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2 - Last updated onDecember 15, 2025

3 - A complex number is expressed as a + ib, where a is the real part and b is the imaginary part, denoted by the imaginary unit i, which satisfies i² = -1. The complex number can be represented as a point on a graph. The x-axis represents a, the real part, and runs horizontally. The y-axis represents b, the imaginary part, and runs vertically. The complex number a + ib is represented as the point (a, b) on a graph, which is known as the complex plane. To plot 3 + 4i, move 3 units to the right along the x-axis and 4 units up along the y-axis Representing complex numbers as points on a plane makes them easier to understand and work with, as it places them in a familiar two-dimensional space.

4 - <h2>Complex Number as a Vector</h2>

5 - What Are Numbers? 🔢 | Fun Explanation with 🎯 Real-Life Examples for Kids | ✨BrightCHAMPS Math

6 - ▶

7 - A<a>complex number</a>a + ib can be represented as a vector from the origin (0, 0) to the point (a, b) on the complex plane. This vector with both direction and<a>magnitude</a>, similar to an arrow in<a>geometry</a>or physics. The length of the vector a + bi is:

8 - Magnitude = \(\sqrt{a² + b²}\). For example, 3 + 4i it will be:

9 - \(\sqrt {3^2 + 4^2 }\)= \(\sqrt {9 + 16}\) =\(\sqrt25\) = 5

10 - Here, the magnitude is 5

11 - Modulus of a Complex Number as a Geometric Property:

12 - A complex number is written as z = a + bi, where a is the real part, b is the imaginary part. The modulus of a number is its distance from zero on the<a>number line</a>.

13 - For any<a>real number</a>x: |x| = x (if x \(\geq\) 0) |x| = x (if x \(\leq\) 0)

14 - For example: |3| = 3 |-3| = 3

15 - Both positive and<a>negative numbers</a>result in positive results. In both cases, the number is 3 units away from 0, Because distance is always positive or zero and never negative

16 - Formula: <a>|z|</a>= \(\sqrt {a^2 + b^2}\)

17 - Example: If you plot the point z = 3 + 4i, it corresponds to the point (3, 4) on the plane. So, |3 + 4i| = \(\sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt25 = 5\) 5 is the length of the line from the origin to the point (3, 4).

18 - Example: If you plot the point z = -2 + 0i, it corresponds to the point (-2, 0) on the plane. So, |-2 + 0i| = \(\sqrt{(-2)^2+0^2} = \sqrt{4+0} = \sqrt4 = 2\) 2 is the length of the line from the origin to the point (2,0).

19 - <h2>Argument of a Complex Number</h2>

20 The argument of a complex<a>number</a>is the angle between the positive real axis and the line joining the origin to the point (a, b) in the complex plane.

1 The argument of a complex<a>number</a>is the angle between the positive real axis and the line joining the origin to the point (a, b) in the complex plane.

21 For a complex number, z = a + bi, the argument of z: = θ = tan⁻¹\(({b \over a})\)

2 For a complex number, z = a + bi, the argument of z: = θ = tan⁻¹\(({b \over a})\)

22 Let us understand how the argument of a complex number z = a + bi is determined in different quadrants.

3 Let us understand how the argument of a complex number z = a + bi is determined in different quadrants.

23 1. First quadrant (a > 0, b > 0): = tan-1\(({b \over a})\)

4 1. First quadrant (a > 0, b > 0): = tan-1\(({b \over a})\)

24 2. Second quadrant (a < 0, b > 0): π - tan-1\(({b \over a})\)

5 2. Second quadrant (a < 0, b > 0): π - tan-1\(({b \over a})\)

25 3. Third quadrant (a < 0, b < 0): θ = - π + tan⁻¹\(({b \over a})\)

6 3. Third quadrant (a < 0, b < 0): θ = - π + tan⁻¹\(({b \over a})\)

26 4. Fourth quadrant (a > 0, b < 0): tan⁻¹\(({b \over a})\)

7 4. Fourth quadrant (a > 0, b < 0): tan⁻¹\(({b \over a})\)

27 The principal argument of a complex number lies in -π < θ <π.

8 The principal argument of a complex number lies in -π < θ <π.

28 The main value of a complex number is restricted to a specific range here. The principal argument is also known as amplitude.

