Determine whether each statement is true or false. If false, explain why. The product of a complex number and its conjugate is always a real number.

검증된 단계별 안내

Recall the definition of a complex number: it is of the form $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit with $i^2 = -1$.

The conjugate of a complex number $z = a + bi$ is given by $\overline{z} = a - bi$.

To find the product of a complex number and its conjugate, multiply $z$ and $\overline{z}$: $z \times \overline{z} = (a + bi)(a - bi)$.

Use the distributive property (FOIL) to expand the product: $(a + bi)(a - bi) = a^2 - abi + abi - b^2 i^2$.

Simplify the expression by combining like terms and using $i^2 = -1$: the middle terms cancel out, and $-b^2 i^2$ becomes $+b^2$, so the product is $a^2 + b^2$, which is a real number.

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주요 개념

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Complex Numbers

A complex number is a number in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. Complex numbers extend the real number system and are used to represent quantities with both real and imaginary parts.

Complex Conjugate

The complex conjugate of a complex number a + bi is a - bi. It reflects the number across the real axis in the complex plane, changing the sign of the imaginary part while keeping the real part the same.

Product of a Complex Number and Its Conjugate

Multiplying a complex number by its conjugate results in a real number equal to a² + b², which is the sum of the squares of the real and imaginary parts. This product is always non-negative and has no imaginary component.

관련 실천

교과서 질문

Use the graph to solve each equation or inequality. Use interval notation where appropriate. 2(x-2) / {(x-1)(x-3)} < 0

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교과서 질문

Solve each problem. During the course of a year, the number of volunteers available to run a food bank each month is modeled by $V (x)$ , where $V (x) = 2 x^{2} - 32 x + 150$ between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, $V (x)$ is modeled by $V (x) = 31 x - 226$ . Find the number of volunteers in each of the following months. Sketch a graph of $y = V (x)$ for January through December. In what month are the fewest volunteers available?