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2 <p>Last updated on<strong>December 2, 2025</strong></p>

3 <p>A complex number is a number that has two parts: a real part (like regular numbers) and an imaginary part, which includes i, where i² = -1. It is written as a + bi, where 'a' represents the real part and 'b' represents the imaginary part. Examples include 3 + 2i and - 5 - i. Complex numbers are useful in math, physics, and engineering to solve problems that real numbers alone cannot.</p>

4 <h2>What are Complex Numbers?</h2>

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

6 <p>▶</p>

7 <p>The idea<a>of</a>complex<a>numbers</a>started long ago with the Greek mathematician Hero of Alexandria in the 1st century. He tried to take the<a>square root of complex number</a>and<a>negative number</a>, but since people at that time didn’t understand negative roots, he changed the negative number into a positive one before solving it. Many years later, in the 16th century, the Italian mathematician Gerolamo Cardano formally introduced complex numbers while working on equations involving negative roots. </p>

8 <p>A complex number is written in the form a + bi, where a is the real part and b is the imaginary part. Complex numbers help us solve problems that include the square root of a negative number-something regular<a>real numbers</a>can’t do. Today, tools like a complex number<a>calculator</a>make it much easier for students and learners to work with these numbers, check their answers, and understand the concept more clearly. </p>

9 <h2>What is i?</h2>

10 <p>The<a>i</a>in the complex number is called ‘iota’. It is used to represent the imaginary part of the complex number. Thereby, it helps to find the<a>square</a>root of negative numbers as the value of i2 = -1. </p>

11 <p><strong>Example:</strong></p>

12 <p>√(-9) = 3i</p>

13 <p>√(-16) = 4i</p>

14 <p>We use i to turn negative square roots into meaningful<a>expressions</a>in complex numbers.</p>

15 <h2>Properties of a Complex Number</h2>

16 <p>In order to understand more about complex numbers, first let’s learn about its properties.</p>

17 <ul><li>If x, y are real numbers and x + iy = 0 then x = 0, y = 0. <p>That is, \( x + iy = 0 = 0 + i0 \)</p>

18 <p>According to the definition of equality for complex numbers, the real parts and imaginary parts must be equal. Therefore, we conclude that:</p>

19 <p>x = 0, y = 0</p>

20 </li>

21 <li>If \(x + iy = u + iv,\) then x = u, and y = v<p>Given that x, y, u and v are real numbers, we know:</p>

22 <p>\( x + iy = u + iv\)</p>

23 <p>By definition of complex number equality, their real and imaginary components must be equal, leading to: </p>

24 <p>x = u, y = v</p>

25 </li>

26 <li>For any three complex numbers u, v, and z, the following fundamental laws hold: </li>

27 </ul><ol><li> Commutative Law: \( u + v = v + u\) \( u . v = v. u\) </li>

28 <li>Associative Law:\( (u + v) + z = u + (v + z)\) \( (u . v) . z = u . (v . z)\) </li>

29 <li>Distributive Law: \(x . (v + z) = x . v + x . z\)</li>

30 </ol><ul><li>The<a>product</a>of two<a>conjugate</a>complex numbers is a real number. Let z = x + iy, where x and y are real. The conjugate is:<p>z = x - iy</p>

31 <p>Multiplying z by its conjugate:</p>

32 <p>\(z . z = (x + iy) (x - iy)\)</p>

33 <p>Expanding using the difference of squares: </p>

34 <p>x² - i²y²</p>

35 <p>Since i² = - 1, this simplifies to: </p>

36 <p>x² + y²</p>

37 <p>Which is a real number.</p>

38 </li>

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41 <h2>Classification of Complex Numbers</h2>

40 <h2>Classification of Complex Numbers</h2>

42 <p>The<a>standard form</a>of a complex number is given by z = a + ib, where a, b ∈ ℝ and i (iota) represents the imaginary unit. Based on the values of a (the real part) and b (the imaginary part), complex numbers can be categorized in</p>

