Title: Course Presentation

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Addition of Complex Numbers in Algebraical Form

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The addition of the complex numbers \(z_{1} = a+bi\) and \(z_{2} = c+di\;\) is done as follows:

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To get the sum of two (or more) complex numbers, the real and imaginary parts must be added separately.

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

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Example 1.

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Find the sum \(z_{1} = 3-18i\) and \(z_{2} = -4+2i\).

\(z_{1}+z_{2} =(3-18i)+(-4+2i)= (3-4)+(-18+2)i = -1-16i\)

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Just like with real numbers, the sum of a complex number and its opposite is equal to zero.

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Example 2.

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

\((1-7i)+(-1+7i)=0\)

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Example 3.

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The co-complex number of \(1-7i\) is \(1+7i\). Therefore

\((1-7i)+(1+7i)=(1+1)+(-7+7)i = 2\)

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The notation \(z +\overline{z}= 2a\) means that the sum of a complex number and its co-complex number is equal to twice the real part of the complex number.

Complex numbers Fill in the Blanks

Calculate \(z_{1}+z_{2}.\)

Q: 1) If \(z_{1} = 3+6i\)  and \(z_{2} = -5+12i\), then \(z_1+z_2=\)[...].

2) If \(z_1=3-7i\) and \(z_2=-5+i\) , then \(z_1+z_2 = \) [...].
A: -2+18i/18i-2; -2-6i/-6i-2
Feedback (0 to 100%): Solution: 1) \((3+6i)+(-5+12i) = 3+6i-5+12i = -2+18i \); 2)\((3-7i)+(-5+i) = 3-7i-5+i = -2-6i\)

Complex numbers Fill in the Blanks

Calculate \((42-3i)+(-2-4i)\)

Q: Answer: [...].
A: 40-7i/-7i+40
Feedback (0 to 100%): Solution: \((42-3i)+(-2-4i) = 42-3i-2-4i = 40-7i\)

Addition Fill in the Blanks

Find the sum of the number \(z\) and its co-complex number if \(z=3-7i\).

Q: \(z+\overline{z} = \) [...]
A: 6
Feedback (0 to 100%): Solution: \( \overline{z} = 3+7i; z+\overline{z} = (3-7i)+(3+7i) = 3-7i+3+7i = 6\)

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Practice