グローバルビジネスで役立つ数学（71）対偶による証明（英語版）

Hi there!

In retrospect, I saw many so-called “Globetrotters” while I was an expatriate; some had a good command of English, and others did not. The former all made themselves understood 100 percent by non-Japanese. … Was it that?

Truth be told, communication skills, as are often the cases, are different from people skills. People who have a good command of English technically can persuade their counterparts to do some actions but cannot convince them that the former persons are trustworthy and worthwhile to doing business with over the long term.

The catch is that we should not get our noses to the grindstone, i.e., make a play, instead establish credibility with people skills by which your counterparts get to think it pays off for them to make a big deal with you.

Fluent English doesn’t always guarantee an excellent career path for a Shoushaman. Let’s think of proof by contraposition in mathematics.

Imagine the proposition “All Shoushaman with a good command of English can get promoted, so Frank Yoshida can also get promotions” is true. Then, in a contrapositive way, “Since Frank Yoshida cannot get promotions, all Shoushaman cannot always get promoted” is also true.

Today, let’s challenge the following word problem regarding proof by contraposition.

Prove the proposition “The equation \(ax + b = 0\) has only one solution for \(x\), so \(a\neq 0\)” by contraposition, given that all the characters are real numbers (Source: Yellow Chart Math Ⅰ＋Ａ P73)

Here is my solution to the problem.

Let’s prove by contraposition of the given proposition as “If \(a = 0\), so the equation \(ax + b = 0\) for \(x\) has either no solutions or two and more.”

We get \(ax = -b\) from \(ax + b = 0\).
If \(a = 0\) and \(b\neq 0\), so we have no solutions.
If \(a = 0\) and \(b = 0\), so we have a myriad of solutions.

Therefore, the contraposition is true, and so is the original proposition.

Once again, special thanks to Yellow Chart Math Ⅰ＋Ａ.

Apparently, I still have a long way to mastering the essence of mathematics.

Stay tuned, and expect to see my next post.

Keep well.

Frank Yoshida

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【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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