41 <p>The<a>standard form</a>of a complex number is given by z = a + ib, where a, b ∈ ℝ and i (iota) represents the imaginary unit. Based on the values of a (the real part) and b (the imaginary part), complex numbers can be categorized in</p>

43 <ul><li><strong>Zero Complex Numbers:</strong>If a = 0 and b = 0, the complex number is called a zero complex number. The only example of this is 0. </li>

42 <ul><li><strong>Zero Complex Numbers:</strong>If a = 0 and b = 0, the complex number is called a zero complex number. The only example of this is 0. </li>

44 <li><strong>Purely Real Numbers:</strong>If a ≠ 0 and b = 0, the number is considered a purely real number, meaning it has no imaginary component. Examples include 2, 3, 5, 7, and all the other real numbers. </li>

43 <li><strong>Purely Real Numbers:</strong>If a ≠ 0 and b = 0, the number is considered a purely real number, meaning it has no imaginary component. Examples include 2, 3, 5, 7, and all the other real numbers. </li>

45 <li><strong>Purely Imaginary Numbers:</strong>If a = 0 and b ≠ 0, the number is considered a purely<a>imaginary number</a>, as it has no real part. Examples include \(- 7i, - 5i, - i, 5i, 7i,\) and other numbers with only an imaginary component. </li>

44 <li><strong>Purely Imaginary Numbers:</strong>If a = 0 and b ≠ 0, the number is considered a purely<a>imaginary number</a>, as it has no real part. Examples include \(- 7i, - 5i, - i, 5i, 7i,\) and other numbers with only an imaginary component. </li>

46 <li><strong>General Complex Numbers:</strong>If a ≠ 0 and b ≠ 0, the number is classified as a complex number that includes both real and imaginary parts. Examples include \((-1 - i), (1 + i), (1 - i), (2 + 3i),\) and similar expressions. </li>

45 <li><strong>General Complex Numbers:</strong>If a ≠ 0 and b ≠ 0, the number is classified as a complex number that includes both real and imaginary parts. Examples include \((-1 - i), (1 + i), (1 - i), (2 + 3i),\) and similar expressions. </li>

47 </ul><h2>Different Forms of Complex Numbers</h2>

46 </ul><h2>Different Forms of Complex Numbers</h2>

48 <p>Complex numbers can be represented in different forms, each with its own advantages depending on the application. The three most common representations are Rectangular (standard) Form, Polar Form, and Exponential Form. These forms allow for easier calculations in various fields.</p>

47 <p>Complex numbers can be represented in different forms, each with its own advantages depending on the application. The three most common representations are Rectangular (standard) Form, Polar Form, and Exponential Form. These forms allow for easier calculations in various fields.</p>

49 <ul><li><strong>Rectangular Form:</strong>Also known as the Standard Form, this representation of a complex number is written as a + ib, where a and b are real numbers.<p>Examples:\( (5 + 5i), - 7i, (- 3 - 4i),\) etc.</p>

48 <ul><li><strong>Rectangular Form:</strong>Also known as the Standard Form, this representation of a complex number is written as a + ib, where a and b are real numbers.<p>Examples:\( (5 + 5i), - 7i, (- 3 - 4i),\) etc.</p>

50 </li>

49 </li>

51 </ul><ul><li><strong>Polar Form of complex number:</strong>In this representation, a complex number is expressed in<a>terms</a>of its<a>magnitude</a>(r) and angle () relative to the positive x-axis. It is written as: <p>r [cos θ + i sin θ]</p>

50 </ul><ul><li><strong>Polar Form of complex number:</strong>In this representation, a complex number is expressed in<a>terms</a>of its<a>magnitude</a>(r) and angle () relative to the positive x-axis. It is written as: <p>r [cos θ + i sin θ]</p>

52 <p>Here, r represents the distance from the origin, and is the angle between the radius vector and the positive x-axis. </p>

