1 added
2 removed
Original
2026-01-01
Modified
2026-02-28
1
-
<p>248 Learners</p>
1
+
<p>274 Learners</p>
2
<p>Last updated on<strong>December 10, 2025</strong></p>
2
<p>Last updated on<strong>December 10, 2025</strong></p>
3
<p>A complex number is written in the form z = a + ib, where i is the imaginary unit (i2 = -1) and a and b are real numbers. It involves applying the distributive property to compute the product of two complex numbers. In this article, we will learn about the multiplication of complex numbers.</p>
3
<p>A complex number is written in the form z = a + ib, where i is the imaginary unit (i2 = -1) and a and b are real numbers. It involves applying the distributive property to compute the product of two complex numbers. In this article, we will learn about the multiplication of complex numbers.</p>
4
<h2>What is the Multiplication of Complex Numbers?</h2>
4
<h2>What is the Multiplication of Complex Numbers?</h2>
5
<p>What Are Numbers? 🔢 | Fun Explanation with 🎯 Real-Life Examples for Kids | ✨BrightCHAMPS Math</p>
5
<p>What Are Numbers? 🔢 | Fun Explanation with 🎯 Real-Life Examples for Kids | ✨BrightCHAMPS Math</p>
6
<p>▶</p>
6
<p>▶</p>
7
<h2>What are the Properties of the Multiplication of Complex Numbers?</h2>
7
<h2>What are the Properties of the Multiplication of Complex Numbers?</h2>
8
<p>The multiplication of complex<a>numbers</a>follows similar properties to the<a>real numbers</a>. The properties are </p>
8
<p>The multiplication of complex<a>numbers</a>follows similar properties to the<a>real numbers</a>. The properties are </p>
9
<ul><li>Commutative property </li>
9
<ul><li>Commutative property </li>
10
<li>Associative property </li>
10
<li>Associative property </li>
11
<li>Distributive property</li>
11
<li>Distributive property</li>
12
</ul><p><strong>Commutative property:</strong>The order of multiplication does not affect the result. For example, \(z_1 \times z_2 = z_2 \times z_1 \) </p>
12
</ul><p><strong>Commutative property:</strong>The order of multiplication does not affect the result. For example, \(z_1 \times z_2 = z_2 \times z_1 \) </p>
13
<p><strong>Associative property:</strong>The<a>associative property</a>of<a>complex numbers</a>states that the order of grouping the complex numbers doesn't change the result. That is, \(z_1 \times (z_2 \times z_3) = (z_1 \times z_2) \times z_3 \) </p>
13
<p><strong>Associative property:</strong>The<a>associative property</a>of<a>complex numbers</a>states that the order of grouping the complex numbers doesn't change the result. That is, \(z_1 \times (z_2 \times z_3) = (z_1 \times z_2) \times z_3 \) </p>
14
<p><strong>Distributive property:</strong>The<a>distributive property</a>states that z1 \(z_1 \times (z_2 + z_3) = z_1 z_2 + z_1 z_3 \)</p>
14
<p><strong>Distributive property:</strong>The<a>distributive property</a>states that z1 \(z_1 \times (z_2 + z_3) = z_1 z_2 + z_1 z_3 \)</p>
15
<h2>How to Multiply Complex Numbers in Cartesian Form</h2>
15
<h2>How to Multiply Complex Numbers in Cartesian Form</h2>
16
<p>Now, let’s learn how to multiply complex numbers in Cartesian form. In this form, we multiply the complex number<a>term</a>by term. </p>
16
<p>Now, let’s learn how to multiply complex numbers in Cartesian form. In this form, we multiply the complex number<a>term</a>by term. </p>
17
<p>That is, if\(z_1 = a + ib \) and \(z_2 = c + id \)</p>
17
