Complex Number Formula

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Complex Number

Complex numbers are helpful in finding the square root of negative numbers. The concept of complex numbers was first referred to in the 1st century by a greek mathematician, Hero of Alexandria when he tried to find the square root of a negative number. But he merely changed the negative into positive and simply took the numeric root value. Further, the real identity of a complex number was defined in the 16th century by Italian mathematician Gerolamo Cardano, in the process of finding the negative roots of cubic and quadratic polynomial expressions.

Complex numbers have applications in many scientific research, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and vibration analysis. Here we can understand the definition, terminology, visualization of complex numbers, properties, and operations of complex numbers.

What are Complex Numbers?

A complex number is the sum of a real number and an imaginary number. A complex number is of the form a + ib and is usually represented by z. Here both a and b are real numbers. The value ‘a’ is called the real part which is denoted by Re(z), and ‘b’ is called the imaginary part Im(z).  Also, ib is called an imaginary number.

Some of the examples of complex numbers are etc.

Power of i

The alphabet i is referred to as the iota and is helpful to represent the imaginary part of the complex number. Further the iota(i) is very helpful to find the square root of negative numbers. We have the value of i2 = -1, and this is used to find the value of √-4 = √i24 = +2i  The value of i2 = -1 is the fundamental aspect of a complex number. Let us try and understand more about the increasing powers of i.

  • i = √-1
  • i2 = -1
  • i = i.i2 = i(-1) = -i
  • i4 = (i2)2 = (-1)2 = 1
  • i4n = 1
  • i4n + 1 = i
  • i4n + 2 = -1
  • i4n + 3 = -i

Graphing of Complex Numbers

The complex number consists of a real part and an imaginary part, which can be considered as an ordered pair (Re(z), Im(z)) and can be represented as coordinates points in the euclidean plane. The euclidean plane with reference to complex numbers is called the complex plane or the Argand Plane, named after Jean-Robert Argand. The complex number z = a + ib is represented with the real part – a, with reference to the x-axis, and the imaginary part-ib, with reference to the y-axis. Let us try to understand the two important terms relating to the representation of complex numbers in the argand plane. The modulus and the argument of the complex number.

Modulus of the Complex Number

The distance of the complex number represented as a point in the argand plane (a, ib) is called the modulus of the complex number. This distance is a linear distance from the origin (0, 0) to the point (a, ib), and is measured as Further, this can be understood as derived from the Pythagoras theorem, where the modulus represents the hypotenuse, the real part is the base, and the imaginary part is the altitude of the right-angled triangle.

Argument of the Complex Number

The angle made by the line joining the geometric representation of the complex number and the origin, with the positive x-axis, in the anticlockwise direction is called the argument of the complex number. The argument of the complex number is the inverse of the tan of the imaginary part divided by the real part of the complex number. Argz (θ) =

Polar Representation of a Complex Number

With the modulus and argument of a complex number and the representation of the complex number in the argand plane, we have a new form of representation of the complex number, called the polar form of a complex number. The complex number z = a + ib, can be represented in polar form as z = r(Cosθ + iSinθ).  Here r is the modulus (r = \sqrt{a^2 + n^2}\), and θ is the argument of the complex number(θ =

Properties of a Complex Number

The following properties of complex numbers are helpful to better understand complex numbers and also to perform the various arithmetic operations on complex numbers.

Conjugate of a Complex Number

The conjugate of the complex number is formed by taking the same real part of the complex number and changing the imaginary part of the complex number to its additive inverse. If the sum and product of two complex numbers are real numbers, then they are called conjugate complex numbers. For a complex number  z = a + ib, its conjugate is ¯z = a – ib.

The sum of the complex number and its conjugate is z+¯z  = ( a + ib) + (a – ib) = 2a, and the product of these complex numbers z.¯z = (a + ib) × (a – ib) = a2 + b2.

Reciprocal of a Complex Number

The reciprocal of complex numbers is helpful in the process of dividing one complex number with another complex number. The process of division of complex numbers is equal to the product of one complex number with the reciprocal of another complex number.. The reciprocal of the complex number z = a + ib is

This also shows that z≠z−1.

