Operations with Complex Numbers

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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 i^2 = -1.

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CCSS.HSN.CN.A.1

Front

What is the imaginary unit?

Back

The imaginary unit, denoted as i, is defined as the square root of -1. It is used to extend the real number system to include solutions to equations that do not have real solutions.

Tags

CCSS.HSN.CN.A.1

Front

How do you add two complex numbers?

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To add two 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

Front

What is the result of (3 + 5i) + (4 - 5i)?

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The result is 7 + 0i, which simplifies to 7.

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CCSS.HSN.CN.A.2

Front

How do you subtract two complex numbers?

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To subtract two complex numbers, subtract 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

Front

What is the product of two complex conjugates?

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The product of two complex conjugates (a + bi)(a - bi) is a^2 + b^2, which is a real number.

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CCSS.HSN.CN.C.8

Front

What is the result of (3 + 5i)(3 - 5i)?

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The result is 34.

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CCSS.HSN.CN.C.8

Front

How do you multiply two complex numbers?

Back

To multiply two complex numbers, use the distributive property (FOIL method): (a + bi)(c + di) = ac + adi + bci + bdi^2, and simplify using i^2 = -1.

Tags

CCSS.HSN.CN.A.2

Front

What is the simplified form of (-9 + 5i) + (3 - 2i)?

Back

The simplified form is -6 + 3i.

Tags

CCSS.HSN.CN.A.2

Front

What is the modulus of a complex number?

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The modulus of a complex number a + bi is given by |a + bi| = √(a^2 + b^2). It represents the distance from the origin to the point (a, b) in the complex plane.