51 <p>Here, r represents the distance from the origin, and is the angle between the radius vector and the positive x-axis. </p>

53 <p>Examples: \([cos 2 + i sin 2], 5 [cos 6 + i sin 6]\), etc.</p>

52 <p>Examples: \([cos 2 + i sin 2], 5 [cos 6 + i sin 6]\), etc.</p>

54 </li>

53 </li>

55 </ul><ul><li><strong>Exponential Form:</strong>This form uses Euler’s Formula to express a complex number as: <p>\(re^{iθ} \space \text{or}\space r·e^{iθ}\)</p>

54 </ul><ul><li><strong>Exponential Form:</strong>This form uses Euler’s Formula to express a complex number as: <p>\(re^{iθ} \space \text{or}\space r·e^{iθ}\)</p>

56 <p>Where r is the<a>magnitude of complex number</a>and θ is the angle. </p>

55 <p>Where r is the<a>magnitude of complex number</a>and θ is the angle. </p>

57 <p>Examples: \(e^{i0}, \quad e^{i\frac{\pi}{2}}, \quad e^{i\frac{5\pi}{6}} \) etc.</p>

56 <p>Examples: \(e^{i0}, \quad e^{i\frac{\pi}{2}}, \quad e^{i\frac{5\pi}{6}} \) etc.</p>

58 </li>

57 </li>

59 </ul><h2>Geometrical Representation of Complex Numbers</h2>

58 </ul><h2>Geometrical Representation of Complex Numbers</h2>

60 <p>A complex number z = a + ib is uniquely represented by the point P (a, b) on the complex plane. Similarly, every point on the complex plane corresponds to a unique complex number.</p>

59 <p>A complex number z = a + ib is uniquely represented by the point P (a, b) on the complex plane. Similarly, every point on the complex plane corresponds to a unique complex number.</p>

61 <p><strong>Complex Plane</strong>: The plane where complex numbers are uniquely represented is known as the Complex Plane, also called the Argand Plane or Gaussian Plane. </p>

60 <p><strong>Complex Plane</strong>: The plane where complex numbers are uniquely represented is known as the Complex Plane, also called the Argand Plane or Gaussian Plane. </p>

62 <p>The Complex Plane has two axes, they are:</p>

61 <p>The Complex Plane has two axes, they are:</p>

63 <ul><li>X-axis or Real Axis </li>

62 <ul><li>X-axis or Real Axis </li>

64 <li>Y-axis or Imaginary Axis</li>

63 <li>Y-axis or Imaginary Axis</li>

65 </ul><p><strong>X-Axis (Real Axis)</strong> </p>

64 </ul><p><strong>X-Axis (Real Axis)</strong> </p>

66 <ul><li>Every purely real complex number is represented by a unique point on this axis. </li>

65 <ul><li>Every purely real complex number is represented by a unique point on this axis. </li>

67 <li>The real part, Re(z), of any complex number is plotted along this axis. </li>

66 <li>The real part, Re(z), of any complex number is plotted along this axis. </li>

68 <li>For this reason, the axis is known as the Real Axis.</li>

67 <li>For this reason, the axis is known as the Real Axis.</li>

69 </ul><p><strong>Y-Axis (Imaginary Axis)</strong> </p>

68 </ul><p><strong>Y-Axis (Imaginary Axis)</strong> </p>

70 <ul><li>Every purely imaginary complex number is represented by a unique point on this axis. </li>

69 <ul><li>Every purely imaginary complex number is represented by a unique point on this axis. </li>

71 <li>The imaginary part, Img(z), of any complex number is plotted along this axis. </li>

70 <li>The imaginary part, Img(z), of any complex number is plotted along this axis. </li>