Equality of Complex Numbers

The equality of complex numbers is similar to the equality of real numbers. Two complex numbers z1=a1+ib1 and z2=a2+ib2 

are said to be equal if the rel part of both the complex numbers are equal a1=a2,  and the imaginary parts of both the complex numbers are equal b1=b2.

Also, the two complex numbers in the polar form are equal, if and only if they have the same magnitude and their argument (angle) differs by an integral multiple of 2π.

Ordering of Complex Numbers

The ordering of complex numbers is not possible. Real numbers and other related number systems can be ordered, but complex numbers cannot be ordered. The complex numbers do not have the structure of an ordered field, and there is no ordering of the complex numbers that are compatible with addition and multiplication. Also, the non-trivial sum of squares in an ordered field is a number , but in a complex number, the non-trivial sum of squares is equal to i2 + 12 = 0. The complex numbers can be measured and represented in a two-dimensional argrand plane by their magnitude, which is its distance from the origin.

Euler’s Formula: As per Euler’s formula for any real value θ we have e = Cosθ + iSinθ, and it represents the complex number in the coordinate plane where Cosθ is the real part and is represented with respect to the x-axis, Sinθ is the imaginary part that is represented with respect to the y-axis, θ is the angle made with respect to the x-axis and the imaginary line, which is connecting the origin and the complex number. As per Euler’s formula and for the functional representation of x and y we have ex + iy = ex(cosy + isiny) = excosy + iexSiny. This decomposes the exponential function into its real and imaginary parts.

Operations on Complex Numbers

The various operations of addition, subtraction, multiplication, division of natural numbers can also be performed for complex numbers also. The details of the various arithmetic operations of complex numbers are as follows.

Addition of Complex Numbers

Th addition of complex numbers is similar to the addition of natural numbers. Here in complex numbers, the real part is added to the real part and the imaginary part is added to the imaginary part. For two complex numbers of the form  and , the sum of complex numbers .  The complex numbers follow all the following properties of addition.

  • Closure Law: The sum of two complex numbers is also a complex number. For two complex numbers  and , the sum of  is also a complex number.
  • Commutative Law: For two complex numbers  we have .
  • Associative Law: For the given three complex numbers  we have .
  • Additive Identity: For a complex number z = a + ib, there exists 0 = 0 + i0, such that z + 0 = 0 + z = 0. 
  • Additive Inverse: For the complex number z = a + ib, there exists a complex number -z = -a -ib such that z + (-z) = (-z) + z = 0.  Here -z is the additive inverse.

Subtraction of Complex Numbers

The subtraction of complex numbers follows a similar process of subtraction of natural numbers. Here for any two complex numbers, the subtraction is separately performed across the real part and then the subtraction is performed across the imaginary part. For the complex numbers z1 = a + ib, z2=c+id, we have z1−z2 = (a – c) + i(b – d)

Multiplication of Complex Numbers

The multiplication of complex numbers is slightly different from the multiplication of natural numbers. Here we need to use the formula of i2=−1.  For the two complex numbers z1 = a + ib, z2 = c + id, the product is z1.z2 = (ca – bd) + i(ad + bc).   

The multiplication of complex numbers is polar form is slightly different from the above mentioned form of multiplication. Here the absolute values of the two complex numbers are multiplied and their arguments are added to obtain the product of the complex numbers. For the complex numbers z1=r1(Cosθ1+iSinθ1), and z2 = z2=r1(Cosθ2+iSinθ2), the product of the complex numbers is z1.z2=r1.r2(Cos(θ1+θ2)+iSin(θ1+θ2)).

Division of Complex Numbers

The division of complex numbers makes use of the formula of reciprocal of a complex number. For the two complex numbers z1= a + ib, z2 = c + id, we have the division as

Algebraic Identities of Complex Numbers

All the algebraic identities apply equally for complex numbers.The addition and subtraction of complex numbers and with exponents of 2 or 3 can be easily solved using algebraic identities of complex numbers.

Complex Numbers Tips and Tricks:

  • 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=a+ib is ¯¯¯z=a−ib.
  • The magnitude of a complex number z=a+ib is |z|=√a2+b2|

Complex Numbers in Maths

Complex numbers are the numbers that are expressed in the form of a+ib where, a,b are real numbers and  ‘i’ is an imaginary number called “iota”. The value of i = (√-1). For example, 2+3i is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im). 

