Learn about complex conjugates and how to identify the real and imaginary parts of complex numbers! This video covers several examples, showing you step-by-step how to find conjugates and separate complex numbers into their real and imaginary components. Perfect for complex number beginners!
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𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Find the real and imaginary parts of complex numbers 2+5i, √2i+7, 5i, -7, (7-5i)/4, (2-3i)²
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
A complex number is generally expressed in the form a + bi, where 'a' is the real part (Re(z)) and 'b' is the imaginary part (Im(z)).
1. Complex number 2 + 5i:
Real part (a) = 2
Imaginary part (b) = 5
2. Complex number √2i + 7:
Rewrite in the form a + bi: 7 + √2i
Real part (a) = 7
Imaginary part (b) = √2
3. Complex number 5i:
Rewrite in the form a + bi: 0 + 5i
Real part (a) = 0
Imaginary part (b) = 5
4. Complex number -7:
Rewrite in the form a + bi: -7 + 0i
Real part (a) = -7
Imaginary part (b) = 0
5. Complex number (7 - 5i) / 4:
Rewrite in the form a + bi: 7/4 - (5/4)i
Real part (a) = 7/4
Imaginary part (b) = -5/4
6. Complex number (2 - 3i)²:
First, expand the expression:
(2 - 3i)² = 2² - 2(2)(3i) + (3i)²
= 4 - 12i + 9i²
Recall that i² = -1:
= 4 - 12i + 9(-1)
= 4 - 12i - 9
= -5 - 12i
Now identify the real and imaginary parts:
Real part (a) = -5
Imaginary part (b) = -12