## 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.

**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 i^{2} = -1, and this is used to find the value of √-4 = √i^{2}4 = +2i The value of i^{2} = 1 is the fundamental aspect of a complex number. Let us try and understand more about the increasing powers of i.

- i = √-1
- i
^{2}= -1 - i
^{3 }= i.i^{2}= i(-1) = -i - i
^{4}= (i^{2})^{2}= (-1)^{2}= 1 - i
^{4n}= 1 - i
^{4n + 1}= i - i
^{4n + 2}= -1 - i
^{4n + 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.

**Terms used in Complex Numbers:**

**Argument**– Argument is the angle we create by the positive real axis and the segment connecting the origin to the plot of a complex number in the complex plane.**Complex Conjugate**– For a given complex number a + bi, a complex conjugate is a – bi.**Complex Plane**– It is a plane which has two perpendicular axis, on which a complex number a + bi is plotted having the coordinate as (a, b).**Imaginary Axis**– The axis in the complex plane that in usual practice coincides with the y-axis of the rectangular coordinate system, and on which the imaginary part bi of the complex number a + bi is plotted.**Modulus**– In the complex plane, the modulus is the distance between the plot of a complex number and the origin.**Polar Form Of A Complex Number**– Let, z be the complex number a + bi,- Polar form of z = r(cos(Θ)+isin(Θ)), where r = | z| and Θ is the argument of z.
**Real Axis**– The real axis in the complex plane is that which coincides with the x-axis of the rectangular coordinate system. On the real axis, the real part of a complex number a + bi is plotted.

### 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 r = |√a2+b2|. Further, this can be understood as derived from the Pythagoras theorem, where the modulus represented the hypotenuse, the real part is the base, and the imaginary part is the altitude of the right-angled triangle.

**De Moivre’s Theorem**

De Moivre’s theorem generalizes the relation to show that to raise a complex number to the n^{th} power, the absolute value is raised to the n^{th} power and the argument is multiplied by n.

Let z=r(cos(Θ)+isin(Θ)

Then zn=[r(cos(Θ)+isin(Θ)]n

= rn(cos(nΘ)+isin(nΘ)

Where, n is any positive integer

**Proof of De Moivre’s Theorem:**

De Moivre’s theorem states that (cosΘ+isinΘ)n=cos(nΘ)+isin(nΘ)

Assuming n = 1

(cosΘ+isinΘ)1=cos(nΘ)+isin(1Θ)

Assume n = k is true

So, (cosΘ+isinΘ)k=cos(nΘ)+isin(kΘ)

Letting n = k + 1

(cosΘ+isinΘ)(k+1)=cos(nΘ)+isin((k+1)Θ)

Assuming n = k, we get

= (cos(kΘ)+isin(kΘ))x(cosΘ+isinΘ)

= cos(kΘ)cos(Θ)+icos(kΘ)sin(Θ)+isin(kΘ)cos(Θ)–sin(kΘ)sin(Θ)

Now we know that,

sin(a+b)=sin(a)cos(b)+sin(b)cos(a)and

cos(a+b)=cos(a)cos(b)–sin(a)sin(b)

=cos(kΘ+Θ)+isin(kΘ+Θ)

=cos((k+1)Θ)+isin((k+1)Θ)

### 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.

## 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

### Complex Modulus and Argument

It should be noted that the process to find the modulus and argument of a complex number is nearly identical to the process of converting Cartesian coordinates to polar coordinates.

The absolute value of a real number is defined as the positive distance from 0 to that number. The absolute value of a complex number is defined in the same way, except this distance is measured on the complex plane.

Since the segment connecting 0 with the complex number is a hypotenuse of a right triangle, the distance of this segment is computed with the Pythagorean theorem. This distance is sometimes called the **modulus** of the complex number.

The angle that the positive real axis makes with the ray that connects 00 with a complex number is called the **argument** of that complex number.

Just like the modulus, the argument of a complex can be computed using triangle relationships.

*θ* can be solved for by taking the inverse tangent of the above equation, but one must take care to account for the quadrant that the complex number is located in.