<p>That is, if\(z_1 = a + ib \) and \(z_2 = c + id \)</p>
18
<p>Then \(z_1 \times z_2 = (a + ib)(c + id) \)</p>
18
<p>Then \(z_1 \times z_2 = (a + ib)(c + id) \)</p>
19
<p>= \(ac + a (id) + ib (c) + i^2 bd \)</p>
19
<p>= \(ac + a (id) + ib (c) + i^2 bd \)</p>
20
<p>= \((ac - bd) + i(ad + bc) \)</p>
20
<p>= \((ac - bd) + i(ad + bc) \)</p>
21
<p>So, \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \)</p>
21
<p>So, \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \)</p>
22
<p>For example, find the product of \(z_1 = 3 + 2i \) and \(z_2 = 1 + 4i \) z1z2 = (3 + 2i) (1 + 4i)</p>
22
<p>For example, find the product of \(z_1 = 3 + 2i \) and \(z_2 = 1 + 4i \) z1z2 = (3 + 2i) (1 + 4i)</p>
23
<p>Using the<a>formula</a>for multiplying complex numbers, \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \)</p>
23
<p>Using the<a>formula</a>for multiplying complex numbers, \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \)</p>
24
<p>\((3 + 2i)(1 + 4i) = ((3 \times 1) - (2 \times 4)) + i((3 \times 4) + (2 \times 1)) \)</p>
24
<p>\((3 + 2i)(1 + 4i) = ((3 \times 1) - (2 \times 4)) + i((3 \times 4) + (2 \times 1)) \)</p>
25
<p>=\((3 - 8) + i (12 + 2) \)</p>
25
<p>=\((3 - 8) + i (12 + 2) \)</p>
26
<p>= \(-5 + 14i\)</p>
26
<p>= \(-5 + 14i\)</p>
27
<h3>Explore Our Programs</h3>
27
<h3>Explore Our Programs</h3>
28
-
<p>No Courses Available</p>
29
<h2>How to Multiply Complex Numbers in Polar Form</h2>
28
<h2>How to Multiply Complex Numbers in Polar Form</h2>
30
<p>When multiplying<a>complex numbers in polar form</a>, we first multiply the moduli and then add the<a>arguments</a>.</p>
29
<p>When multiplying<a>complex numbers in polar form</a>, we first multiply the moduli and then add the<a>arguments</a>.</p>
31
<p>That is, when multiplying \(z_1 = r_1 e^{i\Theta_1} \) and \(z_2 \)= \(r_2 e^{i \Theta_2} \)</p>
30
<p>That is, when multiplying \(z_1 = r_1 e^{i\Theta_1} \) and \(z_2 \)= \(r_2 e^{i \Theta_2} \)</p>
32
<p>\(z_1 \times z_2 = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \)</p>
31
<p>\(z_1 \times z_2 = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \)</p>
33
<p>For example, multiplying \(z_1 = 2 e^{i \pi/6} \) and \(z_2 = 3 e^{i \pi/4} \)</p>
32
<p>For example, multiplying \(z_1 = 2 e^{i \pi/6} \) and \(z_2 = 3 e^{i \pi/4} \)</p>
34
<p>\(z_1 z_2 = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \)</p>
33
<p>\(z_1 z_2 = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \)</p>
35
<p>Here, r1 = 2</p>
34
<p>Here, r1 = 2</p>
36
<p>r2 = 3</p>
35
<p>r2 = 3</p>
37
<p>Θ1 = \(\frac{\pi}{6} \)</p>
36
<p>Θ1 = \(\frac{\pi}{6} \)</p>
38
<p>Θ2 = \(\frac{\pi}{4} \)</p>
37
<p>Θ2 = \(\frac{\pi}{4} \)</p>
39
<p>Multiplying moduli \((r_1 r_2) = 2 \times 3 = 6 \)</p>
38
<p>Multiplying moduli \((r_1 r_2) = 2 \times 3 = 6 \)</p>
40
<p>Adding the arguments = \(\frac{\pi}{6} + \frac{\pi}{4} \)</p>
39
<p>Adding the arguments = \(\frac{\pi}{6} + \frac{\pi}{4} \)</p>
41
<p>= \(\frac{2\pi}{12} + \frac{3\pi}{12} \)</p>
40
<p>= \(\frac{2\pi}{12} + \frac{3\pi}{12} \)</p>
42
<p>= \(\frac{5\pi}{12} \)</p>
41
<p>= \(\frac{5\pi}{12} \)</p>
43
<p>So, \(2 e^{i \pi/6} \times 3 e^{i \pi/4} = 6 e^{i 5\pi/12} .\)</p>
42
<p>So, \(2 e^{i \pi/6} \times 3 e^{i \pi/4} = 6 e^{i 5\pi/12} .\)</p>
44
<h2>Multiplication Of Complex Numbers with Real and Imaginary Numbers</h2>
43
<h2>Multiplication Of Complex Numbers with Real and Imaginary Numbers</h2>
45
