## Complex NumbersThe equation of the form x^2 + 4 = 2 or x^2 + 1 = 0 is not solvable because there exists no rational number whose square is -2 or -1. Euler was the mathematician who first introduced the term Complex numbers signify the combination of real and imaginary parts in a number. Let's first discuss what real and imaginary numbers are. ## Real numbersReal numbers are the numbers in the form of a fraction, integers, and irrational numbers. For example, ## Imaginary NumbersImaginary numbers are the numbers that include For example, It is because the square of iota is equal to -1. The examples of imaginary numbers are: Thus, complex numbers can be represented as: Where, a and b are the two real numbers. The number with the term iota is called an For example, ## Visual Representation of Complex NumbersLet's understand the complex numbers in the visual form. The two axes on a graph are the real axis and imaginary axis, as shown below: The complex number z = a + ib can be represented as follows: Here, a and b are the numbers on the real and imaginary axis. Let's represents some complex numbers on the above graph. 1. 3 + 2i The coordinates of the given complex number are (3, 2). Here, the number 3 lies on the real axis, and 2 lies on the imaginary axis, as shown below: 2. -2 - 4i The coordinates of the given complex number are (-2, -4). Here, the number -2 lies on the real axis, and -4 lies on the imaginary axis, as shown below: Numbers are categorized as real and complex. A number consists of two parts, i.e., real and imaginary. The real numbers consist of only real parts, while the complex number consists of real and imaginary parts. The square of an imaginary part is negative, while the square of the real part is positive. For example, We can also say that the square root of a negative number is an imaginary quantity. ## Power of IotaHere, we will discuss the positive and negative powers of the iota (i). ## Positive powers of i:Thus, we can say that: ## Negative powers of i:Let's evaluate the higher powers of iota. The higher power of iota can be easily calculated with the help of these above four positive powers.
(a)=i (b)=i Solution: (a) We know, i When we divide 999 by 4, we get 3 as a remainder. We can write it as: (b) When we divide 135 by 4, we get 3 as a remainder. We can write it as: ## Equality of Complex NumbersLet the two complex numbers be z1 and z2. If, z1 = a1 + ib1 z2 = a2 +ib2 The above complex numbers are equal, if a1 = a2 and b1 = b2. It means the Real part of z1 = Real part of z2, and the Imaginary part of z1 = imaginary part of z2. Let's understand it with few examples.
Given: z1 = z2. 4 - iy = x + 5i We know, Re (z1) = Re (z2) After comparing, we get, x = 4 Similarly, equating Im (z1) = Im (z2) We get, y = -5 or -y = 5
(a + b) - i (3a + 2b) = 5 + 2i After equating the real and imaginary parts, we get: a + b = 5 and 3a + 2b = 2 Solving the above two equations, we get: a = -12 b = 17 ## Addition of Complex NumbersLet the two complex numbers be z1 and z2, where: z1 = a1 + ib1 z2 = a2 +ib2 The addition of two complex numbers z1 and z2 is equal to the complex number (a1 +a2) + i (b1 + b2). The real part of a first number is added with the real part of the second number. It means that the real part can be added to the real part of the other number. Similarly, the imaginary part of a first number is added with the imaginary part of the second number. Re (z1 + z2) = Re (z1) + Re (z2) Im (z1 + z2) = Im (z1) + Im (z2) Let's consider few examples.
z1 + z2 = (3x - 7) + 2iy + (-5y + (5 + x) i) z1 + z2 = 3x - 7 + 2iy - 5y + 5i + xi z1 + z2 = 3x - 5y -7 + i(x + 2y + 5)
z1 + z2 = 5 + 4i + 6 -2i z1 + z2 = 11 + 2i ## Subtraction of Complex NumbersLet the two complex numbers be z1 and z2, where: z1 = a1 + ib1 z2 = a2 +ib2 The Subtraction of a complex number z2 from z1 is denoted as z1 - z2. We can also define it as the addition of two complex numbers z1 and -z2. Similarly, the Subtraction of a complex number z1 from z2 is denoted as z2 - z1. It is also equal to z2 + (-z1). Let's consider some examples.
z1 - z2 = 5 + 3i - (2 - 7i) = 5 + 3i - 2 + 7i = 3 + 10i z1 - z2 = 3 + 10i
z1 - z2 = 4 + 3i (2i - 7) - [2 - 7i (2i + 6)] z1 - z2 = 4 + 6i^2 - 21i - [2 - 14i^2 - 42i] z1 - z2 = 4 + 6i^2 - 21i - 2 + 14i^2 + 42i z1 - z2 = 4 - 6 - 21i - 2 - 14 + 42i z1 - z2 = -18 + 21i ## Multiplication of Complex NumbersLet's first understand the multiplication of two complex numbers (a +bi) and (c+ di). (a+bi)(c+di)=ac+adi+cbi+bdi (a+bi)(c+di)=ac+adi+cbi-bd The multiplication process of two complex numbers is shown below: Let the two complex numbers be z1 and z2, where: z1 = a1 + ib1 z2 = a2 +ib2 The multiplication of two complex number z1 and z2 is denoted as z1z2. z1z2 = (a1 + ib1)(a2 + ib2) z1z2=a1a2+a1b2i+a2b1i+b1b2i z1z2=a1a2+a1b2i+a2b1i-b1b2 Separating the real and imaginary parts, z1z2=(a1a2-b1b2)+i(a1b2+a2b1) We can write the above equation as: z1z2 = [Re (z1) Re (z2) - Im (z1) Im (z2)] + i[Re (z1) Im (z2) + Re (z2) Im (z1)] Let's understand with the help of few examples.
