Simplifying Radicals and Complex Numbers

Front

Define 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 = √(-1).

Front

What is the imaginary unit 'i'?

Back

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

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Front

How do you multiply two complex numbers (a + bi)(c + di)?

Back

Use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i.

Front

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

Back

The conjugate of a complex number a + bi is a - bi.

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

Back

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

Front

Simplify the expression: (2 + 3i)².

Back

(2 + 3i)² = 4 + 12i + 9i² = 4 + 12i - 9 = -5 + 12i.

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What is the result of multiplying two conjugate complex numbers (a + bi)(a - bi)?

Back

The result is a² + b², which is a real number.