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Dividing Complex Numbers — Easy Worksheet 1
Grade 11 · MathSheets Name: ____________________ Date: __________
Simplify. Write each answer in the form a + bi.
How to solve
Multiply the numerator and the denominator by the conjugate of the denominator (the conjugate of c + di is c - di). The denominator becomes c2 + d2, a real number, because i2 = -1. Then divide each part of the numerator by it.
Example
(1 + 2i)(1 - 2i) = ___
Answer: 5
Not quite. The answer is
💡 A number times its conjugate is a difference of squares: (a + bi)(a − bi) = a2 − (bi)2. (1 - 2i)(1 + 2i) = 12 − (-2i)2 = 1 − 4i2 = 1 + 4 = 5, since i2 = −1.
✓ Correct!
Not quite. The answer is
💡 A number times its conjugate is a difference of squares: (a + bi)(a − bi) = a2 − (bi)2. (1 - 2i)(1 + 2i) = 12 − (-2i)2 = 1 − 4i2 = 1 + 4 = 5, since i2 = −1.
💡 How to solve: Dividing Complex Numbers
Multiply the numerator and the denominator by the conjugate of the denominator (the conjugate of c + di is c - di). The denominator becomes c2 + d2, a real number, because i2 = -1. Then divide each part of the numerator by it.
Try a similar one
(-3 - 3i)(-3 + 3i) = ___
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 A number times its conjugate is a difference of squares: (a + bi)(a − bi) = a2 − (bi)2. (2 - 3i)(2 + 3i) = 22 − (-3i)2 = 4 − 9i2 = 4 + 9 = 13, since i2 = −1.
✓ Correct!
Not quite. The answer is
💡 A number times its conjugate is a difference of squares: (a + bi)(a − bi) = a2 − (bi)2. (2 - 3i)(2 + 3i) = 22 − (-3i)2 = 4 − 9i2 = 4 + 9 = 13, since i2 = −1.
💡 How to solve: Dividing Complex Numbers
Multiply the numerator and the denominator by the conjugate of the denominator (the conjugate of c + di is c - di). The denominator becomes c2 + d2, a real number, because i2 = -1. Then divide each part of the numerator by it.
Try a similar one
(1 + 3i)(1 - 3i) = ___
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The conjugate of 4i is -4i. Multiply the top and the bottom by it. Bottom: (4i)(-4i) = -16i2 = 16, since i2 = −1. Top: (-4 + 4i)(-4i) = 16i − 16i2. Since i2 = −1, -16i2 = 16. Real part: 0 + 16 = 16; imaginary part: 16 + 0 = 16. So the product is 16 + 16i. Divide each part by 16: 1616 = 1 and 1616 = 1, so the answer is 1 + i.
✓ Correct!
Not quite. The answer is
💡 The conjugate of 4i is -4i. Multiply the top and the bottom by it. Bottom: (4i)(-4i) = -16i2 = 16, since i2 = −1. Top: (-4 + 4i)(-4i) = 16i − 16i2. Since i2 = −1, -16i2 = 16. Real part: 0 + 16 = 16; imaginary part: 16 + 0 = 16. So the product is 16 + 16i. Divide each part by 16: 1616 = 1 and 1616 = 1, so the answer is 1 + i.
💡 How to solve: Dividing Complex Numbers
Multiply the numerator and the denominator by the conjugate of the denominator (the conjugate of c + di is c - di). The denominator becomes c2 + d2, a real number, because i2 = -1. Then divide each part of the numerator by it.
Try a similar one
(-1 - 3i)(-1 + 3i) = ___
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The conjugate of 4i is -4i. Multiply the top and the bottom by it. Bottom: (4i)(-4i) = -16i2 = 16, since i2 = −1. Top: (12 - 12i)(-4i) = -48i + 48i2. Since i2 = −1, 48i2 = -48. Real part: 0 − 48 = -48; imaginary part: -48 + 0 = -48. So the product is -48 - 48i. Divide each part by 16: -4816 = -3 and -4816 = -3, so the answer is -3 - 3i.
✓ Correct!
Not quite. The answer is
💡 The conjugate of 4i is -4i. Multiply the top and the bottom by it. Bottom: (4i)(-4i) = -16i2 = 16, since i2 = −1. Top: (12 - 12i)(-4i) = -48i + 48i2. Since i2 = −1, 48i2 = -48. Real part: 0 − 48 = -48; imaginary part: -48 + 0 = -48. So the product is -48 - 48i. Divide each part by 16: -4816 = -3 and -4816 = -3, so the answer is -3 - 3i.
💡 How to solve: Dividing Complex Numbers
Multiply the numerator and the denominator by the conjugate of the denominator (the conjugate of c + di is c - di). The denominator becomes c2 + d2, a real number, because i2 = -1. Then divide each part of the numerator by it.
Try a similar one
(1 - i)(1 + i) = ___
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The conjugate of 4i is -4i. Multiply the top and the bottom by it. Bottom: (4i)(-4i) = -16i2 = 16, since i2 = −1. Top: (12 + 4i)(-4i) = -48i − 16i2. Since i2 = −1, -16i2 = 16. Real part: 0 + 16 = 16; imaginary part: -48 + 0 = -48. So the product is 16 - 48i. Divide each part by 16: 1616 = 1 and -4816 = -3, so the answer is 1 - 3i.
✓ Correct!