9 The main value of a complex number is restricted to a specific range here. The principal argument is also known as amplitude.

29 The \(\theta\) is such that: The principal argument of z lies in -π < Arg(z) ≤ π (in radians), or equivalently -180° < Arg(z) ≤ 180°.

10 The \(\theta\) is such that: The principal argument of z lies in -π < Arg(z) ≤ π (in radians), or equivalently -180° < Arg(z) ≤ 180°.

30 It is the unique angle formed between the positive real axis and the line representing the complex number in the complex plane.

11 It is the unique angle formed between the positive real axis and the line representing the complex number in the complex plane.

31 Example: For z = 1+i = tan-1\(({1 \over 1})\) = 4 Thus, the principal argument of z = 1+ i is \(\pi \over 4\)

12 Example: For z = 1+i = tan-1\(({1 \over 1})\) = 4 Thus, the principal argument of z = 1+ i is \(\pi \over 4\)

32 Let us now understand the principal argument of Z = a =+ ib in different quadrants.

13 Let us now understand the principal argument of Z = a =+ ib in different quadrants.

33 1. First quadrant (a > 0, b > 0): Arg Z = tan⁻¹\(({b \over a})\)

14 1. First quadrant (a > 0, b > 0): Arg Z = tan⁻¹\(({b \over a})\)

34 2. Second quadrant (a < 0, b > 0) : Arg Z = π + tan⁻¹\(({b \over a})\)

15 2. Second quadrant (a < 0, b > 0) : Arg Z = π + tan⁻¹\(({b \over a})\)

35 3. Third quadrant (a < 0, b < 0): Arg Z = - π + tan⁻¹\(({b \over a})\)

16 3. Third quadrant (a < 0, b < 0): Arg Z = - π + tan⁻¹\(({b \over a})\)

36 4. Fourth quadrant (a > 0, b < 0)

17 4. Fourth quadrant (a > 0, b < 0)

37 - <h2>Quadrants in the Complex Plane</h2>

18 +

38 - Quadrant Sign of real(a)Sign of imaginary(b)ExampleI a>0, b>0 + + 2+3i II a<0, b<0 - + -2+3i III a<0, b<0 _ _ -2-3i IV a>0, b<0 + _ 2-3iz = 3 + 0i, written as z = (3, 0) This lies on the positive real axis, which is the x-axis This has no angle, and it lies flat on the axis. The principal argument for this is = 0

39 - Case 2:

40 - z = 0 + 2i, written as z = (0,2) It lies on the positive imaginary axis, that is y-axis There is no angle for this, and it lies along the axis The principal argument of this complex number is \(\pi \over 4\)

41 - A complex number like z = 1 + i is plotted as the point (1, 1). A diagram of the complex plane can show the real axis, imaginary axis, and the vector from (0, 0) to (1, 1), with |z| =\(\sqrt2\), \(\theta\) = π/4

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44 - <h2>Complex Numbers Corresponding to Vector Addition</h2>

45 - Addition or<a>subtraction of complex numbers</a>is similar to<a>adding and subtracting polynomials</a>; we combine real and imaginary parts and then operate.

46 - The<a>formula</a>for adding and subtracting complex numbers is

47 - Adding complex numbers z1= a + bi and z2= c + di is,

48 - z1+ z2 = (a + c) + (b + d)i

49 - For example: Add z1= 3 + 4i and z2= 2 + 5i

50 - Solution:

51 - Given,

52 - z1= 3 + 4i (a=3, b=4)

53 - z2= 2 + 5i (c=2, d=5)

54 - z1+ z2= (3 + 2) + (4 + 5)i

55 - z1+ z2= 5 + 9i

56 - Representing this in a graph below:

57 - Geometrically, adding two complex numbers z1 and z2 corresponds to vector<a>addition</a>using the parallelogram law. The<a>sum</a>is represented by the diagonal of the parallelogram formed by the vectors representing z1 and z2.

58 - Subtracting complex numbers z1 = a + bi and z2 = c + di is performed as follows:

59 - z1- z2 = (a - c) + (b - d)i

60 - For example: Add z1= 3 + 4i and z2= 2 + 5i

61 - Solution:

62 - Given,

63 - z1= 3 + 5i (a=3, b=5)

64 - z2= 2 + 4i (c=2, d=4)

65 - z1+ z2= (3 - 2) + (5 - 4)i

66 - z1+ z2= 1 + 1i

67 - <h2>Multiplication of Complex Numbers Formula </h2>

68 - Multiplying complex numbers follows the same process as<a>multiplying binomials</a>or<a>polynomials</a>. To multiply complex numbers, we use the<a>distributive property</a>.