<p>The multiplication formula for complex numbers is \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). When multiplying the complex number by a real number, let's consider b as 0. That is \(a\) \((c + id) = ac + i a d \). </p>
44
<p>The multiplication formula for complex numbers is \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). When multiplying the complex number by a real number, let's consider b as 0. That is \(a\) \((c + id) = ac + i a d \). </p>
46
<p>For example: \(-5i (2 + 3i) \) </p>
45
<p>For example: \(-5i (2 + 3i) \) </p>
47
<p>Using the formula, \(a (c + id) = ac + i a d \)</p>
46
<p>Using the formula, \(a (c + id) = ac + i a d \)</p>
48
<p>Here, a = -5i</p>
47
<p>Here, a = -5i</p>
49
<p>c = 2</p>
48
<p>c = 2</p>
50
<p>d = 3i</p>
49
<p>d = 3i</p>
51
<p>Substituting the value, \(-5i (2 + 3i) = (-5i \times 2) + (-5i \times 3i) = -10i + (-15 i^2) \)</p>
50
<p>Substituting the value, \(-5i (2 + 3i) = (-5i \times 2) + (-5i \times 3i) = -10i + (-15 i^2) \)</p>
52
<p>Substituting \(i^2 = -1 \)</p>
51
<p>Substituting \(i^2 = -1 \)</p>
53
<p>=\(-10i + 15 = 15 - 10i \). </p>
52
<p>=\(-10i + 15 = 15 - 10i \). </p>
54
<h2>Squaring Complex Numbers</h2>
53
<h2>Squaring Complex Numbers</h2>
55
<p>The formula for multiplying complex numbers is \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). When squaring a complex number, consider a = c and b = d, and then the formula becomes;</p>
54
<p>The formula for multiplying complex numbers is \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). When squaring a complex number, consider a = c and b = d, and then the formula becomes;</p>
56
<p>\((a + ib)^2 = (a \times a - b \times b) + i(ab + ba) \)</p>
55
<p>\((a + ib)^2 = (a \times a - b \times b) + i(ab + ba) \)</p>
57
<p>\((a^2 - b^2) + i (2ab) \)</p>
56
<p>\((a^2 - b^2) + i (2ab) \)</p>
58
<p>For example, squaring \(4 + 5i \)</p>
57
<p>For example, squaring \(4 + 5i \)</p>
59
<p>\((a + ib)^2 = (a^2 - b^2) + i(2ab) \)</p>
58
<p>\((a + ib)^2 = (a^2 - b^2) + i(2ab) \)</p>
60
<p>So, \((4 + 5i)^2 = (4^2 - (5i)^2) + i(2 \times 4 \times 5) \)</p>
59
<p>So, \((4 + 5i)^2 = (4^2 - (5i)^2) + i(2 \times 4 \times 5) \)</p>
61
<p>= \((16 - 25 \times i^2) + 40i \)</p>
60
<p>= \((16 - 25 \times i^2) + 40i \)</p>
62
<p>= \(16 - 25 + 40i \)</p>
61
<p>= \(16 - 25 + 40i \)</p>
63
<p>= \(-9 + 40i \)</p>
62
<p>= \(-9 + 40i \)</p>
64
<h2>Multiplicative Inverse Of Complex Numbers</h2>
63
<h2>Multiplicative Inverse Of Complex Numbers</h2>
65
<p>The<a>multiplicative inverse</a>of a complex number \(z = a + ib\) is another complex number \(z^{-1} \), that is, \(z \times z^{-1} = 1 \). For a complex number \(z = a + ib \), the<a>multiple</a>inverse \(z^{-1} \) = \( \frac {\bar z} {|z|^2}\)</p>
64
<p>The<a>multiplicative inverse</a>of a complex number \(z = a + ib\) is another complex number \(z^{-1} \), that is, \(z \times z^{-1} = 1 \). For a complex number \(z = a + ib \), the<a>multiple</a>inverse \(z^{-1} \) = \( \frac {\bar z} {|z|^2}\)</p>
66
<p>Here, \(z = a - ib \), so, \(|z| = \sqrt{a^2 + b^2} \)</p>
65
<p>Here, \(z = a - ib \), so, \(|z| = \sqrt{a^2 + b^2} \)</p>
67
<p>The formula for the multiplicative inverse: \(z^{-1} \) = \( \frac {\bar z} {|z|^2}\)</p>
66
<p>The formula for the multiplicative inverse: \(z^{-1} \) = \( \frac {\bar z} {|z|^2}\)</p>
68
<p>Where,<a>conjugate</a>:\(z = a - ib \), and the modulus of the complex number is \(|z| = \sqrt{a^2 + b^2}.\)</p>
