Dividing Complex Numbers with conjugates

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What is a complex number?

Back

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

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CCSS.HSN.CN.A.1

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What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part.

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CCSS.HSN.CN.A.3

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Why do we multiply by the conjugate when dividing complex numbers?

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Multiplying by the conjugate helps eliminate the imaginary part from the denominator, making it a real number.

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CCSS.HSN.CN.A.3

Front

What is the formula for multiplying two complex numbers?

Back

If z1 = a + bi and z2 = c + di, then z1 * z2 = (ac - bd) + (ad + bc)i.

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CCSS.HSN.CN.A.2

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How do you simplify a complex fraction?

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To simplify a complex fraction, multiply the numerator and denominator by the conjugate of the denominator.

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What is the result of multiplying (2 + 3i) by its conjugate?

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The result is 2^2 + 3^2 = 4 + 9 = 13.

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CCSS.HSN.CN.C.8

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What is the denominator after multiplying (1 + 2i) by its conjugate (1 - 2i)?

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The denominator is 1^2 + (2)^2 = 1 + 4 = 5.

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CCSS.HSN.CN.A.3

Front

What is the numerator after multiplying (2 - 3i) by (1 + 2i)?

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The numerator is (2)(1) + (2)(-3i) + (-3i)(1) + (-3i)(2i) = 2 - 6i + 3 + 6 = 5 - 3i.

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CCSS.HSN.CN.A.2

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What is the standard form of a complex number?

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The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part.

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CCSS.HSN.CN.A.1

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How do you find the modulus of a complex number?

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The modulus of a complex number a + bi is given by √(a^2 + b^2).