \(\sqrt{-3},\sqrt{-7},\sqrt{-13},\text{ and }i=\sqrt{-6}\) are counted under imaginary numbers.

Examples of complex numbers: \(3+5i\), \(\sqrt{6}-10i\), \(\frac{4}{5}+i\), \(16i\), \(113\).

It is completely possible that \(x\) or \(y\) could be zero and so in \(16i\) the real part is zero. When the real part is zero, we call the complex number a purely imaginary number. In the last example \(113\), here the imaginary part is zero and we actually have a real number.

Representation of Complex Number

As of now, we can say that in a particular complex no. one part is purely real and the other part is purely imaginary. For complex number represented by z = x + iy, x is the real part, iy is the imaginary part and i is the iota value in maths.

Mathematically we can say that z = x + iy where x, y ∈ R and \(i = \sqrt{-1}\)

Power of i

The complex type of numbers is generally employed to solve the \(x^2+2 = 0\) type of equations. As the roots of these types of the equation are of the form\(x = ±\sqrt{-2}\). Such a root is called an imaginary root. The different integral powers of iota i.e i with their respective values are listed below:

\(i=\sqrt{-1}\)

\(i^2 = -1\)

\(i^3 = -i\)

\(i^4= 1\)

In general we can say that;

\( i^{4n}=1\) where n is an integer.

\(i^{4n+1}=i\) ,where n is an integer.

\(i^{4n+2}=-1\) ,where n is an integer.

\(i^{4n+3}=i\) ,where n is an integer.

Graph of Complex Numbers

According to the definition, a complex number can be thought of as an ordered pair of (Re(z)+ Im(z)) numbers. This can also be understood by plotting the coordinates points in a graph.

For z = x + iy the real part i.e. ‘x’ is marked on the x-axis, and the imaginary part i.e. ‘iy’, is marked on the y-axis.

Modulus of Complex Number

Let us step forward and understand the important terms (argument and modulus of a complex number) in the graph for such a system.

The absolute or modulus value of a real number is the number itself. For a number like z = x+iy the modulus of z will be calculated as follows:

\(|z|=\sqrt{\left(x\right)^2+\left(iy\right)^2}\)

\(|z|^2=|x|^2+|iy|^2\)

Here,

\(\left|x\right|^2=x^2\text{ and }\left|iy\right|^2=\left|-y\right|^2=y^2\)

\(|z|^2 = x^2 + y^2\)

Argument of Complex Number

The angle made by the geometric representation of the complex number with the origin, keeping the positive x-axis as the reference in the anticlockwise direction, denotes the argument of the complex number.

Mathematically, the argument of the complex no. is obtained by taking the inverse of the tan of the imaginary part divided by the real part of the. It is denoted by the notation arg(z).

Hence, if z = x + iy

Then;

arg (z) = arg (x+iy) = \(\tanθ=\frac{y}{x}\)

\(θ=\tan^{-1}\left(\frac{y}{x}\right)\)

The argument is represented by the symbol ‘θ’.

Polar Representation of Complex Numbers

The polar form of complex numbers is a way to express a complex number. For z = x + iy, the polar representation is;

\(z=r\left(\cosθ+i\sinθ\right)\)

Where, \(x=r\cosθ\text{ and }y=r\sinθ\)

The modulus is given as; \(r=\sqrt{x^2+y^2}\) and the argument,\(θ=\tan^{-1}\left(\frac{y}{x}\right)\).

Conjugate of Complex Number

Any two complex numbers say \(z_1\text{ and }z_2\)are said to be conjugate of each other if the sum and product of\(z_1\text{ and }z_2\) both are real.

i.e If z = a + ib then its conjugate is given by;

\(\overline{z}=a-ib\)

Theoretically, we can say that the conjugate of a given complex no. is created by keeping the real part the same as the original one and modifying the imaginary part of the complex number to its additive inverse.

