Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

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Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 9

Capitolo 5, Problema 9

Perform the indicated operations and write the result in standard form. (−2 + √−100)²

Guida verificata passo dopo passo

Recognize that the expression involves a complex number because of the square root of a negative number: $\sqrt{-100}$. Recall that $\sqrt{-1} = i$, where $i$ is the imaginary unit.

Rewrite $\sqrt{-100}$ as $\sqrt{100} \times \sqrt{-1} = 10i$. So the expression becomes $(-2 + 10i)^2$.

Use the formula for squaring a binomial: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = -2$ and $b = 10i$.

Calculate each term separately: $a^2 = (-2)^2$, $2ab = 2 \times (-2) \times 10i$, and $b^2 = (10i)^2$.

Combine the results and simplify, remembering that $i^2 = -1$, to write the final expression in standard form $a + bi$, where $a$ and $b$ are real numbers.

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Complex Numbers and Imaginary Unit

Complex numbers consist of a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to interpret and manipulate expressions involving √-100 requires recognizing that √-100 = 10i.

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Introduction to Complex Numbers

Operations with Complex Numbers

Performing operations like addition, multiplication, and exponentiation on complex numbers follows algebraic rules, treating i as a variable but applying i² = -1 to simplify. Squaring a complex number involves expanding the binomial and simplifying using this property.

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Dividing Complex Numbers

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a and b are real numbers. After performing operations, the result should be simplified and expressed clearly in this form, separating the real and imaginary parts.

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Complex Numbers In Polar Form

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