69 - (a + ib) (c + id) = ac + iad + ibc + i2bd

70 - (a + ib) (c + id) = (ac - bd) + i(ad + bc) [because i2 = -1]

71 - Multiplying complex numbers formula

72 - (a + ib) (c + id) = (ac - bd) + (ad + bc)i

73 - <h2>Multiplication of Complex Numbers in Polar Form</h2>

74 - The polar form of a complex number is z = r (cos + i sin), where r is the complex number's modulus and its argument. The formula for<a>multiplying complex numbers</a>would be:

75 - z1= r1(cosθ1+ i sinθ1)

76 - z2= r2(cosθ2+ i sinθ2)

77 - z1z2 = [r1(cosθ1+ i sinθ1)] [r2(cosθ2+ i sinθ2)]

78 - = r1r2 (cosθ1cosθ2 + i cosθ1 sinθ2 + i sinθ1cosθ2 + i2 sinθ1 sinθ2)

79 - = r1r2 (cosθ1cos2 + i cosθ1 sinθ2 + i sinθ1cosθ2 - sinθ1 sinθ2) {because i2= -1}

80 - = r1r2 [cosθ1cosθ2 - sinθ1 sinθ2+ i (cosθ1 sinθ2 + sinθ1cosθ2)]

81 - = r1r2 [cos (θ1+ θ2) + i sin(θ1+ θ2)] {because cos a cos b - sin a sin b = cos (a + b) and sin a cos b + sin b cos a = sin (a + b)}

82 - Hence, [r1(cosθ1+ i sinθ1)] [r2(cosθ1+ i sinθ2)] = r1r2 [cos (θ1+ θ2) + i sin(θ1+ θ2)]

83 - <h2>The Geometric Effect of the Conjugation of Complex Numbers</h2>

84 - Z = a + ib represents (a, b) in the complex plane

85 - z = a - ib represents (a, -b), which is the<a>conjugate</a>on the complex plane

86 - Conjugation geometrically reflects a point across the real axis by changing the sign of the imaginary part, thus creating a mirror image

87 - Below is a graphical representation of the above condition:

88 - <h2> Polar Form Connection of a Complex Number</h2>

89 - Any complex number z = a + bi is written in polar form as

90 - z = r (cosθ + i sinθ)

91 - And using Euler's formula, it is written as:

92 - z = reiθ

93 - And how are they connected?

94 - Algebra and geometry: In the complex plane, z = a + bi represents a point, which can also be viewed geometrically as a vector from the origin (0, 0) to the point (a, b).

95 - This will be in rectangular form, which is expressed as z = a + bi, and this represents coordinates (a, b).

96 - Modulus and Argument: The modulus r = |z| = \(\sqrt {a^2 + b^2}\) represents the length of the vector, while the argument θ=arg(z) =tan-1(b/a) represents the angle the vector makes with the real axis, thus describing the same point using polar coordinates in<a>terms</a>of distance and direction.

97 - <h2>Trigonometric Principles and Euler’s Formula of Complex Numbers</h2>

98 - Trigonometry: applying trigonometric principles to a right triangle

99 - cos θ = \(a \over r\)

100 - sin θ = \(b \over r\)

101 - Then a = r cos θ and b = r sin θ.

102 - Therefore, z = a + bi = r cos θ + i r sin θ = r(cos θ + i sin θ), which is the polar form.

103 - This is in polar form, which is expressed as z = r(cos θ + i sin θ), and this represents length and angle

104 - Euler’s formula connecting to complex numbers

105 - Using Euler’s formula, cos θ + i sin θ = \(re^{i\theta}\),

106 - We can write z = r (cos θ + i sin θ) = \(re^{i\theta}\).

107 - <h2> Distance Between Complex Numbers</h2>

108 - The distance between two complex numbers, z1 = a + bi and z2 = c + di, is the same as the distance between their corresponding points (a, b) and (c, d) in the complex plane.