Combination of both the real number and imaginary number is a complex number.

Examples of complex numbers:

  • 1 + j
  • -13 – 3i
  • 0.89 + 1.2 i
  • √5 + √2i

An imaginary number is usually represented by ‘i’ or ‘j’, which is equal to √-1. Therefore, the square of the imaginary number gives a negative value. 

Since, i = √-1, so, i2 = -1

The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc., which rely on sine or cosine waves, etc.

Definition

The complex number is basically the combination of a real number and an imaginary number. The complex number is in the form of a+ib, where a = real number and ib = imaginary number. Also, a,b belongs to real numbers and i = √-1. 

Hence, a complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number. One part of it is purely real and the other part is purely imaginary.

Notation

An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = ib.

Z = a + i b

Examples

See the table below to differentiate between a real number and an imaginary number.

Complex NumberReal NumberImaginary Number
-1+2i-12i
7-9i7-9i
-6i0-6i (Purely Imaginary)
660i (Purely Real)

Is 0 a complex Number?

As we know, 0 is a real number. And real numbers are part of complex numbers. Therefore, 0 is also a complex number and can be represented as 0+0i.

Graphical representation

In the graph below, check the representation of complex numbers along the axes. Here we can see, the x-axis represents real part and y represents the imaginary part.

Let us see an example here. If we have to plot a graph of complex number 3 + 4i, then:

Absolute Value

The absolute value of a real number is the number itself. The absolute value of x is represented by modulus, i.e. |x|. Hence, the modulus of any value always gives a positive value, such that;

|3| = 3

|-3| = 3

Now, in case of complex numbers, finding the modulus has a different method.

Suppose, z = x+iy is a complex number. Then, mod of z, will be:

|z| = √(x2+y2)

This expression is obtained when we apply the Pythagorean theorem in a complex plane. Hence, mod of complex number, z is extended from 0 to z and mod of real numbers x and y is extended from 0 to x and 0 to y respectively. Now these values form a right triangle, where 0 is the vertex of the acute angle. Now, applying Pythagoras theorem,

|z|2 = |x|2+|y|2

|z|2 = x2 + y2

|z| = √(x2+y2)

Algebraic Operations on Complex Numbers

There can be four types of algebraic operation on complex numbers which are mentioned below. Visit the linked article to know more about these algebraic operations along with solved examples. The four operations on the complex numbers include:

  • Addition
  • Subtraction
  • Multiplication
  • Division

Roots of Complex Numbers

When we solve a quadratic equation in the form of ax2 +bx+c = 0, the roots of the equations can be determined in three forms;

  • Two Distinct Real Roots
  • Similar Root
  • No Real roots (Complex Roots)

Complex Number Formulas

While performing the arithmetic operations of complex numbers such as addition and subtraction, combine similar terms. It means that combine the real number with the real number and imaginary number with the imaginary number.

Addition

(a + ib) + (c + id) = (a + c) + i(b + d)

Subtraction

(a + ib) – (c + id) = (a – c) + i(b – d)

Multiplication

When two complex numbers are multiplied by each other, the multiplication process should be similar to the multiplication of two binomials. It means that the FOIL method (Distributive multiplication process) is used.

(a + ib). (c + id) = (ac – bd) + i(ad + bc)

Division

The division of two complex numbers can be performed by multiplying the numerator and denominator by its conjugate value of the denominator, and then applying the FOIL Method.

(a + ib) / (c + id) = (ac+bd)/ (c2 + d2) + i(bc – ad) / (c2 + d2)

Power of Iota (i)

Depending upon the power of “i”, it can take the following values;

i4k+1 = i.i4k+2 = -1 i4k+3 =  -i.i4k = 1

Where k can have an integral value (positive or negative).

Similarly, we can find for the negative power of i, which are as follows;

i-1 = 1 / i

Multiplying and dividing the above term with i, we have;

i-1 = 1 / i  ×  i/i  × i-1  = i / i2 = i / -1 = -i / -1 = -i

Note: √-1 × √-1 = √(-1  × -1) = √1 = 1 contradicts to the fact that i2 = -1.

Therefore, for an imaginary number, √a × √b is not equal to √ab.

Identities

Let us see some of the identities.