A full rotation of a complex number of 2*π* radians will produce an image that is co-terminal with the complex number. Therefore, each complex number has infinitely many arguments.

If *θ* is the argument of a complex number, then *θ*+2*kπ* is also an argument of that complex number, where k is an integer.

## 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) = a^{2} + b^{2}.

### 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.

Take this equation into consideration. a+bi=c+di. Here, real part is equal with each other and imaginary parts are equal i.e. a=c and b=d

**Addition of Complex Numbers:**(a+bi)+(c+di) = (a+c) + (b+d)i**Subtraction of Complex Numbers:**(a+bi)−(c+di) = (a−c) + (b−d)i**Multiplication of Complex Numbers:**(a+bi)×(c+di) = (ac−bd) + (ad+bc)i**Multiplication Conjugates:**(a+bi) × (a+bi) = a^{2}+b^{2}

### Complex Conjugates

The **complex conjugate** of a complex number a+bi*a*+*bi* is a-bi*a*−*bi*.

Name the complex conjugates of the following numbers:

- -5+6i−5+6
*i* - 8/3−
*i* - -2i−2
*i* - 17.17.

Their complex conjugates are as follows:

- -5+6i ⟹−5−6
*i* - 8/3−
*i*⟹8/3+*i* - -2i⟹2
*i*: the complex conjugate of an imaginary number is the negation of that number. - 17 ⟹17: the complex conjugate of a real number is the number itself.

A complex conjugate can also be thought of as the reflection of a complex number about the real axis in the complex plane.

Complex conjugates are useful for rationalizing denominators that contain complex numbers. The process of rationalizing a complex denominator is very similar to how the process works for radicals.

Rationalize the denominator and write in standard form:

The conjugate of the denominator is 5+2i. Multiply both the numerator and denominator by this number:

In addition, the complex conjugate root theorem states how complex roots of polynomials always come in conjugate pairs.

**Complex Conjugate Root Theorem:**

If a+bi is a root of a polynomial with rational coefficients, then a-bi is also a root of that polynomial.

Given that b and c*c* are integers, the other root must be the conjugate of 1+i. Writing the polynomial in factored form gives

We have

### Gaussian Integers

A **Gaussian integer** is a complex number a+bi, where both a*a* and b*b* are integers. It should be noted that a Gaussian integer is *not actually* an integer unless the imaginary part is equal to 0.

Gaussian integers are of interest in number theory because the problems of quadratic, cubic, and quartic reciprocity are more conveniently stated as problems about Gaussian integers.

### 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 ≠0, but in a complex number, the non-trivial sum of squares is equal to i^{2} + 1^{2} = 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^{iθ} = Cosθ + iSinθ, and it represented 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 e^{x + iy} = e^{x}(cosy + isiny) = e^{x}cosy + ie^{x}Siny. 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 complete 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 z1=a+id and z2=c+id, the sum of complex numbers z1+z2=(a+c)+i(b+d). The complex numbers follow all the following properties of addition.

### 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 *z*_{2} = 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.

1. Real number system

- Natural Number (N): N = {1,2,3, …………..}
- Whole Number (W): W = {0, 1, 2, …………..} = {N} + {0}
- Integers (Z or I): Z or I = {………-3, -2, -1, 0, 1, 2, 3, ………}
- Rational Numbers (Q): The numbers which are in the form of p/q (Where p, q ∈ I, q ≠ 0)
- Irrational Numbers: The numbers which are not rational i.e. which can not be expressed in p/q form or whose decimal part is non terminating

Real Numbers (R): The set of Rational and Irrational Number is called as set of Real Numbers i.e. N ⊂ W ⊂ Z ⊂ Q ⊂ R

4. Complex Number

A number of the form z = x + iy where x, y ∈ R and

8. Square root of a complex number

The square root of z = a + ib is

9. Triangle Inequalities

- |z
_{1}± z_{2}| ≤ |z_{1}| + |z_{2}| - |z
_{1}± z_{2}| ≥ |z_{1}| – |z_{2}|

Condition of equality, equality sign holds if z_{1}, z_{2} and origin are colinear.