109 - You can calculate this distance using either the standard distance formula from geometry:

110 - Distance = \(\sqrt { (a - c)^2 + (b-d)^2}\)

111 - Or by finding the modulus (<a>absolute value</a>) of their difference:

112 - Distance = | z1- z2| = |(a - c) + (b - d)i| = \(\sqrt { (a - c)^2 + (b-d)^2}\)

113 - Both methods yield the same result, rooted in the Pythagorean theorem.

114 - Subtracting two complex numbers, z1 - z2, and then taking the modulus, |z1 - z2|, directly calculates the distance between the points representing those complex numbers in the complex plane.

115 - <h2>Tips and Tricks to master Operating with Complex Numbers</h2>

116 - To grasp operations with complex numbers, you will need an<a>understanding of</a>structures and characteristics, and the rules of algebraic manipulation. Here are the follow tips and tricks to work effectively and accurately with complex numbers:

117 - <ul><li>Know the Standard Form: Remember that every complex number is expressed in the form of a + bi, with a being the real number and b being the<a>imaginary number</a>. </li>

118 - <li>Stay Straight on the Arithmetic Rules: Remember the exact rules for addition,<a>subtraction</a>,<a>multiplication</a>, and<a>division</a>, and pay careful attention to the real parts and imaginary parts. </li>

119 - <li>Use Conjugates When Dividing: If you are required to divide by a complex number, multiply both the<a>numerator</a>and the<a>denominator</a>by the conjugate of the denominator (it will result in you removing the imaginary unit from the denominator). </li>

120 - <li>Get Used to Powers of i: Keep in mind that i2 = -1, and be familiar with the cycle for taking powers of i to help with simplification. </li>

121 - <li>Draw a Geometric Interpretation: When working in the complex number system, interpret complex numbers as either points or vectors of the complex plane to help visualize the magnitude or direction. </li>

122 - <li>Teachers can start teaching the topic by introducing the concepts of a Cartesian plane and by teaching them “what is a complex number?” Strengthen their basic knowledge of mathematics. </li>

123 - <li>Parents can keep the language parallel. Use the words “real direction” instead of x-direction and “i-direction” instead of y-direction for a while, so that your children would feel that it is the same with the new labels. </li>

124 - <li>Teachers can use different color codes for the real and imaginary axes. Also, remember to maintain the same colors across all lessons. Visual consistency helps children build their long-term memory. </li>

125 - <li>Parents can help their children learn it simply by pointing out the sign patterns that quadrants use.Quadrant I \(\rightarrow\) complex number with form (+a+bi) Quadrant II \(\rightarrow\) complex number with form (-a+bi) Quadrant III \(\rightarrow\) complex number with form (-a-bi) Quadrant IV \(\rightarrow\) complex number with form (+a-bi)

126 - </li>

127 - </ul><h2>Common Mistakes and How to Avoid Them in Operating with Complex Numbers</h2>

128 - Students often make mistakes when working with complex numbers, such as confusing real and imaginary parts, among other errors. Here are some common mistakes and tips on how to avoid them.

129 - <h2>Real-life applications of complex numbers</h2>

130 - Complex numbers are helpful in various real-world applications, in fields like oscillations, waves, rotations, and many more. Here are some examples of it

131 - 1. Signal Processing:In Signal Processing (Image and Audio), beyond basic signal analysis, complex numbers are used in advanced image and audio processing techniques, including compression and restoration.

132 - 2. Medical Facilities:When medical imaging techniques like MRI (Magnetic Resonance Imaging) and CT scans utilize Fourier analysis with complex numbers to reconstruct detailed images of the human body.

133 - 3. Quantum Mechanics:In Quantum Mechanics, the wave<a>function</a>in quantum mechanics, which describes the state of a particle, is inherently complex-valued. Complex numbers are not just a mathematical convenience here; they are fundamental to the theory.

134 - 4. Finance:Complex numbers are used in Financial Mathematics, where it is applied in modeling stochastic processes, such as stock price movements, and the analysis of financial derivatives.

135 - 5. Navigation:In Navigation Systems (GPS), Signal processing algorithms in GPS receivers use techniques involving complex numbers to determine precise location and timing information.