  1. (z+ z2)2 = (z1)+ (z2)2 + 2 z1 × z2
  2. (z– z2)2 = (z1)+ (z2)2 – 2 z1 × z2
  3. (z1)– (z2)2 =  (z+ z2)(z– z2)
  4. (z+ z2)3 = (z1)3 + 3(z1)2 z +3(z2)2 z1 + (z2)3
  5. (z– z2)3 = (z1)3 – 3(z1)2 z +3(z2)2 z1 – (z2)3

Properties

The properties of complex numbers are listed below:

  • The addition of two conjugate complex numbers will result in a real number
  • The multiplication of two conjugate complex number will also result in a real number
  • If x and y are the real numbers and x+yi =0, then x =0 and y =0
  • If p, q, r, and s are the real numbers and p+qi = r+si, then p = r, and q=s
  • The complex number obeys the commutative law of addition and multiplication.

          z1+z2  = z2+z1

          z1. z2  = z2. z1

  • The complex number obeys the associative law of addition and multiplication.

         (z1+z2) +z= z1 + (z2+z3)

          (z1.z2).z= z1.(z2.z3)

  • The complex number obeys the distributive law

           z1.(z2+z3) = z1.z+ z1.z3

  • If the sum of two complex number is real, and also the product of two complex number is also real, then these complex numbers are conjugate to each other.
  • For any two complex numbers, say zand z2, then |z1+z2| ≤ |z1|+|z2|
  • The result of the multiplication of two complex numbers and its conjugate value should result in a complex number and it should be a positive value.

Modulus and Conjugate

Let z = a+ib be a complex number.

The Modulus of z is represented by |z|.

Mathematically,

Argand Plane and Polar Form

Similar to the XY plane, the Argand(or complex) plane is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b.

We find the real and complex components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis. Check out the detailed argand plane and polar representation of complex numbers in this article and understand this concept in a detailed way along with solved examples.

Complex Numbers Examples

Example 1: Can we help Sophia express the roots of the quadratic equation x2+x+1=0 as complex numbers?

Solution:

Comparing the given equation with ax2+bx+c=0

,a=1

b=1

c=1

(This equation is as same as the one we saw in the beginning of this page).

Substitute these values in the quadratic formula:

Thus the roots of the given quadratic equation are:

Example 2: Express the sum, difference, product, and quotient of the following complex numbers as a complex number.

Solution:

Sum:

Difference:

Quotient:

Therefore, we have:

Sum = -1 – i
Difference = -3 + 3i
Product = 5i
Division  = -4/5 – 3i/5

Example 1: Simplify

a) 16i + 10i(3-i)

b) (7i)(5i)

c) 11i + 13i – 2i

Solution: 

a) 16i + 10i(3-i)

= 16i + 10i(3) + 10i (-i)

= 16i +30i – 10 i2

= 46 i – 10 (-1)

= 46i + 10

b) (7i)(5i) = 35  i2 = 35 (-1) = -35

c) 11i + 13i – 2i = 22i

Example 2: Express the following in a+ib form:

(5+√3i)/(1-√3i).

Solution:

Given: (5+√3i)/(1-√3i)

FAQs on Complex Numbers

What are Complex Numbers in Math?

A complex number is a combination of real values and imaginary values. It is denoted by z = a + ib, where a, b are real numbers and i is an imaginary number. i = √−1−1 and no real value satisfies the equation i2 = -1, therefore, I is called the imaginary number.

What are Complex Numbers Used for?

The complex number is used to easily find the square root of a negative number. Here we use the value of i2 = -1 to represent the negative sign of a number, which is helpful to easily find the square root. Here we have √-4 = √i24 = + 2i. Further to find the negative roots of the quadratic equation, we used complex numbers.

What is Modulus in Complex Numbers?

The modulus of a complex number z = a + ib is the distance of the complex number in the argand plane, from the origin. It is represented by |z| and is equal to If a complex number is considered as a vector representation in the argand plane, then the module of the complex number is the magnitude of that vector.

What is Argument in Complex Numbers?

The argument of a complex number is the angle made by the line joining the origin to the geometric representation of the complex number, with the positive x-axis in the anticlockwise direction. The argument of the complex number is the inverse of the tan of the imaginary part divided by the real part of the complex number. Argz (θ) =

How to Perform Multiplication of Complex Numbers?