Reciprocal of Complex Number

The reciprocal of a given complex number, say, z = x + iy is obtained as follows:

If z is a number then its reciprocal is \(\frac{1}{z}\)

Here, z = x + iy

Therefore the reciprocal \(=\frac{1}{x+iy}\)

Multiply both numerator and denominator by the conjugate of x + iy

\(\frac{1}{x+iy}\times\frac{x−iy}{x−iy}\)

Using the property \((x-y)(x+y)=(x^2-y^2)\)

=\(\frac{x−iy}{x^2−i^2y^2}\)

\(\frac{x−iy}{x^2+y^2}=\frac{x}{x^2+y^2}+\frac{i\left(-y\right)}{x^2+y^2}\)

Hence, the reciprocal z = x + iy is \(\frac{x}{x^2+y^2}+\frac{i\left(-y\right)}{x^2+y^2}\).

Equality of Complex Numbers

Consider \(z_1=x_1+iy_1\text{and}\ z_2=x_2+iy_2\) be any two complex no, where\(x_1,x_2,y_1,y_2\) belong to R and \(i=\sqrt{-1}\).

Then we can say, \(z_1=z_2\) if \(x_1=x_2\) and \(y_1=y_2\).

This implies that if \(z_1=z_2\) then \(Re z_1= Re z_2\) and \(Im z_1= Im z_2\).

Theoretically, we can say that two complex no’s. are equal if the real part of both the numbers is equal plus the imaginary parts of both numbers are also equal.

Order of Complex Numbers

When compared to a real or natural number, complex numbers cannot be ordered, as the structure of these numbers is not defined. Therefore the ordering of complex numbers is not possible. Also, there is no such ordering for these numbers that are consistent with addition and multiplication.

Euler’s Formula: According to Euler’s formula for any real value \(\theta\) we have 

\(e^{i\theta}=\cos(\theta)+i\sin(\theta)\), 

and it represents the complex number in the coordinate plane, where 

• \(\cos(\theta)\) is the real part and is represented with respect to the \(x\)-axis, 
• \(\sin(\theta)\) is the imaginary part that is represented with respect to the \(y\)-axis, 
• \(\theta\) is the angle made with respect to the \(x\)-axis and the imaginary line, which is connecting the origin and the complex number. 

From Euler’s formula and for the functional representation of \(x\) and \(y\), we have 

\(e^{x+iy}=e^{x}(\cos(y)+i\sin(y))=e^{x}\cos(y)+ie^{x}\sin(y)\). 

This decomposes the exponential function into its real and imaginary parts.

Operations on Complex Numbers

Let us now learn the different operations on complex numbers:

Addition of Complex Numbers: The addition of a complex type of number is somewhat similar to the addition of natural numbers. Herein, the real part is added to the real part and the imaginary part is added to the imaginary part.

Consider \(z_1=x_1+iy_1\text{ and }z_2=x_2+iy_2\) be any two complex numbers where \(x_1,x_2,y_1,y_2\in R\text{ and}\ i=\sqrt{-1}\)

then \(z_1+z_2=\left(x_1+x_2\right)+i\left(y_1+y_2\right)\).

The complex numbers follow all the following properties of addition.

1. Closure Law: The sum of two complex numbers is also a complex number. For two complex numbers \(z_{1}\) and \(z_{2}\), the sum of \(z_{1}+z_{2}\) is also a complex number.
2. Commutative Law: For two complex numbers \(z_{1}\) and \(z_{2}\), we have \(z_{1}+z_{2}=z_{2}+z_{1}\).
3. Associative Law: For the given three complex numbers \(z_{1}\), \(z_{2}\), and \(z_{3}\), we have \(z_{1}+(z_{2}+z_{3})=(z_{1}+z_{2})+z_{3}\).
4. Additive Identity: For a complex number \(z = x + iy\), there exists \(0 = 0 + i0\), such that \(z + 0 = 0 + z = 0\).
5. Additive Inverse: For the complex number \(z = x + iy\), there exists a complex number \(-z = -x – iy\) such that \(z + (-z) = (-z) + z = 0\). Here \(-z\) is the additive inverse.

Subtraction of Complex Numbers: In the subtraction of these numbers, the real part is subtracted separately and the same is for the imaginary part.

If \(z_1=x_1+iy_1\text{ and }z_2=x_2+iy_2\) be any two complex numbers where \(x_1,x_2,y_1,y_2\in R\text{ and}\ i=\sqrt{-1}\)

then \(z_1-z_2=\left(x_1-x_2\right)+i \left(y_1-y_2\right)\).

