Theorem 372 (Landau 240). (x/y) • y = x.
(x/y) • y = y • (x/y) = x.
Division
Theorems 372–388
Wherever a quotient is written, its divisor is supposed not to be o. Each proof shows that the left side satisfies the equation that defines the right side (Theorem 361).
Theorem 373 (Landau 241). (x • y)/y = x.
y • x = x • y.
Theorem 374 (Landau 242). If x ≠ o, then x/(x/y) = y.
x/y ≠ o by Theorem 353, since (x/y) • y = x ≠ o. And (x/y) • y = x.
Theorem 375 (Landau 243). (x/y)/z = x/(y • z).
(y • z) • ((x/y)/z) = y • (z • ((x/y)/z)) = y • (x/y) = x.
Theorem 376 (Landau 244). (x • y)/z = x • (y/z).
z • (x • (y/z)) = x • (z • (y/z)) = x • y.
Theorem 377 (Landau 245). If y ≠ o and z ≠ o, then (x/y) • z = x/(y/z).
(y/z) • ((x/y) • z) = (x/y) • (z • (y/z)) = (x/y) • y = x.
Theorem 378 (Landau 246). (x • z)/(y • z) = x/y.
(y • z) • (x/y) = ((x/y) • y) • z = x • z.
Theorem 379 (Landau 247). (x/y) • (z/u) = (x • z)/(y • u).
(y • u) • ((x/y) • (z/u)) = ((x/y) • y) • ((z/u) • u) = x • z.
Theorem 380 (Landau 248). If y, z, u are not o, then (x/y)/(z/u) = (x • u)/(y • z).
By Theorems 379 and 378, ((x • u)/(y • z)) • (z/u) = ((x • u) • z)/((y • z) • u) = (x • (u • z))/(y • (z • u)) = x/y.
Theorem 381 (Landau 249). x/e = x.
e • x = x.
Theorem 382 (Landau 250). If x ≠ o, then x/x = e.
x • e = x.
Theorem 383 (Landau 251). x/y = e if and only if x = y.
If x = y, Theorem 382. If x/y = e, then x = (x/y) • y = e • y = y.
Theorem 384 (Landau 252). x/y = z/u if and only if x • u = y • z.
Proof: (x/y) • (y • u) = ((x/y) • y) • u = x • u, and (z/u) • (y • u) = y • ((z/u) • u) = y • z. So if the quotients are the same, so are the products. Conversely, if x • u = y • z, then by Theorem 378, x/y = (x • u)/(y • u) = (z • y)/(u • y) = z/u.
This is the sameness of fractions of Definition 29 (cross multiplication), now proved of complex numbers.
Theorem 385 (Landau 253). x/y + z/y = (x + z)/y.
y • (x/y + z/y) = y • (x/y) + y • (z/y) = x + z.
Theorem 386 (Landau 254). x/y + z/u = (x • u + y • z)/(y • u).
By Theorems 378 and 385, x/y + z/u = (x • u)/(y • u) + (y • z)/(y • u).
Theorem 387 (Landau 255). x/y − z/y = (x − z)/y.
y • (x/y − z/y) = y • (x/y) − y • (z/y) = x − z.
Theorem 388 (Landau 256). x/y − z/u = (x • u − y • z)/(y • u).
By Theorems 378 and 387.