4.48 Dividing Complex Numbers

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What is a complex number?

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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?

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The conjugate of a complex number a + bi is a - bi. It is used in division to eliminate the imaginary part from the denominator.

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CCSS.HSN.CN.A.3

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How do you multiply complex numbers?

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To multiply complex numbers, use the distributive property (FOIL method) and combine like terms. For example, (a + bi)(c + di) = ac + adi + bci + bdi^2.

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CCSS.HSN.CN.A.2

Front

What is the formula for dividing complex numbers?

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To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, to divide (a + bi) by (c + di), multiply by (c - di)/(c - di).

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CCSS.HSN.CN.A.3

Front

What is the result of multiplying a complex number by its conjugate?

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The result of multiplying a complex number a + bi by its conjugate a - bi is a^2 + b^2, which is a real number.

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CCSS.HSN.CN.C.8

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What is the significance of the denominator in complex division?

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The denominator in complex division must be a real number after multiplying by the conjugate, allowing for simplification of the expression.

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CCSS.HSN.CN.A.3

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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 and simplify the resulting expression.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, and it is used to express complex numbers.

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CCSS.HSN.CN.A.1

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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

Front

How do you add complex numbers?

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To add complex numbers, add their real parts and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

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CCSS.HSN.CN.A.2