The multiplication of complex numbers is slightly different from the multiplication of natural numbers. Here we need to use the formula of i2 = -1.  For the two complex numbers z1 = a + ib, z2 = c + id, the product is z1.z2 = (ca – bd) + i(ad + bc). 

What is Polar Form in Complex Numbers?

The polar form of a complex number is another form of representing and identifying a complex number in the argand plane. The polar form makes the use of the modulus and argument of a complex number, to represent the complex number. The complex number z = a + ib, can be represented in polar form as z = r(Cosθ + iSinθ).  Here r is the modulus (r = \sqrt{a^2 + n^2}\), and θ is the argument of the complex number

What Are Real and Complex Numbers?

Complex numbers are a part of real numbers. Certain real numbers with a negative sign are difficult to compute and we represent the negative sign with an iota ‘i’, and this representation of numbers along with ‘i’ is called a complex number. Further complex numbers are useful to find the square root of a negative number, and also to find the negative roots of a quadratic or polynomial expression.

How To Divide Complex Numbers?

The division of complex numbers makes use of the formula of reciprocal of a complex number. For the two complex numbers z1 = a + ib, z2 = c + id, we have the division as

How to Graph Complex Numbers?

The complex number of the form z = a + ib can be represented in the argand plane. The complex number z = a + ib can be presented as the coordinates of a point as (Re(z), Im(z)) = (a, ib). Here the real part is presented with reference to the x-axis, and the imaginary part is presented with reference to the y-axis.

How to Convert Complex Numbers to Polar Form?

The complex number can be easily converted into a polar form. The complex number z = a + ib has a modulus (r = \sqrt{a^2 + n^2}\), and an argument With the help of modulus and argument the polar form of the complex number is written as z = r(Cosθ + iSinθ).

How to Write Complex Numbers in Standard Form?

The standard form of writing a complex number is z = a + ib. The standard form of the complex number has two parts, the real part, and the imaginary part. In the complex number z = a + ib, a is the real part and ib is the imaginary part. 

How Complex Numbers were Invented?

The inverse of a complex numbers is helpful in the process of dividing one complex number with another complex number. The process of division of complex numbers is equal to the product of one complex number with the inverse of another complex number.. The inverse of the complex number z = a + ib is

This also shows that z≠z−1.

What are Real Numbers?

Any number which is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers. It is represented as Re(). For example: 12, -45, 0, 1/7, 2.8, √5, etc., are all real numbers.

What are Imaginary Numbers?

The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. It is represented as Im(). Example: √-2, √-7, √-11 are all imaginary numbers.

The complex numbers were introduced to solve the equation x2+1 = 0. The roots of the equation are of form x = ±√-1 and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.

We denote √-1 with the symbol ‘i’, which denotes Iota (Imaginary number).

What is meant by complex numbers?

The complex number is the combination of a real number and imaginary number. An example of a complex number is 4+3i. Here 4 is a real number and 3i is an imaginary number.

How to divide the complex numbers?

To divide the complex number, multiply the numerator and the denominator by its conjugate. The conjugate of the complex number can be found by changing the sign between the two terms in the denominator value. Then apply the FOIL method to simplify the expression.

Mention the arithmetic rules for complex numbers.

The arithmetic rules of complex numbers are:
Addition Rule: (a+bi) + (c+di) = (a+c)+ (b+d)i
Subtraction Rule: (a+bi) – (c+di) = (a-c)+ (b-d)i
Multiplication Rule: (a+bi) . (c+di) = (ac-bd)+(ad+bc)i

Write down the additive identity and inverse of complex numbers.

The additive identity of complex numbers is written as (x+yi) + (0+0i) = x+yi. Hence, the additive identity is 0+0i.
The additive inverse of complex numbers is written as (x+yi)+ (-x-yi) = (0+0i). Hence, the additive inverse is -x-yi.

Write down the multiplicative identity and inverse of the complex number.

The multiplicative identity of complex numbers is defined as (x+yi). (1+0i) = x+yi. Hence, the multiplicative identity is 1+0i.
The multiplicative identity of complex numbers is defined as (x+yi). (1/x+yi) = 1+0i. Hence, the multiplicative identity is 1/x+yi.

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