# Quod Erat Demonstrandum

## 2008/09/03

### Prerequisite of F.4 algebra

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 5:17 下午

As usual, we start with basic algebraic computation drilling at the beginning of F.4 mathematics lessons. But, I just did some useless mathematical chatting and gave something about infinity (e.g. the discussion of 1 – 1 + 1 – 1 + … ) and they showed their excitment with hands clapping. Well, of course, I switched to serious matter very soon. As expected, most of the students did not know the fact that “zero is an even number" (even, they did not know that zero is an integer). I went through some ‘prerequisite’ questions with them, one is:

(a) $\sqrt{9} = \pm3$ 　[True or false?]
(b) If $x^2 = 9$, then $x = 3$.　[True or false?]
(c) The square roots of 9 are 3 and -3.　[True or false?]
(d) If $x^2 = 9$, then $x = 3$ and $x = -3$.　[True or false?]

F.4 students, here are the answers.

(a) False
(b) False
(c) True
(d) False

Here are some explanations.

(a) The symbol $\sqrt{9}$ represents “the positive square root of 9″. $\sqrt{9} = 3$ only, not $\pm3$.

(b) If $x^2 = 9$, then $x = 3$ or $x = -3$. We may also write “If $x^2 = 9$, then $x = \pm3$“. It is because, both $3$ and $-3$ satisfy the equation $x^2 = 9$.

(c) To make it clear, we write:
The positive square root of 9 is 3. [In symbol, $\sqrt{9}$]
The negative square root of 9 is -3. [In symbol, $-\sqrt{9}$]
The square roots of 9 are 3 and -3.

(d) We should write “If $x^2 = 9$, then $x = 3$ or $x = -3$." On writing “$x = 3$ and $x = -3$“, it means, the value of $x$ is 3 and -3 at the same time, which is impossible. Also, you may read (c) again, we used the conjunction “and" there, it is possible because we are not talking about the single value of $x$, but the “square roots of 9″, and there are 2 different values of “square roots of 9″, namely 3 and -3.

suppose $\alpha$ and $\beta$ are roots of $x^2 - 3x + 2 = 0$, then, can we say something like

$\alpha = 1$ or $\beta = 2$“?

It is not that OK to express the solution as above (why?), instead, you may write

$(\alpha = 1$ and $\beta = 2)$ or $(\alpha = 2$ and $\beta = 1)$

or

$(\alpha , \beta) = (1,2)$ or $(2,1)$

or simply

$\{\alpha, \beta\} = \{1 , 2\}$

(note: the above two ways involve set notations)

## 3 則迴響 »

1. Oh, the F.4 students this year are not as good as those in F.5.
Actually, which F.4 class are you talking about?

迴響 由 lone — 2008/09/03 @ 9:15 下午 | 回應

2. 今日我上數學堂，professor都問了同一問題，"square root of 9 equal what??"，結果有為數不少的人答+3 or -3，不同的是，我們都是經歷了兩個會考的大學生…

迴響 由 lanven — 2008/09/04 @ 1:51 上午 | 回應

3. @lone

How to draw the conclusion that ‘F.4 students this year are not as good as those in F.5’? This year, I’m the math teacher of F.4E.

@lanven

Urm, may be I should say this is a kind of failure of secondary math teaching, I need to work harder…

迴響 由 johnmayhk — 2008/09/04 @ 7:38 下午 | 回應