136 - <h3>Problem 1</h3>

137 - Find the magnitude of the complex number 6 + 8i

138 - Okay, lets begin

139 - The magnitude of 6 + 8i = 10

140 - <h3>Explanation</h3>

141 - Given 6 + 8i

142 - We know that, Magnitude |z| = \(\sqrt{a^2 + b^2}\)

143 - Here, a = 6 and b = 8

144 - 6 + 8i = \(\sqrt{6^2 + 8^2}\) = \( \sqrt{36=64} = \sqrt{100}\) = 10.

145 - Well explained 👍

146 - <h3>Problem 2</h3>

147 - Find the argument of z = 2 + 2i

148 - Okay, lets begin

149 - The argument of z = 2 + 2i is π/4

150 - <h3>Explanation</h3>

151 - Given complex number is 2 + 2i

152 - We know that, arg (z) = : = tan-1\(({b \over a})\)

153 - a = 2, b = 2 = tan-1\(({2 \over 2})\) = tan-1 (1) = \(\pi \over 4\)

154 - z lies in the first quadrant and = \(\pi \over 4\)

155 - Well explained 👍

156 - <h3>Problem 3</h3>

157 - Add complex numbers (3 + 2i) and (4 + 5i)

158 - Okay, lets begin

159 - The sum is 7 + 7i

160 - <h3>Explanation</h3>

161 - Given (3 + 2i) and (4 + 5i)

162 - We know that, z1= a + bi and z2= c + di is, z1+ z2 = (a + c) + (b + d)i

163 - a = 3, b = 2, c = 4, and d = 5.

164 - z1+ z2 = (3 + 4) + (2 + 5)i z1+ z2 = 7 + 7i

165 - Well explained 👍

166 - <h3>Problem 4</h3>

167 - Multiply the complex numbers 1 - 3i and 4 + 2i

168 - Okay, lets begin

169 - The product is 10 - 10i

170 - <h3>Explanation</h3>

171 - Given 1 - 3i and 4 + 2i

172 - We know that, (a + bi) (c + di) = (ac - bd) + (ad + bc)i

173 - Here, a = 1, b = -3, c = 4 and d = 2

174 - (1 - 3i)(4 + 2i) = 1 × 4 + 1 × 2i - 3i × 4 - 3i × 2i = 4 + 2i - 12i - 6i² = 4 - 10i -6 (-1) (Because i2 = -1) =10 - 10i

175 - Well explained 👍

176 - <h3>Problem 5</h3>

177 - Find the distance between the complex numbers z1= -3 + 2i and z2= -1 +5i

178 - Okay, lets begin

179 - The distance between the complex number is √13

180 - <h3>Explanation</h3>

181 - Given z1= -3 + 2i and z2= -1 +5i

182 - We know that, Distance = | z1- z2| = |(a - c) + (b - d)i| = \(\sqrt{ (a-c)^2 + (b-d)^2}\)

183 - z1- z2 = (-3 + 2i) - (-1 + 5i)= (-3 +2i + 1 - 5i) = (-2 - 3i)

184 - |z1- z2| = ∣-2 - 3i∣ =\(\sqrt{ (-2)^2 + (-3)^2} = \sqrt{4+9} = \sqrt{13}\)

185 - Well explained 👍

186 - <h2>FAQ’s on Complex Numbers</h2>

187 - <h3>1.What are complex numbers?</h3>

188 - The numbers that are expressed in the form a + bi, where a and b are the real numbers and i is the imaginary part.

189 - <h3>2.What is the imaginary unit 'i'?</h3>

190 - The imaginary unit i is defined as the<a>square root</a>of -1, denoted as i = -1, which implies that i² = -1, allowing us to work with square roots of negative numbers.

191 - <h3>3.What is the argument of a complex number?</h3>

192 - The argument of a complex number is the angle between the positive real axis and the line segment connecting the origin to the point (a, b) in the complex plane. Which can be determined using = tan-1 ba and considering the quadrant of the complex number.

193 - <h3>4.What is the magnitude or modulus of a complex number?</h3>

194 - The modulus or magnitude of a complex number is given by |z| = a2+b2

195 - <h3>5.What is the conjugate of a complex?</h3>

196 - For a complex number z = a + bi, the conjugate is a - bi

197 - <h2>Hiralee Lalitkumar Makwana</h2>

198 - <h3>About the Author</h3>

199 - Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

200 - <h3>Fun Fact</h3>

201 - : She loves to read number jokes and games.