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Chapter 6: Problem 50

Suppose \(w\) and \(z\) are complex numbers such that the real part of \(w z\) equals the real part of \(w\) times the real part of \(z\). Explain why either \(w\) or \(z\) must be a real number.

Short Answer

Expert verified
Given the complex numbers \(w = a + bi\) and \(z = c + di\), where \(a, b, c,\) and \(d\) are real numbers, the real part of their product is \(Re(wz) = ac - bd\). Given that \(Re(wz) = Re(w) \cdot Re(z)\), we have \(ac - bd = a \cdot c\). This implies that either \(b = 0\) or \(d = 0\). If \(b = 0\), then \(w\) is a real number; if \(d = 0\), \(z\) is a real number. Thus, either \(w\) or \(z\) must be a real number.

Step by step solution

01

Express the complex numbers in general form

Let \(w = a + bi\) and \(z = c + di\), where \(a, b, c,\) and \(d\) are real numbers.
02

Multiply the complex numbers and find the real part

Now, multiply the complex numbers \(w\) and \(z\): \((a + bi)(c + di) = ac + adi + bci + bdi^2\) Since \(i^2 = -1\), we can write the product as: \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\). The real part of this product is: \(Re(wz) = ac - bd\).
03

Show the given relationship

According to the exercise, the real part of the product of w and z equals the product of their real parts, i.e., \(Re(wz) = Re(w) \cdot Re(z)\) Substitute the expressions for the real parts from Step 2: \(ac - bd = a \cdot c\) Now we'll solve for b or d.
04

Solve for one of the imaginary parts

Rearrange the equation: \(bd = ac - ac\) or \(bd = 0\) This implies that either \(b = 0\) or \(d = 0\).
05

Conclusion

If \(b = 0\), then \(w = a\), which means \(w\) is a real number. If \(d = 0\), then \(z = c\), which means that \(z\) is a real number. Thus, either \(w\) or \(z\) must be a real number.

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Most popular questions from this chapter

Write $$ \left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right)\left(\cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\right) $$ in polar form.

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Show that if \(p\) is a polynomial with real coefficients, then $$ p(\bar{z})=\overline{p(z)} $$ for every complex number \(z\).

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