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

第一年在濟記教書,同學泉泉問我有關「俄羅斯乘法」的背後原理,先介紹一下:例如:

35 ╳ 52 = 1820

上式自然不過。讓我運用所謂「俄羅斯乘法」處理,方法如下:不斷把左數乘 2,右數除 2;若除不盡者,只寫商數。到右數為 1 或 0 便停止。

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

把右數為單數的左數加起,便是原式的答案:即

35 ╳ 52 = 140 + 560 + 1120 = 1820。

再舉一例:

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

故此,

110 ╳ 17 = 110 + 1760 = 1870

雖然在科技千里的今天,這個方法有點「無聊」,但大家可否找出這個數法之所以有效的背後原因,明白古人之智慧呢?

(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.

1 則迴響 »

  1. 1 hexadecimal digit -> 4 binary digits as 16=2^4
    A (16) -> 1010(2), B -> 1011(2), C -> 1100(2)
    ABC (16)-> 1010 1011 1100 (2)
    easier to do than to convert to denary first
    ABC (16) -> 2748 (10)

    迴響 由 y.u. — 2008/03/21 @ 7:46 下午 | 回覆


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