67
<p>Where,<a>conjugate</a>:\(z = a - ib \), and the modulus of the complex number is \(|z| = \sqrt{a^2 + b^2}.\)</p>
69
<h2>Tips and Trips to Master Multiplying Complex Numbers</h2>
68
<h2>Tips and Trips to Master Multiplying Complex Numbers</h2>
70
<p>Learn effective strategies to multiply complex numbers accurately, both in standard and polar forms, and understand their geometric interpretations.</p>
69
<p>Learn effective strategies to multiply complex numbers accurately, both in standard and polar forms, and understand their geometric interpretations.</p>
71
<ul><li>Always use the formula \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). </li>
70
<ul><li>Always use the formula \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). </li>
72
<li>Separate real and imaginary parts before multiplying. </li>
71
<li>Separate real and imaginary parts before multiplying. </li>
73
<li>Practice squaring complex numbers to see patterns. </li>
72
<li>Practice squaring complex numbers to see patterns. </li>
74
<li>Use the Argand plane to visualize multiplication geometrically. </li>
73
<li>Use the Argand plane to visualize multiplication geometrically. </li>
75
<li>Convert to polar form to simplify multiplication: \(r_1 e^{i\Theta_1} \cdot r_2 e^{i\Theta_2} = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \). </li>
74
<li>Convert to polar form to simplify multiplication: \(r_1 e^{i\Theta_1} \cdot r_2 e^{i\Theta_2} = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \). </li>
76
<li>Teachers should help students build a foundation before diving deep into multiplying complex numbers. Making sure that they are fluent with the distributive property and<a>binomial</a>multiplication (FOIL), since we apply the same ideas while multiplying complex numbers. </li>
75
<li>Teachers should help students build a foundation before diving deep into multiplying complex numbers. Making sure that they are fluent with the distributive property and<a>binomial</a>multiplication (FOIL), since we apply the same ideas while multiplying complex numbers. </li>
77
<li>Parents should review the key rule \(i^2 = -1\) and have them practice rewriting<a>expressions</a>such as \(5i^2 = -5\) to avoid later sign mistakes. </li>
76
<li>Parents should review the key rule \(i^2 = -1\) and have them practice rewriting<a>expressions</a>such as \(5i^2 = -5\) to avoid later sign mistakes. </li>
78
<li>Parents can help young learners by encouraging them to write intermediate steps instead of doing everything mentally. It is easier for them to track the source of each term, and they can find errors more easily. </li>
77
<li>Parents can help young learners by encouraging them to write intermediate steps instead of doing everything mentally. It is easier for them to track the source of each term, and they can find errors more easily. </li>
79
<li>Teachers can note some of the structures students may often see and highlight them to learners, so they can easily understand and practice them until they feel almost automatic in calculating.</li>
78
<li>Teachers can note some of the structures students may often see and highlight them to learners, so they can easily understand and practice them until they feel almost automatic in calculating.</li>
80
</ul><h2>Common Mistakes and How to Avoid Them in Multiplying Complex Numbers</h2>
79
