I like this a lot … feels so uncomplicated … expanding the brackets (in this situation) just feels overkill. #UKMathsChat
This 'revaluing' approach (Hackenberg calls it 'transformation') is taught at Primary … e.g. 53−19=54−20 ... but I've never seen it applied to Secondary algebra ... but if it was, things like 20−(13−2x)=15 then become possible with number knowledge rather than brackets knowledge. Hmm?!
3/3
Pre-algebra simultaneous equations. Clever!
盈不足術
/yíng bù zú shù/
rule of too much and not enough (!)
حساب الخطأين
/ḥisāb al-khaṭa'ayn/
reckoning from two falsehoods
regula duorum falsorum
rule of double false position
Subtract if both guesses are deficits/excesses.
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So I'm reading about the origins of algebra ... and I find this from Abū Kāmil in about 900CE ... and of course Kāmil's algebra (al-jabr!) could solve this. Pretty amazing really.
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• goal: revalue the minuend and subtrahend
• so 'restore' the 10 by adding x to [10 less x]
• and complete the goal by adding x to [14]
giving:
then: [14 and x] less [10]
then: = [x and 14 less 10]
then: = [x and 4]
now: 14 + x − 10
now: = x + 4
2/3
Piecing together thought processes for 9th century algebra … in the absence of a) symbolic algebra and b) negative numbers:
now: simplify 14 − (10 − x)
then: simplify [14] less [10 less x]
minuend: [14]
subtrahend: [10 less x]
• goal: ...
1/3
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Semiotic observation (!) ... when you think of the Inscribed Angle Theorem, which arc do you mentally highlight? I've always done (a) but actually Euclid presents it as (b) … which makes sense because the angles are indeed in that segment. Time on my hands? Maybe ;-)
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I'm looking for a publisher.
240 pages of old-world trigonometry, geometry focused & algebra free – where it came from, who built it, how they built it, how they used it, and how to do trigonometry in this way today.
Any suggestions gratefully received.
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Euclid Elements
Book II
Proposition 5
Intriguingly this was used by the Egyptian mathematician Abū Kāmil in the 10th century to solve quadratic equations of the form x² + 21 = 10x
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