72 <li>Therefore, the Y-axis is referred to as the imaginary Axis. </li>

71 <li>Therefore, the Y-axis is referred to as the imaginary Axis. </li>

73 </ul><h2>Steps to Represent Complex Numbers on a Complex Plane</h2>

72 </ul><h2>Steps to Represent Complex Numbers on a Complex Plane</h2>

74 <p>To plot a complex number z = a + ib on the complex plane, follow these rules:</p>

73 <p>To plot a complex number z = a + ib on the complex plane, follow these rules:</p>

75 <ul><li>The real part of z, denoted as Re(z) = a, is assigned as the X-coordinate of the point P. </li>

74 <ul><li>The real part of z, denoted as Re(z) = a, is assigned as the X-coordinate of the point P. </li>

76 <li>The imaginary part of z, denoted as Im(z) = b, is assigned as the Y-coordinate of the point P. </li>

75 <li>The imaginary part of z, denoted as Im(z) = b, is assigned as the Y-coordinate of the point P. </li>

77 <li>Thus, the complex number z = a + ib corresponds to the point P (a, b) on the complex plane. </li>

76 <li>Thus, the complex number z = a + ib corresponds to the point P (a, b) on the complex plane. </li>

78 </ul><h2>Modulus and Argument of Complex Number</h2>

77 </ul><h2>Modulus and Argument of Complex Number</h2>

79 <p>The modulus of a complex number represents its<a>absolute value</a> of a complex number and is defined as the distance from the origin to the corresponding point on the complex plane. It is also known as the magnitude of the complex number. </p>

78 <p>The modulus of a complex number represents its<a>absolute value</a> of a complex number and is defined as the distance from the origin to the corresponding point on the complex plane. It is also known as the magnitude of the complex number. </p>

80 <p>For a complex number z = a + ib, the modulus is calculated as:</p>

79 <p>For a complex number z = a + ib, the modulus is calculated as:</p>

81 <p>z = √(a² + b²)</p>

80 <p>z = √(a² + b²)</p>

82 <p>The argument of a complex number is the angle θ between its vector and the positive real (x) axis. </p>

81 <p>The argument of a complex number is the angle θ between its vector and the positive real (x) axis. </p>

83 <p>Mathematically, for z = a + ib, it is given by:</p>

82 <p>Mathematically, for z = a + ib, it is given by:</p>

84 <p> \(\theta = \arctan\left(\frac{b}{a}\right) \)</p>

83 <p> \(\theta = \arctan\left(\frac{b}{a}\right) \)</p>

85 <h2>Operations on Complex Numbers</h2>

84 <h2>Operations on Complex Numbers</h2>

86 <p>Complex numbers follow specific mathematical operations similar to real numbers, but with additional rules for handling the imaginary unit i. These operations include<a>addition</a>,<a>subtraction</a>,<a>multiplication</a>, and<a>division</a>, are carried out using fundamental algebraic properties. Mastering these operations is crucial for handling complex numbers in both mathematical and real-world applications. </p>

85 <p>Complex numbers follow specific mathematical operations similar to real numbers, but with additional rules for handling the imaginary unit i. These operations include<a>addition</a>,<a>subtraction</a>,<a>multiplication</a>, and<a>division</a>, are carried out using fundamental algebraic properties. Mastering these operations is crucial for handling complex numbers in both mathematical and real-world applications. </p>

87 <ul><li><strong>Addition:</strong>To<a>add complex numbers</a>,<a>sum</a>their real and imaginary parts separately.</li>

86 <ul><li><strong>Addition:</strong>To<a>add complex numbers</a>,<a>sum</a>their real and imaginary parts separately.</li>

88 </ul><p> Example: (3 + 2i) + (1 + 4i) = 4 + 6i.</p>

87 </ul><p> Example: (3 + 2i) + (1 + 4i) = 4 + 6i.</p>

89 <ul><li><strong>Subtraction:</strong>Subtract the real parts and the imaginary parts separately.</li>

88 <ul><li><strong>Subtraction:</strong>Subtract the real parts and the imaginary parts separately.</li>

