Complex Number Operations

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 the square root of -1, and it is used to extend the real number system to include complex numbers.

Tags

CCSS.HSN.CN.A.1

Front

How do you simplify powers of 'i'?

Back

Powers of 'i' cycle through four values: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. For any integer n, i^n can be simplified by finding n mod 4.

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CCSS.HSN.CN.A.2

Front

What is the sum of two complex numbers (a + bi) and (c + di)?

Back

The sum is (a + c) + (b + d)i, where you add the real parts and the imaginary parts separately.

Tags

CCSS.HSN.CN.A.2

Front

How do you subtract complex numbers?

Back

To subtract complex numbers (a + bi) - (c + di), subtract the real parts and the imaginary parts: (a - c) + (b - d)i.

Tags

CCSS.HSN.CN.A.2

Front

What is the product of two complex numbers (a + bi) and (c + di)?

Back

The product is (ac - bd) + (ad + bc)i, obtained by using the distributive property and combining like terms.

Tags

CCSS.HSN.CN.A.2

Front

What is the conjugate of a complex number a + bi?

Back

The conjugate of a complex number a + bi is a - bi. It is used to simplify division of complex numbers.

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CCSS.HSN.CN.A.3

Front

How do you divide complex numbers?

Back

To divide (a + bi) by (c + di), multiply the numerator and denominator by the conjugate of the denominator: (a + bi)(c - di) / (c^2 + d^2).

Tags

CCSS.HSN.CN.A.3

Front

What is the modulus of a complex number a + bi?

Back

The modulus (or absolute value) of a complex number a + bi is given by √(a^2 + b^2). It represents the distance from the origin in the complex plane.

Front

What is the argument of a complex number?

Back

The argument of a complex number a + bi is the angle θ formed with the positive real axis, calculated as θ = arctan(b/a).