10. Some important points

(i) If ABC is an equilateral triangle having vertices z_{1}, z_{2}, z_{3} then z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1} or

(ii) If z_{1}, z_{2}, z_{3}, z_{4} are vertices of parallelogram then z_{1} + z_{3} = z_{2} + z_{4}

(iii) Amplitude of a complex number:

The amplitude or argument of a complex number z is the inclination of the directed line segment representing z, with real axis. The amplitude of z is generally written as amp z or arg z, thus if x = x + iy then amp z = tan^{-1}(y/x).

(iv) While finding the solution of equation of form x^{2} + 1 = 0, x^{2} + x + 1 = 0, the set of real number was extended into set of complex numbers. First of all ‘Euler’ represented √-1 by the symbol i and proved that the roots of every algebraic equation are number of the form a + ib where a, b ∈ R. A number of this form called complex Number.

(v) Distance formulae:

The distance between two points P(z_{1}) and Q(z_{2}) is given by

PQ = |z_{2} – z_{1}|

= |affix of Q – affix of P|

Section formula:

If Re(z) divides the line segment joining P(z_{1}) and Q(z_{2}) in the ratio m_{1} : m_{2} (m_{1}, m_{2} > 0)

Then,

(vi) Some particular locus

(a) Equation of the line joining complex number z_{1} and z_{2} is z = tz_{1} + (1 – t)z_{1}, t ∈ R or

(b) General equation of a line in complex plane a¯z+az¯+b = 0 where b ∈ R and a is a fixed non zero complex number.

(c) Equation of circle in central form is |z – z_{0}| = r where z_{0} is the center and r is the radius of the circle further on squaring, we get

11. Let z_{1}, z_{2} are two complex no.’s then at complex plane

(i) |z_{1} – z_{2}| is distance between two complex no.’s

(ii)

(iii) Let “z” is any variable point then |z – z_{1}| + |z – z_{2}| = 2a where |z_{1} – z_{2}| < 2a then locus of z is an ellipse, z_{1}, z_{2} are two foci.

(iv) If |z – z_{1}| – |z – z_{2}| = 2a where |z_{1} – z_{2}| > 2a then z describes a hyperbola where z_{1}, z_{2} are two foci.

(viii) Circle may given in any one of following manner

- |z – z
_{0}| = k where z_{0}is centre and k(real no.) is radius

## 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.

**Complex Numbers Examples**

**Example 1:** Can we help Sophia express the roots of the quadratic equation as complex numbers?

**Solution:**

Comparing the given equation with ,

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

Substitute these values in the quadratic formula:

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

z1=−2+i

z2=1−2i

**Solution:**

Sum:

z1+z2=(−2+i)+(1−2i)

=(−2+1)+(i−2i)

=−1−i

Difference:

z1−z2=(−2+i)−(1−2i)

=(−2−1)+(i+2i)

=−3+3i

Product:

z1⋅z2=(−2+i)(1−2i)

=−2+4i+i−2i2

=−2+4i+i+2[∵i2=−1]

=5i

Quotient:

Therefore, we have:

Sum = -1 – i

Difference = -3 + 3i

Product = 5i

Division = -4/5 – 3i/5

**Solved Examples for Complex Number Formula**

Q.1: Simplify 6i + 10i(2-i)

Solution: 6i + 10i(2-i)

= 6i + 10i(2) + 10i (-i)

= 6i +20i – 10 i^{2}

= 26 i – 10 (-1)

= 26i + 10

**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 and no real value satisfies the equation i^{2} = -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 i^{2} = -1 to represent the negative sign of a number, which is helpful to easily find the square root. Here we have √-4 = √i^{2}4 = + 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 r = √a2+b2. 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.

### 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 i^{2} = -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}

## Important Concepts and Formulas – Complex Numbers

✅ 12th grade math formulas ⭐️⭐️⭐️⭐️⭐

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