Algebraic Laws

Commutative laws for addition and multiplication: If $a$ and $b$ are arbitrary real numbers then

$$a+b = b+a,$$ (C.1)

$$ab = ba.$$ (C.2)

Associative laws for addition and multiplication: If $a,b,$ and $c$ are arbitrary real numbers then

$$(a + b) + c = a + (b + c),$$ (C.3)

$$(ab)c = a(bc).$$ (C.4)

As a consequence of equations C.3 and C.4 we usually omit the parentheses in triple sums or products, and write $a+b+c$ or $abc.$ We know that all meaningful ways of inserting parentheses yield the same result.

Distributive laws: If $a,b$ and $c$ are arbitrary real numbers, and $d$ is an arbitrary non-zero real number then

$$c(a + b) = ca + cb,$$ (C.5)

$$c(a - b) = ca - cb,$$ (C.6)

$$(a + b)c = ac + bc,$$ (C.7)

$$(a - b)c = ac - bc,$$ (C.8)

$$(a + b)/d = a/d + b/d ,$$ (C.9)

$$(a-b )/d = a/d - b/d.$$ (C.10)

Properties of zero and one: The rational numbers $0$ and $1$, have the property that for all real numbers $a$

$$a + 0 = a,$$ (C.11)

$$0 + a = a,$$ (C.12)

$$a \cdot 1 = a,$$ (C.13)

$$1 \cdot a = a,$$ (C.14)

$$0 \cdot a = 0,$$ (C.15)

$$a \cdot 0 = 0.$$ (C.16)

Moreover

$$0 \not= 1,$$ (C.17)

and

$$\mbox{ if }\ ab = 0 \mbox{ then }\ a = 0 \mbox{ or }\ b=0 \mbox{ (or both)}.$$ (C.18)

Additive and multiplicative inverses: For each real number $a$ there is a real number $-a$ (called the additive inverse of $a$ ) and for each non-zero real number $b$ there is a real number $b^{-1}$ (called the multiplicative inverse of $b$) such that

$$a + (-a) = 0,$$ (C.19)

$$(-a) + a = 0,$$ (C.20)

$$b \cdot b^{-1} = 1,$$ (C.21)

$$b^{-1} \cdot b = 1,$$ (C.22)

$$-0 = 0,$$ (C.23)

$$1^{-1} = 1.$$ (C.24)

Moreover for all real numbers $a,c$ and all non-zero real numbers $b$

$$-(-a) = a,$$ (C.25)

$$a - c = a + (-c),$$ (C.26)

$$a/b = a \cdot b^{-1},$$ (C.27)

$$b^{-1} = 1/b ,$$ (C.28)

$$(ab)^{-1} = a^{-1}b^{-1}$$ (C.29)

$$-a = (-1) \cdot a,$$ (C.30)

$$(b^{-1})^{-1} = b,$$ (C.31)

$$(-a)(-c) = ac,$$ (C.32)

$$(-a)c = a(-c) = -(ac),$$ (C.33)

$$-(\frac{a}{b} ) = \frac{-a}{b} = \frac{a}{-b}.$$ (C.34)

Note that by equation C.33, the expression $-xy$ without parentheses is unambiguous, i.e. no matter how parentheses are put in the result remains the same.

Cancellation laws: Let $a,b,c$ be real numbers. Then

$$\mbox{if } a+b = a+c, \mbox{ then }\ b = c.$$ (C.35)

$$\mbox{if } b+a = c+a, \mbox{ then }\ b = c.$$ (C.36)

$$\mbox{if } ab = ac \mbox{ and $a \not= 0$} \mbox{ then }\ b = c.$$ (C.37)

$$\mbox{if } ba = ca \mbox{ and $a \not= 0$} \mbox{ then }\ b = c.$$ (C.38)

Some miscellaneous identities: For all real numbers $a,b,c,d,x$

$$a^2 - b^2 = (a-b)(a+b),$$ (C.39)

$$(a + b)^2 = a^2 + 2ab + b^2,$$ (C.40)

$$(a - b)^2 = a^2 - 2ab + b^2,$$ (C.41)

$$(x + a)(x+b) = x^2 +(a+b)x + ab,$$ (C.42)

$$(a+b)(c+d) = ac + ad + bc + db.$$ (C.43)

Moreover, if $b \not= 0$ and $d \not= 0$ then

$$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}.$$ (C.44)

$$\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}.$$ (C.45)

$$\frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}.$$ (C.46)

If $w,x,y,z$ are real numbers, then

$$w-x-y+z+y \mbox{ means } w+((-x) + ((-y) + (z+y)))$$ (C.47)

i.e. the terms of the sum are associated from right to left. It is in fact true that all meaningful ways of introducing parentheses into a long sum yield the same result, and we will assume this. I will often make statements like

$$w-x-y+z+y = w+z-x$$ (C.48)

without explanation. Equation C.48 can be proved from our assumptions, as is shown below, but we will usually take such results for granted.

Proof of equation C.48. Let $w,x,y,z$ be real numbers. Then

$$w-x-y+z+y = w +( (-x) + ((-y)+(z+y))) \mbox{ by \ref{lrassoc}}$$

$$= w + ( (-x) + ((-y)+(y + z))) \mbox{ by \ref{comlaw}}$$

$$= w + ( (-x) + (( (-y) + y)+ z)) \mbox{ by \ref{ass1}}$$

$$= w + ( (-x) + ( 0 + z))\mbox{ by \ref{mapa2}}$$

$$= w + ( (-x) + z) \mbox{ by \ref{zprop}}$$

$$= w + ( z + (-x)) \mbox{ by \ref{comlaw}}$$

$$= w + z - x \mbox{ by \ref{lrassoc}}.$$