# Quod Erat Demonstrandum

## 2008/03/03

### 俄羅斯乘法@濟濟一堂, denary and binary

Filed under: Junior Form Mathematics — johnmayhk @ 5:27 下午

This is my first time to teach the topic about conversion among denary (十進制), binary (二進制) and hexadecimal (十六進制) equivalence in form two mathematics. The first question asked by students was “what is the use of learning this?", especially their calculators can give answers swiftly and precisely.

Based on my poor mathematics history knowledge, apart from computer languages and some games derived from the binary system, I cannot figure out the vivid use of learing the stuff. For me, I prefer saying that it may be something about presentation of a number. Denary system may not be perfect in the presentation of the number like $0.333..._{(10)}$ (say) but this number is neat in ternary system (三進制), namely $0.333..._{(10)} = 0.1_{(3)}$. When talking about Cantor set in elementary set theory, ternary system may help. OK, not going too deep, just read the following to see an ‘application’ of binary system.

Form 濟濟一堂學術討論區 2003-10-06 20:03:47

35 ╳ 52 = 1820

35 ╳ 52
70 ╳ 26
140 ╳ 13
280 ╳ 6
560 ╳ 3
1120 ╳ 1

35 ╳ 52 = 140 + 560 + 1120 = 1820。

110 ╳ 17
220 ╳ 8
440 ╳ 4
880 ╳ 2
1760 ╳ 1

110 ╳ 17 = 110 + 1760 = 1870

（注：唔好答我『咪二進制囉』，我都知，但我想要的是同學能詳盡地把背後原理清楚解釋。）

= = = = = = = = = =

The second question from students was “can we do the conversion between hexadecimal system and binary system directly?". I just use denary system as the ‘bridge’, could you give any good idea to resolve the problem? For ‘simple’ numbers, the conversion is quick, see

$11_{16} = 10001_{2}$
$111_{16} = 100010001_{2}$
$1111_{16} = 1000100010001_{2}$
$11111_{16} = 10001000100010001_{2}$

Just inserting three zeros between ones. Do the same, even there are zeros, like

$101_{16} = 100000001_{2}$
$1011_{16} = 1000000010001_{2}$

Why? Too easy to explain.