Title: Course Presentation

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Subtraction of Complex Numbers

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The subtraction of the complex numbers \(z_1=a+bi\) and \(z_2=c+di\) is done as follows:

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\(z_{1}-z_{2} =(a+bi)-(c+di)= (a-c)+(b-d)i\)

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To obtain the difference of two (or more) complex numbers, the real and imaginary parts must be subtracted.

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Example 1

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Find the difference between the complex numbers \(z_{1} = 3-18i\) and \(z_{2} = -4+2i\).

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\(z_{1}-z_{2} = (3-18i)-(-4+2i)=(3-(-4))+(-18-2)i = 7-20i\)

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Example 2

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Subtraction with the opposite number

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The opposite of \(1-7i\;\) is \(\;-1+7i\). Therefore \((1-7i)-(-1+7i)=2-14i\)

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Example 3

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Subtraction with a conjugate number

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The conjugate number to \(1-7i\) is \(1+7i\). Therefore

\((1-7i)-(1+7i)=(1-1)+(-7-7)i=14i\)

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Practice

Addition Fill in the Blanks

Find the difference between the complex numbers \(z_{1}\) and \(z_{2}\).

Q: 1) If \(z_{1} = 2-4i\) and \(z_{2} = 6+i\) then \(z_{1}-z_{2} = \) [...].

2) If \(z_{1} = -2+5i\) and \(z_{2} = -8-i\) then \(z_{1}-z_{2} = \) [...].
A: -4-5i/-5i-4; 6+6i/6i+6
Feedback (0 to 100%): Solution: \(1) (2-4i)-(6+i) = (2-6)+(-4i-i) = -4-5i\); \(2) (-2+5i)-(-8-i) = (-2-(-8))+(5i-(-i)) = (-2+8)+(5i+i) = 6+6i\)

Addition Fill in the Blanks

Calculate.

Q: \((12+i)-(-5-3i)= \)[...]
A: 17+4i/4i+17
Feedback (0 to 100%): Solution: \((12+i)-(-5-3i)= (12-(-5))+(i-(-3i)) =(12+5)+(i+3i)= 17+4i\)

Addition Fill in the Blanks

Find the difference between the number \(z\) and its conjugate number if \(z=-4-5i\).

Q: \(z-\overline{z} = \) [...]
A: -10i
Feedback (0 to 100%): \(\overline{z} = -4+5i; z-\overline{z} = (-4-5i)-(-4+5i) = (-4-(-4)) +(-5i-5i) = -10i\)