</ul><h2>Common Mistakes and How to Avoid Them in Multiplying Complex Numbers</h2>
81
<p>The students usually make mistakes when multiplying complex numbers, and mostly they repeat the same mistakes. In this section, we will discuss some common mistakes and ways to avoid them in multiplying complex numbers. </p>
80
<p>The students usually make mistakes when multiplying complex numbers, and mostly they repeat the same mistakes. In this section, we will discuss some common mistakes and ways to avoid them in multiplying complex numbers. </p>
82
<h2>Real-World Applications of Multiplying Complex Numbers</h2>
81
<h2>Real-World Applications of Multiplying Complex Numbers</h2>
83
<p>Complex number multiplication is used in various fields, like engineering, quantum mechanics, computer graphics, etc. In this section, we will discuss them in detail. </p>
82
<p>Complex number multiplication is used in various fields, like engineering, quantum mechanics, computer graphics, etc. In this section, we will discuss them in detail. </p>
84
<ul><li><strong>Electrical engineering - </strong>Multiplying complex numbers is used to calculate impedance, voltage, and current in alternating current (AC) circuits, where signals have both magnitude and phase. </li>
83
<ul><li><strong>Electrical engineering - </strong>Multiplying complex numbers is used to calculate impedance, voltage, and current in alternating current (AC) circuits, where signals have both magnitude and phase. </li>
85
<li><strong>Signal Processing - </strong>Complex number multiplication helps in analyzing and manipulating signals in communication systems, such as modulation and filtering. </li>
84
<li><strong>Signal Processing - </strong>Complex number multiplication helps in analyzing and manipulating signals in communication systems, such as modulation and filtering. </li>
86
<li><strong>Control Systems - </strong>Engineers use complex numbers to represent system dynamics; multiplying them helps in understanding system stability and response. </li>
85
<li><strong>Control Systems - </strong>Engineers use complex numbers to represent system dynamics; multiplying them helps in understanding system stability and response. </li>
87
<li><strong>Quantum mechanics - </strong>Complex numbers represent quantum states, and their multiplication is used in computing<a>probability</a>amplitudes and transformations. </li>
86
<li><strong>Quantum mechanics - </strong>Complex numbers represent quantum states, and their multiplication is used in computing<a>probability</a>amplitudes and transformations. </li>
88
<li><strong>Rotations in 2D space - </strong>Multiplying complex numbers is used to rotate vectors or points in a plane, which is useful in computer graphics, robotics, and navigation.</li>
87
<li><strong>Rotations in 2D space - </strong>Multiplying complex numbers is used to rotate vectors or points in a plane, which is useful in computer graphics, robotics, and navigation.</li>
89
</ul><h3>Problem 1</h3>
88
</ul><h3>Problem 1</h3>
90
<p>Find the product of (2 + 3i) × (4 + 5i)</p>
89
<p>Find the product of (2 + 3i) × (4 + 5i)</p>
91
<p>Okay, lets begin</p>
90
<p>Okay, lets begin</p>
92
<p>The product of\( (2 + 3i) × (4 + 5i) is -7 + 22i.\)</p>
91
<p>The product of\( (2 + 3i) × (4 + 5i) is -7 + 22i.\)</p>
93
<h3>Explanation</h3>
92
<h3>Explanation</h3>
94
<p>To multiply the complex number, we use the formula;</p>
93