90 </ul><p>Example: (3 + 2i) - (1 + 4i) = 2 - 2i</p>

89 </ul><p>Example: (3 + 2i) - (1 + 4i) = 2 - 2i</p>

91 <ul><li><strong>Multiplication:</strong>Multiply two complex numbers using the<a>distributive property</a>and the rule i2 = -1.</li>

90 <ul><li><strong>Multiplication:</strong>Multiply two complex numbers using the<a>distributive property</a>and the rule i2 = -1.</li>

92 </ul><p>Example: (3 + 2i) (1 + 4i) = 3 + 12i + 2i + 8i2 = 3 + 14i - 8 = - 5 + 14i.</p>

91 </ul><p>Example: (3 + 2i) (1 + 4i) = 3 + 12i + 2i + 8i2 = 3 + 14i - 8 = - 5 + 14i.</p>

93 <ul><li><strong>Division:</strong>To divide a complex number by another, multiply both numerator and denominator by the conjugate of the denominator, then simplify. </li>

92 <ul><li><strong>Division:</strong>To divide a complex number by another, multiply both numerator and denominator by the conjugate of the denominator, then simplify. </li>

94 </ul><p>Example: (3 + 2i) / (1 + 4i) = (3 + 2i) (1 - 4i) / (1 + 4i)(1 - 4i) = (11 - 10i)/17.</p>

93 </ul><p>Example: (3 + 2i) / (1 + 4i) = (3 + 2i) (1 - 4i) / (1 + 4i)(1 - 4i) = (11 - 10i)/17.</p>

95 <h2>Trips and Tricks to Master Complex Numbers</h2>

94 <h2>Trips and Tricks to Master Complex Numbers</h2>

96 <p>Solving complex numbers can be difficult, but with the right approach, it can be easy and simple. Here are some tips and tricks to master the complex numbers:</p>

95 <p>Solving complex numbers can be difficult, but with the right approach, it can be easy and simple. Here are some tips and tricks to master the complex numbers:</p>

97 <ol><li>Memorize the<a>powers</a>of i: \(i = √-1\) \(i2 = -1\) \(i3 = -i \) \(i4 = 1\) </li>

96 <ol><li>Memorize the<a>powers</a>of i: \(i = √-1\) \(i2 = -1\) \(i3 = -i \) \(i4 = 1\) </li>

98 <li>When adding or subtracting two complex numbers, add or subtract the real parts and the imaginary parts separately and then combine both. </li>

97 <li>When adding or subtracting two complex numbers, add or subtract the real parts and the imaginary parts separately and then combine both. </li>

99 <li>When solving for large power of i, simplify the power in<a>factor</a>of 4, for easy calculation. </li>

98 <li>When solving for large power of i, simplify the power in<a>factor</a>of 4, for easy calculation. </li>

100 <li>Remember the values of some common expressions: \((1 + i)2 = 2i\) \((1 - i)2 = -2i\) \(w3 = 1\) \(1 + w + w2 = 0\) </li>

99 <li>Remember the values of some common expressions: \((1 + i)2 = 2i\) \((1 - i)2 = -2i\) \(w3 = 1\) \(1 + w + w2 = 0\) </li>

101 <li>Always<a>rationalize the denominator</a>by multiplying both the<a>numerator and denominator</a>by the conjugate of the denominator.</li>

100 <li>Always<a>rationalize the denominator</a>by multiplying both the<a>numerator and denominator</a>by the conjugate of the denominator.</li>

102 </ol><h2>Common Mistakes of Complex Numbers and How to Avoid Them</h2>

101 </ol><h2>Common Mistakes of Complex Numbers and How to Avoid Them</h2>

103 <p>Complex numbers can be a bit tricky to learn, and making mistakes while learning about them is quite normal for students. Here are the top five mistakes of complex numbers and how to avoid them.</p>

102 <p>Complex numbers can be a bit tricky to learn, and making mistakes while learning about them is quite normal for students. Here are the top five mistakes of complex numbers and how to avoid them.</p>

