Page:The Algebra of Mohammed Ben Musa (1831).djvu/38

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units, the units by the greater numbers, and the units by the units.

If the units, combined with the greater numbers, are positive, then the last multiplication is positive; if they are both negative, then the fourth multiplication is likewise positive. But,if one of them is positive, and one (16) negative, then the fourth multiplication is negative. ^[1]

For instance, “ten and one to be multiplied by ten and two.”^[2] Ten times ten is a hundred; once ten is ten positive; twice ten is twenty positive, and once two is two positive; this altogether makes a hundred and thirty-two.

But if the instance is “ten less one, to be multiplied by ten less one,”^[3] then ten times ten is a hundred; the

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1. ${\begin{aligned}{\text{In multiplying}}\;(x\pm a)\;{\text{by}}\;(y\pm b)\\+a\times +b=+ab\\-a\times -b=+ab\\+a\times -b=-ab\\-a\times +b=-ab\end{aligned}}$
2. ${\begin{aligned}&(10+1)\times (10+2)\\&=10\times 10\ldots 100\\&+1\times 10\ldots 10\\&+2\times 10\ldots 20\\&+1\times 2\ldots 2\\&+132\end{aligned}}$
3. ${\begin{aligned}&(10-1)(10-1)\\&=10\times 10\cdot \cdot +100\\&+1\times 10\cdot \cdot -10\\&+1\times 10\cdot \cdot -10\\&+1\times -1\cdot \cdot +1\\&+81\end{aligned}}$