Multiplication of Complex Numbers: In the Multiplication of Complex Numbers, no’s every element of the given two numbers is multiplied with every other element.

If \(z_1=x_1+iy_1\text{ and }z_2=x_2+iy_2\) be any two complex numbers where \(x_1,x_2,y_1,y_2\in R\text{ and}\ i=\sqrt{-1}\)

then;

\(z_1.z_2=\left(x_1+iy_1\right).\left(x_2+iy_2\right)\)

\(=\left(x_1.x_2-y_1.y_2\right)+i\left(x_1.y_2+x_2.\ y_1\right)\)

Division of Complex Numbers: The division for complex types of numbers is as shown:

Consider\(\ z_1=x_1+iy_1\ and\ \ z_2=x_2+iy_{2\ }\)be any two complex numbers

where\(\ x_1,\ x_2,\ y_{1,\ }\ y_{2,\ }\ \in\ R\ and\ i=\sqrt{-1}\)

Then\(\ \frac{z_1}{z_2}=\frac{\left(x_1+iy_1\right)}{\left(x_2+iy_2\right)}=\frac{\left(x_1.\ x_2+y_1.\ y_2\right)}{x_2^2\ +y_2^2}+i\frac{\left(x_2.\ y_1-x_1.\ y_2\right)}{x_2^2\ +y_2^2}\)

Algebraic Identities of Complex Numbers

In the previous header, you learnt about the various operations on complex numbers ranging from addition, subtraction, multiplication and division followed by the related formulas, now let us understand the different algebraic identities related to the topic.

\(\left(𝑧_a+𝑧_b\right)^2=\left(𝑧_a\right)^2+\left(𝑧_b\right)^2+2\times𝑧_a\times𝑧_b\)

\(\left(𝑧_a-𝑧_b\right)^2=\left(𝑧_a\right)^2+\left(𝑧_b\right)^2-2\times𝑧_a\times𝑧_b\)

\(\left(𝑧_a\right)^2−\left(𝑧_b\right)^2=\left(𝑧_a+𝑧_b\right)\left(𝑧_a−𝑧_b\right)\)

\(\left(\ z_a+z_b\right)^3=z_a^3+3z_a^2z_b+3z_az_b^2+z_b^3\)

\(\left(\ z_a-z_b\right)^3=z_a^3-3z_a^2z_b+3z_az_b^2-z_b^3\)

\((z_{a}+z_{b}+z_{c})^{2}=z_{a}^{2}+z_{b}^{2}+z_{c}^{2}+2z_{a}z_{b}+2z_{b}z_{c}+2z_{c}z_{a}\).

Similarly, you can understand the other related properties as well.

Important Points on Complex Numbers

• All real numbers are complex numbers but all complex numbers don’t need to be real numbers.
• All imaginary numbers are complex numbers but all complex numbers don’t need to be imaginary numbers.
• The conjugate of a complex number \(z=x+iy\) is \(\overline{z}=x-iy\).
• The reciprocal of a complex number \(z=x+iy\) is \(z^{-1}=\frac{1}{z}=\frac{x}{x^{2}+y^{2}}+\frac{i(-y)}{x^{2}+y^{2}}\).
• The magnitude of a complex  \(z=x+iy\) is \(|z|=\sqrt{x^{2}+y^{2}}\).

Solved Examples of Complex Numbers

Let us practice some solved examples of complex numbers to understand these concepts more clearly.

Example 1: Obtain the sum and difference for the given set of complex numbers.
\(z_1=−3+2i,\ z_2=5−3i\).

Solution: Given;

\(z_1=−3+2i,\ z_2=5−3i\)

Summation or addition of \(z_1\text{and }z_2\) is as follows;

\(z_1+z_2=\left(−3+2i\right)+\left(5−3i\right)\)

\(=\left(−3+5\right)+\left(2i−3i\right)\)

\(=2-i\)

Hence, \(z_1+z_2=2-i\)

Subtraction of \(z_1\text{and }z_2\) is as follows;

\(z_1-z_2=\left(−3+2i\right)-\left(5−3i\right)\)

Using the formula;

\(z_1-z_2=\left(x_1-x_2\right)+i \left(y_1-y_2\right)\)

\(z_1-z_2=\left(−3-5\right)+i\left(2+3\right)\)

\(z_1-z_2=-8+i5\)

Example 2: Obtain the conjugate of 5+i4.