104 <h2>Real Life Applications of Complex Numbers</h2>

103 <h2>Real Life Applications of Complex Numbers</h2>

105 <p>Complex numbers are more than just numbers or theoretical<a>math</a>, they are essential in technology, science, and engineering. Here are some of the real applications of complex numbers. </p>

104 <p>Complex numbers are more than just numbers or theoretical<a>math</a>, they are essential in technology, science, and engineering. Here are some of the real applications of complex numbers. </p>

106 <ul><li><strong>Electrical Engineering:</strong>Complex numbers are used in AC circuit analysis, where voltage and current are represented as complex numbers to simplify calculations involving phase differences. </li>

105 <ul><li><strong>Electrical Engineering:</strong>Complex numbers are used in AC circuit analysis, where voltage and current are represented as complex numbers to simplify calculations involving phase differences. </li>

107 <li><strong>Signal processing:</strong>They help in processing audio, radio, and TV signals by representing waves mathematically. They are also used in fluid dynamics. </li>

106 <li><strong>Signal processing:</strong>They help in processing audio, radio, and TV signals by representing waves mathematically. They are also used in fluid dynamics. </li>

108 <li><strong>Computer graphics and animation:</strong>They are used to performing transformations, rotations, and scaling of objects in 2D and 3D graphics. </li>

107 <li><strong>Computer graphics and animation:</strong>They are used to performing transformations, rotations, and scaling of objects in 2D and 3D graphics. </li>

109 <li><strong>Physics:</strong>Complex numbers are used in quantum mechanics to describe wave<a>functions</a>, in electromagnetism to solve field equations, and in fluid dynamics. </li>

108 <li><strong>Physics:</strong>Complex numbers are used in quantum mechanics to describe wave<a>functions</a>, in electromagnetism to solve field equations, and in fluid dynamics. </li>

110 <li><strong>Control systems:</strong>Engineers use complex numbers to evaluate stability and performance in systems such as airplane autopilot and robotic motion control. </li>

109 <li><strong>Control systems:</strong>Engineers use complex numbers to evaluate stability and performance in systems such as airplane autopilot and robotic motion control. </li>

111 </ul><h3>Problem 1</h3>

110 </ul><h3>Problem 1</h3>

112 <p>What is (3 + 2i) + (4 - 5i)?</p>

111 <p>What is (3 + 2i) + (4 - 5i)?</p>

113 <p>Okay, lets begin</p>

112 <p>Okay, lets begin</p>

114 <p> 7 - 3i </p>

113 <p> 7 - 3i </p>

115 <h3>Explanation</h3>

114 <h3>Explanation</h3>

116 <p>Add the real parts: </p>

115 <p>Add the real parts: </p>

117 <p>\(3 + 4 = 7\)</p>

116 <p>\(3 + 4 = 7\)</p>

118 <p>Then add the imaginary parts: </p>

117 <p>Then add the imaginary parts: </p>

119 <p>\(2i + (- 5i) = - 3i\)</p>

118 <p>\(2i + (- 5i) = - 3i\)</p>

120 <p>Well explained 👍</p>

119 <p>Well explained 👍</p>

121 <h3>Problem 2</h3>

120 <h3>Problem 2</h3>

122 <p>What is (2 + 3i) x (1 - i)?</p>

121 <p>What is (2 + 3i) x (1 - i)?</p>

123 <p>Okay, lets begin</p>

122 <p>Okay, lets begin</p>

124 <p>5 + i </p>

123 <p>5 + i </p>

125 <h3>Explanation</h3>

124 <h3>Explanation</h3>

126 <p>Use the distributive property: </p>

125 <p>Use the distributive property: </p>

127 <p> \((2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i\).</p>

126 <p> \((2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i\).</p>

128 <p>Well explained 👍</p>

127 <p>Well explained 👍</p>

129 <h3>Problem 3</h3>

128 <h3>Problem 3</h3>

130 <p>What is the conjugate of 4 - 7i?</p>

129 <p>What is the conjugate of 4 - 7i?</p>

131 <p>Okay, lets begin</p>

130 <p>Okay, lets begin</p>

132 <p> 4 + 7i</p>

131 <p> 4 + 7i</p>

133 <h3>Explanation</h3>

132 <h3>Explanation</h3>

134 <p>The conjugate of a complex number a + bi is found by changing the sign of the imaginary part, so 4 - 7i becomes 4 + 7i. </p>

