...methods for finding the fum depend upon other principles. IV. Properties of Numbers. THEOR. I. The fum of two quantities multiplied by their difference is equal to the difference of their fquares. Let _ Let the quantities be reprefented by a and £, thena + £Xa — b=az — bit as appears...

...denominator ; that is by -V- when that is — ; and by — when that is -f- ; because the product of the sum of two quantities multiplied by their difference, is equal to the difference of their squares. DIVISION ot" ftatw$, or Divided Jtatio, is when of tour proportional quantities, the differences of...

...168 a2 64 + 144 a2 b* 3. To multiply a + 6 hy a — 6, we have (a + b) (a — J) = o2 _ 62 Whence, the sum of two quantities multiplied by their difference is equal to the difference of their squares. Thus, (7 +4) (7 —4) =49— 16 = 33 So also (8 a2 + 7 a 62) (8 a2 — 7 a 62) = 64 a6 — 49 a2 64...

...and byp+ 'jq, when the denominator is p— Vq. For multiplying in this manner, and recollecting that the sum of two quantities, multiplied by their difference, is equal to the difference of their squares, (5), we have aa(P~ (p+Vq)(p-Vq) p3-q " p3-q i- Vq)_ap + aVq j p3-q in which the denominators are rational....

...by p+ ^/q, when the denominator is p— ^/q. For multiplying in this manner, and recollecting that the sum of two quantities, multiplied by their difference, is equal to the difference of their squares, we have a _ a(p— Vq) _a(p— Vq)_ap—aVq aa(p+y/g) a(p+ Jq) P- V<(P in which the denominators are...

...formula (a + 6) x (a - 6) = axa — bxb, which in common algebra expresses that the sum of two numbers multiplied by their difference is equal to the difference of their squares, will now be a compendious representation of the following geometrical theorem : — Let there be two...

...jj-j. i/qt when the denominator is p — */q. For multiplying in this manner, and recollecting that the sum of two quantities, multiplied by their difference, is equal to the difference of their squares, we have aa(p— Vq) a(p—i/q)_ap—dVq _ p+Vq~(p+Vq)(p-Vq)~~ fq ~ fq _ _ p- Vq~(p- Vq)(p+ V<L)~~ fq...

...quantities equal to ? 40. Let it be required to multiply a+6 by a — b. We have (a+6)x(a-6)=<z»-62. Hence, the sum of two quantities, multiplied by their difference, is equal to the difference of their squares. 1. Multiply 2c+5 by 2c— b. We have (2c+6)x(2c-6)=4c=-62. 2. Multiply 9<zc+36c by 9ac—3bc. We have...

...believes anything at all, and knows the first elements of Algebra, ever doubted that the sum of any two quantities, multiplied by their difference, is equal to the difference of their squares : because no one was ever interested in this principle being supposed false. It is a principle which...

...and magnitude. Thus, for example, whilst Arithmetic teaches that the sum of the two numbers 6 and 4 multiplied by their difference is equal to the difference of their squares, Algebra teaches that the same is true for any two numbers whatever, whole or fractional.] 58. Known...