<p>To multiply the complex number, we use the formula;</p>
95
<p>\((a + ib) (c + id) = (ac - bd) + i(ad + bc)\). </p>
94
<p>\((a + ib) (c + id) = (ac - bd) + i(ad + bc)\). </p>
96
<p>Here, a = 2, b = 3, c = 4, and d = 5</p>
95
<p>Here, a = 2, b = 3, c = 4, and d = 5</p>
97
<p>\((ac - bd) = 2 × 4 - 3 × 5 = 8 - 15 = -7\)</p>
96
<p>\((ac - bd) = 2 × 4 - 3 × 5 = 8 - 15 = -7\)</p>
98
<p>\((ad + bc) = 2 × 5 + 3 × 4 = 10 + 12 = 22\)</p>
97
<p>\((ad + bc) = 2 × 5 + 3 × 4 = 10 + 12 = 22\)</p>
99
<p>So, the product is \(-7 + 22i\)</p>
98
<p>So, the product is \(-7 + 22i\)</p>
100
<p>Well explained 👍</p>
99
<p>Well explained 👍</p>
101
<h3>Problem 2</h3>
100
<h3>Problem 2</h3>
102
<p>Find the product of (1 + i) and (1-i) using the polar form</p>
101
<p>Find the product of (1 + i) and (1-i) using the polar form</p>
103
<p>Okay, lets begin</p>
102
<p>Okay, lets begin</p>
104
<p>The product is 2.</p>
103
<p>The product is 2.</p>
105
<h3>Explanation</h3>
104
<h3>Explanation</h3>
106
<p>To find the product in polar form, we first convert each complex number to polar form. </p>
105
<p>To find the product in polar form, we first convert each complex number to polar form. </p>
107
<p>For 1 + i, </p>
106
<p>For 1 + i, </p>
108
<p>\(r = \sqrt{a^2 + b^2} \)</p>
107
<p>\(r = \sqrt{a^2 + b^2} \)</p>
109
<p>= \(\sqrt{1^2 + 1^2} = \sqrt{2} \)</p>
108
<p>= \(\sqrt{1^2 + 1^2} = \sqrt{2} \)</p>
110
<p>\(\Theta = \tan^{-1}\left(\frac{b}{a}\right) \)</p>
109
<p>\(\Theta = \tan^{-1}\left(\frac{b}{a}\right) \)</p>
111
<p>= \(\tan^{-1}\left(\frac{1}{1}\right) = 45^\circ \)</p>
110
<p>= \(\tan^{-1}\left(\frac{1}{1}\right) = 45^\circ \)</p>
112
<p>Therefore, \(1 + i = \sqrt{2} \big(\cos 45^\circ + i \sin 45^\circ\big) \)</p>
111
<p>Therefore, \(1 + i = \sqrt{2} \big(\cos 45^\circ + i \sin 45^\circ\big) \)</p>
113
<p>For 1 - i, </p>
112
<p>For 1 - i, </p>
114
<p>\(r = \sqrt{1^2 + (-1)^2} \)</p>
113
<p>\(r = \sqrt{1^2 + (-1)^2} \)</p>
115
<p>=√2</p>
114
<p>=√2</p>
116
<p>\(\Theta = \tan^{-1}\left(\frac{-1}{1}\right) = -45^\circ \)</p>
115
<p>\(\Theta = \tan^{-1}\left(\frac{-1}{1}\right) = -45^\circ \)</p>
117
<p>Therefore,\(1 - i = \sqrt{2} \big(\cos(-45^\circ) + i \sin(-45^\circ)\big) \)</p>
116
<p>Therefore,\(1 - i = \sqrt{2} \big(\cos(-45^\circ) + i \sin(-45^\circ)\big) \)</p>
118
<p>Here, \(r_1 r_2 = \sqrt{2} \times \sqrt{2} = 2 \)</p>
117
<p>Here, \(r_1 r_2 = \sqrt{2} \times \sqrt{2} = 2 \)</p>
119
<p>\(\Theta_1 + \Theta_2 = 45^\circ + (-45^\circ) = 0^\circ \)</p>
118
<p>\(\Theta_1 + \Theta_2 = 45^\circ + (-45^\circ) = 0^\circ \)</p>
120
<p>\((1 + i)(1 - i) = 2 \big(\cos 0^\circ + i \sin 0^\circ\big) = 2 \).</p>
119
<p>\((1 + i)(1 - i) = 2 \big(\cos 0^\circ + i \sin 0^\circ\big) = 2 \).</p>
121
<p>Well explained 👍</p>
120
<p>Well explained 👍</p>
122
<h3>Problem 3</h3>
121
<h3>Problem 3</h3>
123
<p>Find the product of √5 (cos15° + i sin15°) and √5 (cos 30° + i sin30°)</p>
122
<p>Find the product of √5 (cos15° + i sin15°) and √5 (cos 30° + i sin30°)</p>
124
<p>Okay, lets begin</p>
123
<p>Okay, lets begin</p>
125
<p>The product of \(\sqrt{5} \big(\cos 15^\circ + i \sin 15^\circ\big) \) and \(\sqrt{5} \big(\cos 30^\circ + i \sin 30^\circ\big) \) = \(5 \big(\cos 45^\circ + i \sin 45^\circ\big).\)</p>
124
<p>The product of \(\sqrt{5} \big(\cos 15^\circ + i \sin 15^\circ\big) \) and \(\sqrt{5} \big(\cos 30^\circ + i \sin 30^\circ\big) \) = \(5 \big(\cos 45^\circ + i \sin 45^\circ\big).\)</p>