z1z2 = (3 + 4i) (6 + 7i) z1z2 = 18 + 21i + 24i + 28i^2 z1z2 = 18 + 21i + 24i - 28 z1z2 = -10 + 45i
z1z2 = (3 - 4 (i + 4)) (6 - 2i) z1z2 = (3 - 4i -16) (6 - 2i) z1z2 = (-13 - 4i) (6 - 2i) z1z2 = -78 + 26i -24i + 8i^2 z1z2 = -78 + 26i -24i - 8 z1z2 = - -86 + 2i ## Division of Complex NumbersLet the complex number be z = a + ib. The division of a complex number z by a non-zero complex number z1 can be represented as z/z1. We can also multiply z with the multiplicative inverse of z1. It can also be represented as z/z1. Let z = a + ib and z1 = a1 + ib1. We can write it as: Let's consider an example.
## Conjugate of Complex NumbersLet the complex number be z = a + bi. Conjugate of a complex number z is a - bi. Similarly, the conjugate of a complex number 2 + 4i is 2 - 4i. Hence, the conjugate is defined as the number obtained by replacing the i by -i. ## Modulus of a Complex NumberThe modulus of a complex number z is denoted by |z|. It is defined as: Let's consider some examples.
We can write the above complex number as: z = 5 + 6i + 3 i z = 8 + 6i ## Reciprocal of a Complex NumberLet z = a + ib We will multiply the denominator and numerator of a given complex number with its conjugate, as shown below: Hence, the multiplication inverse of complex number z is 1/z. We can say that the multiplicative inverse of a non-zero complex number z is same as it's It means conjugate of a complex number divide by the square of its modulus. Let's consider some examples.
Thus, the reciprocal of the given complex number is: We can also calculate the value directly using the formula: Where, |z| = 13
Thus, the reciprocal of the given complex number is: ## Polar form of a complex numberLet z be the complex number, where z = x +iy. It is represented by a point P (x, y) in the plane, as shown below: We know, z = x + iy. Putting the value of x and y in the above equation, we get: z=|z|cosθ+|z|isinθ z=|z|(cosθ+isinθ) We can write it as: z=r(cosθ+isinθ) Where, r=|z| Such a form of z is known as polar form. Thus, the polar form of z is given by z=r[cos(2πn+θ)+isin(2πn+θ)] Where, r=|z| n = integer θ=arg(z) The arg (z) signifies the angle measured in counter clockwise direction. It depends on the point that lies in one of the four quadrants. The four quadrants are shown below: Let, anglefordifferentqudrants=α For a point lying in the θ=α For a point lying in the θ=π-α For a point lying in the θ=-(π-α) For a point lying in the θ=-α Let's consider some examples.
The polar form of a complex number is given by The x and y of the given complex number (1 + i) is (1, 1). It means that the point (1, 1) lies in the first quadrant. The arg (z) for the first quadrant is: θ=α The polar form of the given complex number is:
The polar form of a complex number is given by z=r(cosθ+isinθ) Where, The x and y of the given complex number (1 + i) is (-1, -1). It means that the point (1, 1) lies in the third quadrant. The arg (z) for the third quadrant is: The polar form of the given complex number is: z=√2 cos(-θ)=cos(θ)∧sin(-θ)=-sin(θ) Hence, the above polar form can be written as: z=√2
The polar form of a complex number is given by The x and y of the given complex number (1 + i) is (1, -1). It means that the point (1, 1) lies in the fourth quadrant. The arg (z) for the fourth quadrant is: The polar form of the given complex number is: z=√2 cos(-θ)=cos(θ)∧sin(-θ)=-sin(θ) Hence, the above polar form can be written as: z=√2 ## Square root of a complex numberLet a + ib be a complex number. √a+ib=x+iy Where, x and y are the real numbers. The above equation can be evaluated as: √a+ib=x+iy Squaring both sides, we get: After equation imaginary and real parts, we get: a=x b=2xy We can write the above equation as: Let's consider few more examples.
Let us assume, Now, comparing the real and imaginary values, we get:
Let us assume, Now, comparing the real and imaginary values, we get: Next TopicDiagonal formula |