グローバルビジネスで役立つ数学（45）複素数から実部を求める（英語版）

Hi there!

Once upon a time, a president who had no reservations about giving a fart or cutting the cheese in his office worked at a small and medium-sized company. When he got his buttocks up, that was the time for him to let a fart.

His employees desperately needed to keep their distance from him by saying, “Sorry, President. Nature calls me. I’ve got to go!” and whatnot even when they were being spurred or motivated to sell more by the sometimes being crop-dusting.

I’m still wondering why he had such the nerve to practice ecologically unfriendly actions. Was the act of audacity originated from his bygone glory as the best salesperson in an electronic parts company? To be candid, I used to view his “misdemeanor” as the devil’s work.

His first chutzpah or nerve was to give a fart in public.
The second one was to flirt with a young female employee.
The third was to pluck his nose hair even in front of the customers.

The last fourth chutzpah would better be undisclosed; however, the fourth power cannot be neglected. Here is a word problem related to complex numbers in mathematics.

When you make a complex number \(z = a + i\) to the fourth power, it becomes a real number \(r\), in which \(a\) is a real number and \(i\) is the imaginary unit. Find all pairs of \(a\) and \(r\).(Source: The Mathematics Certification Institute of Japan [Slightly changed the expressions in the word problem concerned])

Here is my solution to the problem.

First of all, let me square \(z = a + i\).

\(z^{2} = (a + i)^{2} = a^{2} – 1 + 2ai\)
Thus,
\(z^{4} = (a^{2} – 1 + 2ai)^{2} = (a^{2} – 1)^{2} – 4a^{2} + 4a(a^{2} -1)i\), which is equivalent to the real number \(r\).

\(r = (a^{2} – 1)^{2} – 4a^{2} = a^{4} – 2a^{2} + 1 – 4a^{2} = a^{4} – 6a^{2} + 1\)・・・（Ⅰ）

The imaginary part must be 0, so

\(a(a^{2} – 1) = 0\)・・・（Ⅱ）

Solving (Ⅱ), \(a(a + 1)(a – 1) = 0\).
Thus,
\(a = 0, -1, 1\).

Now let me assign each of the values \(a\) to (Ⅰ).

\(r = 1\) at \(a = 0\),
\(r = -4\) at \(a = -1\), and
\(r = -4\) at \(a = 1\).

Therefore,
the answer is \((a, r) = (0, 1), (-1, -4), (1, -4)\)

I’ve discovered the fourth power is a dark horse. Making the best use of factorization was the key to success in solving the word problem.

Stay tuned, and expect to see my next post.

Keep well.

Frank Yoshida

￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣
【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
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