Complex Number Division:
by Dividing complex numbers,
1. Write down the problem in fractional form,
2. Reduce the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.
3. Keep in mind that conjugate time its conjugate will give a real number.
4. This process will eliminate the I from the denominator.
The conjugate of a + bi is the complex number a – bi.
Dividing using the complex number:
Divide Complex Number – Example 1:
Solve
((4 + 2i) / (3-i)) = ((4+2i) / (3-i)) * ((3 + i) / (3 + i))
Solution:
Step 1:
= ((12+4i+6i+2i^2) / (9+3i-3i-i^2))
Step 2:
= ((12+10i+2(-1)) / (9 – (-1))
Step 3:
= (10 + 10i) / (10) = (1 + i) / (1)
Answer:
= 1+i
Divide Complex Number – Example 2:
(3+4i) / (5-i)The bottom: (5-i) (5+i) = 25+1=26. Step 1: (11+23i)/26 = 11/26 +23i/26. (3+4i) / (4-i)The bottom: (4-i) (4+i) = 16+1=17. Step 2:(8+19i)/17 = 8/17 +19i/17.The bottom: (4-i) (4+i) = 16+1=17. Step 3:(9+15i)/17 = 9/17 +15i/17.The bottom: (4-i) (4+i) = 16+1=17. Step 3:(5+14i)/17 = 5/17 +14i/17.
Solution:
Step 1:
Write down the complex conjugate of the number on the bottom. That’s 5+i.
= ((3+4i) / (5-i)) * ((5+i) / (5+i))
Step 2:
Multiply both top and bottom by that number.
The top: (3+4i) (5+i) = 15+3i+20i-4=11+23i.
Step 4:
Carry out the division, now that the bottom is real.
Answer:
0.423+ 0.884i
Divide Complex Number – Example 3:
Solution:
Step 1:
Write down the complex conjugate of the number on the bottom. That’s 4+i.
= ((3+4i) / (4-i)) * ((4+i) / (4+i))
Step 2:
Multiply both top and bottom by that number.
The top: (3+4i) (4+i) = 12+3i+16i-4=8+19i.
Step 4:
Carry out the division, now that the bottom is real.
Answer:
0.470 + 1.117 i
Divide Complex Number – Example 4:
((3+3i) / (4-i))
Step 1:
Write down the complex conjugate of the number on the bottom. That’s 4+i.
= ((3+3i) / (4-i)) * ((4+i) / (4+i))
Step 2:
Multiply both top and bottom by that number.
The top: (3+3i) (4+i) = 12+3i+12i-3= 9+15i
Step 3:
Carry out the division, now that the bottom is real.
Answer:
0.529 + 0.882i
Divide Complex Number – Example 5:
((2+3i) / (4-i))
Step 1:
Write down the complex conjugate of the number on the bottom. That’s 4+i.
= ((2+3i) / (4-i)) * ((4+i) / (4+i))
Step 2:
Multiply both top and bottom by that number.
The top: (2+3i) (4+i) = 8+2i+12i-3= 5+14i
Step 3:
Carry out the division, now that the bottom is real.
Answer:
0.294 + 0.823i
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