4.4 Operations on Complex Numbers

Front

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

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as i = √(-1). It is used to represent the square root of negative numbers.

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CCSS.HSN.CN.A.1

Front

How do you add complex numbers?

Back

To add complex numbers, combine their real parts and their imaginary parts separately. For example, (-9 + 5i) + (3 - 2i) = (-9 + 3) + (5i - 2i) = -6 + 3i.

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CCSS.HSN.CN.A.2

Front

What is the formula for multiplying two complex numbers?

Back

To multiply two complex numbers (a + bi)(c + di), use the distributive property: ac + adi + bci + bdi², and remember that i² = -1.

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CCSS.HSN.CN.A.2

Front

What is the result of (11 + 5i)(11 - 5i)?

Back

The result is 146, calculated as (11 + 5i)(11 - 5i) = 11² - (5i)² = 121 - (-25) = 121 + 25 = 146.

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CCSS.HSN.CN.C.8

Front

How do you multiply a complex number by a real number?

Back

To multiply a complex number by a real number, multiply the real part and the imaginary part by that real number. For example, 3i(4 - i) = 3i * 4 - 3i * i = 12i + 3 = 3 + 12i.

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CCSS.HSN.CN.A.2

Front

What is the value of i^2?

Back

The value of i^2 is -1.

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CCSS.HSN.CN.A.1

Front

What is the pattern of powers of i?

Back

The powers of i cycle every four: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then it repeats.

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CCSS.HSN.CN.A.2

Front

What is the simplified form of i^13?

Back

The simplified form of i^13 is i, as 13 mod 4 = 1.

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CCSS.HSN.CN.A.2

Front

How do you square a complex number?

Back

To square a complex number (a + bi)², use the formula: (a + bi)² = a² + 2abi + (bi)² = a² - b² + 2abi.

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CCSS.HSN.CN.C.8