126
<h3>Explanation</h3>
125
<h3>Explanation</h3>
127
<p>When multiplying complex numbers in polar form, we first multiply the moduli and then add the arguments. </p>
126
<p>When multiplying complex numbers in polar form, we first multiply the moduli and then add the arguments. </p>
128
<p>Here, \(r_1 = \sqrt{5}, \quad r_2 = \sqrt{5} \)</p>
127
<p>Here, \(r_1 = \sqrt{5}, \quad r_2 = \sqrt{5} \)</p>
129
<p>\(\Theta_1 = 15^\circ, \quad \Theta_2 = 30^\circ \)</p>
128
<p>\(\Theta_1 = 15^\circ, \quad \Theta_2 = 30^\circ \)</p>
130
<p>\(r_1 r_2 = \sqrt{5} \times \sqrt{5} = 5 \)</p>
129
<p>\(r_1 r_2 = \sqrt{5} \times \sqrt{5} = 5 \)</p>
131
<p>\(\Theta_1 + \Theta_2 = 15^\circ + 30^\circ = 45^\circ \)</p>
130
<p>\(\Theta_1 + \Theta_2 = 15^\circ + 30^\circ = 45^\circ \)</p>
132
<p>Therefore,\(\sqrt{5} \big(\cos 15^\circ + i \sin 15^\circ\big) \text{ and } \sqrt{5} \big(\cos 30^\circ + i \sin 30^\circ\big) = 5 \big(\cos 45^\circ + i \sin 45^\circ\big).\)</p>
131
<p>Therefore,\(\sqrt{5} \big(\cos 15^\circ + i \sin 15^\circ\big) \text{ and } \sqrt{5} \big(\cos 30^\circ + i \sin 30^\circ\big) = 5 \big(\cos 45^\circ + i \sin 45^\circ\big).\)</p>
133
<p>Well explained 👍</p>
132
<p>Well explained 👍</p>
134
<h3>Problem 4</h3>
133
<h3>Problem 4</h3>
135
<p>Find the product of (3 -4i) and (1 +2i)</p>
134
<p>Find the product of (3 -4i) and (1 +2i)</p>
136
<p>Okay, lets begin</p>
135
<p>Okay, lets begin</p>
137
<p>The product of (3 -4i) and (1 +2i) is \(11 + 2i.\)</p>
136
<p>The product of (3 -4i) and (1 +2i) is \(11 + 2i.\)</p>
138
<h3>Explanation</h3>
137
<h3>Explanation</h3>
139
<p>To multiply the complex number, we use the formula;</p>
138
<p>To multiply the complex number, we use the formula;</p>
140
<p>\((a + ib) (c + id) = (ac - bd) + i(ad + bc)\). </p>
139
<p>\((a + ib) (c + id) = (ac - bd) + i(ad + bc)\). </p>
141
<p>Here, a = 3, b = -4, c =1, d = 2</p>
140
<p>Here, a = 3, b = -4, c =1, d = 2</p>
142
<p>\((ac - bd) = ((3 × 1) - (-4 × 2)) = 3 + 8 = 11\)</p>
141
<p>\((ac - bd) = ((3 × 1) - (-4 × 2)) = 3 + 8 = 11\)</p>
143
<p>\((ad + bc) = ((3 × 2) + (-4 × 1)) = 6 - 4 = 2\)</p>
142
<p>\((ad + bc) = ((3 × 2) + (-4 × 1)) = 6 - 4 = 2\)</p>
144
<p>Therefore, the product of (3 -4i) and (1 +2i) is \(11 + 2i.\)</p>
143
<p>Therefore, the product of (3 -4i) and (1 +2i) is \(11 + 2i.\)</p>
145
<p>Well explained 👍</p>
144
<p>Well explained 👍</p>
146
<h3>Problem 5</h3>
145
<h3>Problem 5</h3>
147
<p>Find the square of the complex number z = 3 + 4i</p>
146
<p>Find the square of the complex number z = 3 + 4i</p>
148
<p>Okay, lets begin</p>
147
<p>Okay, lets begin</p>
149
<p>\((3 + 4i)^2 = -7 + 24i .\)</p>
148
<p>\((3 + 4i)^2 = -7 + 24i .\)</p>
150
<h3>Explanation</h3>
149
<h3>Explanation</h3>
151
<p>\((a + ib)^2 = (a^2 - b^2) + i (2ab) \)</p>
150
<p>\((a + ib)^2 = (a^2 - b^2) + i (2ab) \)</p>
152
<p>= \((3^2 - (4i)^2) + i (2 \times 3 \times 4) \)</p>
151
<p>= \((3^2 - (4i)^2) + i (2 \times 3 \times 4) \)</p>
153
<p>= \((9 - 16) + i (24) \)</p>
152
<p>= \((9 - 16) + i (24) \)</p>
154
<p>= \(-7 + 24i.\)</p>
153
<p>= \(-7 + 24i.\)</p>
155
<p>Well explained 👍</p>
154
<p>Well explained 👍</p>
156
<h2>FAQs on Multiplying Complex Numbers</h2>
155
<h2>FAQs on Multiplying Complex Numbers</h2>
157
<h3>1.What is a complex number?</h3>
156
<h3>1.What is a complex number?</h3>
158
<p>The complex number is a number that has a real and an imaginary part. It can be written in the form a + ib.</p>