133 <p>The conjugate of a complex number a + bi is found by changing the sign of the imaginary part, so 4 - 7i becomes 4 + 7i. </p>

135 <p>Well explained 👍</p>

134 <p>Well explained 👍</p>

136 <h3>Problem 4</h3>

135 <h3>Problem 4</h3>

137 <p>Find the magnitude (modulus) of - 3 + 4i.</p>

136 <p>Find the magnitude (modulus) of - 3 + 4i.</p>

138 <p>Okay, lets begin</p>

137 <p>Okay, lets begin</p>

139 <p>5</p>

138 <p>5</p>

140 <h3>Explanation</h3>

139 <h3>Explanation</h3>

141 <p>The modulus of a + bi is given by </p>

140 <p>The modulus of a + bi is given by </p>

142 <p>a2+ b2</p>

141 <p>a2+ b2</p>

143 <p>Substituting a = - 3 and b = 4, we get </p>

142 <p>Substituting a = - 3 and b = 4, we get </p>

144 <p>\((-3)2 + (42) = 9 + 16 = 25 = 5\)</p>

143 <p>\((-3)2 + (42) = 9 + 16 = 25 = 5\)</p>

145 <p>Well explained 👍</p>

144 <p>Well explained 👍</p>

146 <h3>Problem 5</h3>

145 <h3>Problem 5</h3>

147 <p>Solve 3 + 2i / 1 - i.</p>

146 <p>Solve 3 + 2i / 1 - i.</p>

148 <p>Okay, lets begin</p>

147 <p>Okay, lets begin</p>

149 <p>(1 + 5i) / 2 </p>

148 <p>(1 + 5i) / 2 </p>

150 <h3>Explanation</h3>

149 <h3>Explanation</h3>

151 <p>Multiply by the conjugate of the denominator</p>

150 <p>Multiply by the conjugate of the denominator</p>

152 <p>\((3 + 2i) (1 + i) / (1 - i) (1 + i)\)</p>

151 <p>\((3 + 2i) (1 + i) / (1 - i) (1 + i)\)</p>

153 <p>The denominator simplifies to 12 - (-1) = 2. Expanding the numerator:</p>

152 <p>The denominator simplifies to 12 - (-1) = 2. Expanding the numerator:</p>

154 <p>\(3 + 3i + 2i + 2i²\)</p>

153 <p>\(3 + 3i + 2i + 2i²\)</p>

155 <p>= \(3 + 5i - 2\)</p>

154 <p>= \(3 + 5i - 2\)</p>

156 <p>=\( 1 + 5i.\)</p>

155 <p>=\( 1 + 5i.\)</p>

157 <p>Dividing by 2 gives (1 + 5i) / 2. </p>

156 <p>Dividing by 2 gives (1 + 5i) / 2. </p>

158 <p>Well explained 👍</p>

157 <p>Well explained 👍</p>

159 <h2>Hiralee Lalitkumar Makwana</h2>

158 <h2>Hiralee Lalitkumar Makwana</h2>

160 <h3>About the Author</h3>

159 <h3>About the Author</h3>

161 <p>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.</p>

160 <p>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.</p>

162 <h3>Fun Fact</h3>

161 <h3>Fun Fact</h3>

163 <p>: She loves to read number jokes and games.</p>

162 <p>: She loves to read number jokes and games.</p>