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.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit '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.

Front

How do you add complex numbers?

Back

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

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How do you subtract complex numbers?

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

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What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by the formula |a + bi| = √(a² + b²). It represents the distance of the complex number from the origin in the complex plane.

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What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part.

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How do you multiply complex numbers?

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To multiply complex numbers, use the distributive property (FOIL method). For example, (a + bi)(c + di) = ac + adi + bci + bdi², and since i² = -1, combine like terms.

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What is the significance of the end behavior of a function involving complex numbers?

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The end behavior of a function describes how the function behaves as the input approaches positive or negative infinity. It helps in understanding the limits and asymptotic behavior of complex functions.

Front

What is the solution to the equation x² = 16?

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The solutions to the equation x² = 16 are x = ±4, which means x can be either 4 or -4.

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What does it mean for a complex number to be in standard form?

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A complex number is in standard form when it is expressed as a + bi, where a is the real part and b is the imaginary part.