Algebraic Proofs
The various multiplication etc. devices shown in this book can be proved by arithmetical and geometrical methods, but here for brevity we give algebraic proofs.
Chapter 3
Examples
1-5 (ax + 5)² = a(a + 1)x² + 25 x =10
6-9 (ax + b)(ax + 10-b) = a(a + 1)x² + b(10 - b) x=10
10 as above with x=100
Chapter 4
Examples
11-12 (ax + b)((10-a)x + b) = (a(10-a) + b)x² x=10
Chapter 5
Examples
1-15 (x + a)(x + b) = x(x + a + b) +ab, x=10n
Subtraction
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Examples
16-19 (nx + a)(nx + b) = nx(nx + a + b) + ab
20-22 (x + a)(y + b) = (x + a)y + bx + ab, x=10m, y=10n
23-24 (x + a)(x + b)(x + c) = x²(x+a+b+c) + x(ab+ac+bc) + abc
25-26 (x + a)² = (x + 2a) + a²
27 (nx + a)² = n(nx + a)x + a²
28-29 (50 + a)² = 100(25 + a) + a²
30-31 a(xn - 1) = (a - 1)xn + (xn - a)
Chapter 6
(axn + bxn-1 + cxn-2 + . . . + zx0)(Axm + Bxm-1 + Cxm-2 + . . . + Zx0)
= aAxn+m + (aB+bA)xn+m-1 + (aC+bB+cA)xn+m-2 + . . . + zZ, x=10
For grouping 2, 3 etc. figures on the right of the numbers x=10², 103 etc.
Division near a base. Since bx = (x - a)b + ab
therefore bx/(x-a)= b remainder ab. x=10, a < x
Chapter 7
(a + b)(a - b) = a² - b², where a = average, b < a
Chapter 8
(a + p)² = a² + p(2a + p)
(a + 3p)(a + 2p) - (a + p)a = p(a+3p + a+2p + a+p + a)
