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Chapter 2: Problem 7

Find real numbers \(a\) and \(b\) such that the equation is true. $$a+b i=-12+7 i$$

Short Answer

Expert verified
The real numbers \(a\) and \(b\) that make the equation \(a + b i = -12 + 7 i\) true are \(a = -12\) and \(b = 7\).

Step by step solution

01

Analyze Complex Numbers

A complex number is expressed in the form \(a + b i\), where \(a\) and \(b\) represent real numbers, and \(i\) is the imaginary unit. In the equation \(a + b i = -12 + 7 i\), \(a\) and \(b\) are unknown real numbers, and -12 and 7 are their corresponding real numbers.
02

Equate Real Parts on Both Sides

Both sides of the equation must be equal to each other. Hence the real part \(a\) on the left side should be equal to real part -12 on the right side. So we have \(a = -12\). This gives the first solution for the equation.
03

Equate Imaginary Parts on Both Sides

Similarly from above step, compare the imaginary parts on both sides. Hence \(b i\) should be equal to \(7 i\), which gives the equation \(b = 7\). So, the second real number is 7.

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Most popular questions from this chapter

Sketch the graph of each polynomial function. Then count the number of real zeros of the function and the numbers of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe? (a) \(f(x)=-x^{3}+9 x\) (b) \(f(x)=x^{4}-10 x^{2}+9\) (c) \(f(x)=x^{5}-16 x\)

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