Solution: Given;

As per the definition If z = a + ib then its conjugate is given by;  \(\overline{z}=a-ib\)

Here z = a + ib= 5+i4, hence conjugate of z is equal to \(\overline{z}=5-i4\).

Example 3: Multiply 3 and (2 + 4i).

Solution: Given;

As per the algebra of complex numbers, we can write 3 as (3 + 0i)

No according to the multiplication rule:

\((3+0i)\times(2+4i)=3\times(2+4i)\)

\(=6+12i\)

Example 4: Find the result of the \(\frac{3}{2+i}\).

Solution: For the given data we have to find the division result of \(\frac{3}{2+i}\)

According to the division rule;

\(\frac{3}{2+i}\times\frac{2-i}{2-i}\)

\(=\frac{6-3i}{5}\)

\(=\frac{6}{5}-\frac{3i}{5}\)

Questions to Ace Your Exams

Q1. Which pair of complex numbers has a real number product? How to write complex numbers in polar form?
Complex numbers with conjugate pairs have actual number products. When the imaginary parts are negatives of each other, the product results in a factual number. Hence, to express a complex number in polar form, convert it to polar coordinates using the modulus (distance from the origin) and argument (angle with the positive real axis). 
To express a complex number \(a+bi\) in polar form, use \(r(cosθ + i sinθ)\), where \(r\) is the magnitude calculated as \(a2+b2\), and \(\θ\) is the angle measured in radians, given by \(\arctan(ba)\). This polar form provides a concise representation useful in trigonometric operations.

Q2. How to divide complex numbers? How do you graph complex numbers?
To divide complex numbers, you multiply the numerator and denominator by the conjugate of the denominator. The conjugate is obtained by changing the sign of the imaginary part. Simplify the result to obtain the quotient. This process eliminates the imaginary part from the denominator, facilitating division.
Graphing complex numbers involves plotting the real part on the x-axis and the imaginary part on the y-axis, creating a two-dimensional coordinate system. The resulting point represents the complex number in the complex plane. This visualization helps us understand the geometric interpretation of complex numbers and their relationships. Graphing provides insights into the magnitude and phase angle, aiding in the analysis of complex number properties.

Q3. What is “i” in complex numbers? How do you simplify complex numbers? How do you write complex numbers in standard form?
In complex numbers, "i" is the imaginary unit, representing the square root of -1. To simplify complex numbers, combine like terms and express the real and imaginary parts separately. The standard form of a complex number is a+bi, where "a" is the real part, "b" is the coefficient of the imaginary unit, and "i" satisfies the equation i2 = -1. 
To write in standard form, ensure both the real and imaginary parts are clearly stated, and if possible, simplify further by factoring out fundamental terms. Therefore, this representation is crucial in various mathematical applications, including electrical engineering and physics. 

Q4. How to multiply complex numbers? How do you add complex numbers? How do you simplify complex numbers?
A: To multiply complex numbers, use the distributive property and combine like terms. Multiply the real parts together, then the imaginary parts, and simplify accordingly.
Adding complex numbers involves adding their real parts and imaginary parts separately. Simply combine the real and imaginary components to obtain the sum.
Therefore, to simplify a complex number, express it in the form "a+bi," where "a" is the real part and "bi" is the imaginary part. Combine like terms and simplify the expression by performing any necessary arithmetic operations, ensuring the final result is in its simplest form. These methods follow straightforward algebraic rules and are essential for working with complex numbers in various mathematical contexts.

Conclusion

In summary, complex numbers are a basic extension of real numbers, enabling the solution of problems and equations that would otherwise be impossible. Their full mathematical complexity, such as graphical illustration, modulus, and argument, makes them universally useful across disciplines. Complex numbers are invaluable tools in engineering, physics, and computer science, respectively, for modeling and analysis. By understanding their principles and operations, one appreciates mathematics more deeply and its widespread applications in the contemporary world.