157
<p>The complex number is a number that has a real and an imaginary part. It can be written in the form a + ib.</p>
159
<h3>2.What is the multiplication of complex numbers formula?</h3>
158
<h3>2.What is the multiplication of complex numbers formula?</h3>
160
<p>The formula to multiply complex numbers, a + ib and c + id is (a + ib)(c + id) = (ac -bd) + i(ad + bc) </p>
159
<p>The formula to multiply complex numbers, a + ib and c + id is (a + ib)(c + id) = (ac -bd) + i(ad + bc) </p>
161
<h3>3.What is the formula to multiply complex numbers in polar form?</h3>
160
<h3>3.What is the formula to multiply complex numbers in polar form?</h3>
162
<p>The formula to multiply the complex numbers in polar form is r1r2(cos(Θ1 + Θ2) + sin(Θ1 + Θ2))</p>
161
<p>The formula to multiply the complex numbers in polar form is r1r2(cos(Θ1 + Θ2) + sin(Θ1 + Θ2))</p>
163
<h3>4.What is an argument in polar form?</h3>
162
<h3>4.What is an argument in polar form?</h3>
164
<p>The argument is the angle between the positive real axis and the line connecting the origin. The argument can be calculated using the formula, Θ = tan-1(b/a)</p>
163
<p>The argument is the angle between the positive real axis and the line connecting the origin. The argument can be calculated using the formula, Θ = tan-1(b/a)</p>
165
<h3>5.What is the value of i²?</h3>
164
<h3>5.What is the value of i²?</h3>
166
<h3>6.What real-life applications can I explain to my child?</h3>
165
<h3>6.What real-life applications can I explain to my child?</h3>
167
<p>Applications include calculating AC circuit currents, analyzing signals in communication systems, representing rotations in computer graphics, and understanding transformations in quantum mechanics.</p>
166
<p>Applications include calculating AC circuit currents, analyzing signals in communication systems, representing rotations in computer graphics, and understanding transformations in quantum mechanics.</p>
168
<h3>7.How can I help my child practice multiplying complex numbers at home?</h3>
167
<h3>7.How can I help my child practice multiplying complex numbers at home?</h3>
169
<p>Encourage them to solve step-by-step using the formula \((a+ib)(c+id)=(ac-bd)+i(ad+bc)\), start with simple numbers, then move to larger numbers or polar forms.</p>
168
<p>Encourage them to solve step-by-step using the formula \((a+ib)(c+id)=(ac-bd)+i(ad+bc)\), start with simple numbers, then move to larger numbers or polar forms.</p>
170
<h3>8.At what stage should my child learn to multiply complex numbers?</h3>
169
<h3>8.At what stage should my child learn to multiply complex numbers?</h3>
171
<p>Typically, students learn about complex numbers in higher secondary (grades 11-12) or during early college courses in mathematics, physics, or engineering.</p>
170
<p>Typically, students learn about complex numbers in higher secondary (grades 11-12) or during early college courses in mathematics, physics, or engineering.</p>
172
<h2>Hiralee Lalitkumar Makwana</h2>
171
<h2>Hiralee Lalitkumar Makwana</h2>
173
<h3>About the Author</h3>
172
<h3>About the Author</h3>
174
<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>
173
<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>
175
<h3>Fun Fact</h3>
174
<h3>Fun Fact</h3>
176
<p>: She loves to read number jokes and games.</p>
175
<p>: She loves to read number jokes and games.</p>