Gottlieb

Rules of Algebra

Addition

Let a, b, c be any real numbers

1) a + b = b + a

2) a + (b + c) = (a + b) + c

3) a + (−1)a = 0 (also written a − a = 0)

4) a + 0 = a

Multiplication

5) ab = ba

6) a(bc) = (ab)c

7) aa^{−1} = 1 (also written \frac{a}{a} = 1) [if a ≠ 0]

8) 1 · a = a, 0 · a = 0, (−a)(−b) = ab

9) \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} = (ad + bc)(bd)^{−1}

10) a(b + c) = ab + ac

Exponents

11) a^{n} = a · a … a (n–times) where n is a positive integer 12) a^{0} = 1

13) a^{−n}: = \frac{1}{a^{n}} [a ≠ 0]

14) a^{n/m}: = \sqrt[m/n]{a} (i.e. (a^{n/m})^{m} = a^{n}) [a ≥ 0 if m is even]

15) a^{r+s} = a^{r}a^{s} [for r, s any numbers and a > 0]

16) (a^{r})^{s} = a^{rs} [for r, s any numbers and a > 0]

17) (ab)^{r} = a^{r}b^{r} [for r any number and a > 0, b > 0]

18) 1^{r} = 1

19) 0^{r} = 0 [if r > 0] Note that

20) a − b: = a + (−1)b

21) \frac{a}{b}: = ab^{−1}

22) a^{m/n}: = \sqrt[m/n]{a} where (a^{m/n})^{n} = a^{m} So √ , subtraction, and \frac{a}{a} are not necessary.