Not quite. The answer is
💡 The conjugate of 4i is -4i. Multiply the top and the bottom by it. Bottom: (4i)(-4i) = -16i2 = 16, since i2 = −1. Top: (12 + 4i)(-4i) = -48i − 16i2. Since i2 = −1, -16i2 = 16. Real part: 0 + 16 = 16; imaginary part: -48 + 0 = -48. So the product is 16 - 48i. Divide each part by 16: 1616 = 1 and -4816 = -3, so the answer is 1 - 3i.
💡 How to solve: Dividing Complex Numbers
Multiply the numerator and the denominator by the conjugate of the denominator (the conjugate of c + di is c - di). The denominator becomes c2 + d2, a real number, because i2 = -1. Then divide each part of the numerator by it.
Try a similar one
(3 + 3i)(3 - 3i) = ___
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Multiply the top and the bottom by -2 + i, the conjugate of the denominator. Bottom: (-2 - i)(-2 + i) = (-2)2 − (-i)2 = 4 − i2 = 4 + 1 = 5, since i2 = −1. Top: (1 + 3i)(-2 + i) = -2 + i − 6i + 3i2. Since i2 = −1, 3i2 = -3. Real part: -2 − 3 = -5; imaginary part: 1 − 6 = -5. So the product is -5 - 5i. Divide each part by 5: -55 = -1 and -55 = -1, so the answer is -1 - i.
✓ Correct!
Not quite. The answer is
💡 Multiply the top and the bottom by -2 + i, the conjugate of the denominator. Bottom: (-2 - i)(-2 + i) = (-2)2 − (-i)2 = 4 − i2 = 4 + 1 = 5, since i2 = −1. Top: (1 + 3i)(-2 + i) = -2 + i − 6i + 3i2. Since i2 = −1, 3i2 = -3. Real part: -2 − 3 = -5; imaginary part: 1 − 6 = -5. So the product is -5 - 5i. Divide each part by 5: -55 = -1 and -55 = -1, so the answer is -1 - i.
💡 How to solve: Dividing Complex Numbers
Multiply the numerator and the denominator by the conjugate of the denominator (the conjugate of c + di is c - di). The denominator becomes c2 + d2, a real number, because i2 = -1. Then divide each part of the numerator by it.
Try a similar one
(3 - 2i)(3 + 2i) = ___
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Multiply the top and the bottom by -1 - 2i, the conjugate of the denominator. Bottom: (-1 + 2i)(-1 - 2i) = (-1)2 − (2i)2 = 1 − 4i2 = 1 + 4 = 5, since i2 = −1. Top: (9 - 3i)(-1 - 2i) = -9 − 18i + 3i + 6i2. Since i2 = −1, 6i2 = -6. Real part: -9 − 6 = -15; imaginary part: -18 + 3 = -15. So the product is -15 - 15i. Divide each part by 5: -155 = -3 and -155 = -3, so the answer is -3 - 3i.
✓ Correct!
Not quite. The answer is
💡 Multiply the top and the bottom by -1 - 2i, the conjugate of the denominator. Bottom: (-1 + 2i)(-1 - 2i) = (-1)2 − (2i)2 = 1 − 4i2 = 1 + 4 = 5, since i2 = −1. Top: (9 - 3i)(-1 - 2i) = -9 − 18i + 3i + 6i2. Since i2 = −1, 6i2 = -6. Real part: -9 − 6 = -15; imaginary part: -18 + 3 = -15. So the product is -15 - 15i. Divide each part by 5: -155 = -3 and -155 = -3, so the answer is -3 - 3i.
💡 How to solve: Dividing Complex Numbers
Multiply the numerator and the denominator by the conjugate of the denominator (the conjugate of c + di is c - di). The denominator becomes c2 + d2, a real number, because i2 = -1. Then divide each part of the numerator by it.
Try a similar one
(1 - 2i)(1 + 2i) = ___
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Multiply the top and the bottom by 1 - 3i, the conjugate of the denominator. Bottom: (1 + 3i)(1 - 3i) = 12 − (3i)2 = 1 − 9i2 = 1 + 9 = 10, since i2 = −1. Top: (4 - 8i)(1 - 3i) = 4 − 12i − 8i + 24i2. Since i2 = −1, 24i2 = -24. Real part: 4 − 24 = -20; imaginary part: -12 − 8 = -20. So the product is -20 - 20i. Divide each part by 10: -2010 = -2 and -2010 = -2, so the answer is -2 - 2i.
✓ Correct!
Not quite. The answer is
💡 Multiply the top and the bottom by 1 - 3i, the conjugate of the denominator. Bottom: (1 + 3i)(1 - 3i) = 12 − (3i)2 = 1 − 9i2 = 1 + 9 = 10, since i2 = −1. Top: (4 - 8i)(1 - 3i) = 4 − 12i − 8i + 24i2. Since i2 = −1, 24i2 = -24. Real part: 4 − 24 = -20; imaginary part: -12 − 8 = -20. So the product is -20 - 20i. Divide each part by 10: -2010 = -2 and -2010 = -2, so the answer is -2 - 2i.
💡 How to solve: Dividing Complex Numbers
Multiply the numerator and the denominator by the conjugate of the denominator (the conjugate of c + di is c - di). The denominator becomes c2 + d2, a real number, because i2 = -1. Then divide each part of the numerator by it.
Try a similar one
(-3 - 3i)(-3 + 3i) = ___
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
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Dividing Complex Numbers — Easy Worksheet 1
Simplify. Write each answer in the form a + bi.