03. Properties of arithmetic in ℝ

Properties of arithmetic in $\mathbb{R}$

Let $a,b,c,x,y$ be real numbers

1. Uniqueness of $0$: If $a+x=a$ for all $a$ then $x=0$

2. Cancellation for $+$: If $a+x=a+y$ then $x=y$

3. $-0=0$

4. $-(-a)=a$

5. $-(a+b)=(-a)+(-b)$

6. Uniqueness of $1$: If $a\cdot x=a$ for all $a\ne 0$ then $x=1$

7. Cancellation of $\cdot$: If $a\ne 0$ and $a\cdot x=a\cdot y$ then $x=y$

8. If $a\ne 0$ then $\frac{1}{\frac{1}{a}}=a$

9. $(a+b)\cdot c=a\cdot c+b\cdot c$

10. $a\cdot 0=0$

11. $a\cdot (-b)=-(a\cdot b)$
    In particular, $(-1)\cdot a=-a$

12. $(-1)\cdot (-1)=1$

13. If $a\cdot b=0$ then $a=0$ or $b=0$
    If $a\ne 0$ and $b\ne 0$ then $\frac{1}{a\cdot b}=\frac{1}{a}\cdot \frac{1}{b}$

Proof

1. Suppose that $a+x=a$ for all $a$ then

$$\begin{array}{rl}x& =x+0(\text{additive identity})\\ & =0+x(+\text{ is commutative})\\ & =0(\text{by hypothesis, with }a=0)\end{array}$$

2. Suppose that $a+x=a+y$ then

$$\begin{array}{rl}y& =y+0(\text{additive identity})\\ & =y+(a+(-a))(\text{additive inverses})\\ & =(y+a)+(-a)(+\text{ is associative})\\ & =(a+y)+(-a)(+\text{ is commutative})\\ & =(a+x)+(-a)(\text{hypothesis})\\ & =(x+a)+(-a)(+\text{is commutative})\\ & =x+(a+(-a))(+\text{ is associative})\\ & =x+0(\text{additive inverse})\\ & =x(\text{additive identity})\end{array}$$

3. We have that

$$\begin{array}{r}0+0=0(\text{additive identity})\\ 0+(-0)=0(\text{additive inverses})\end{array}$$

Hence

$$0+0=0+(-0)\text{ hence }0=-0(\text{cancellation for }+(2))$$

4. We have

$$\begin{array}{rl}(-a)+a& =a+(-a)(+\text{ is commutative})\\ & =0(\text{additive inverses})\end{array}$$

and $(-a)+(-(-a))=0(\text{additive inverses})$
so $(-a)+a=(-a)+(-(-a))$,
so $a=(-(-a))(\text{cancellation for }+(2))$

5. TODO onwards… (also refer to lecture notes for some of them)

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Notation on operations for arithmetic

$$\begin{array}{rl}a-b& \text{ for }a+(-b)\\ ab& \text{ for }a\cdot b\\ \frac{a}{b}& \text{ for }a\cdot \left(\frac{1}{b}\right)\\ a^{-1}& \text{ sometimes for }\frac{1}{a}\end{array}$$

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Positive Powers

Take $a\in \mathbb{R}\setminus \{0\}$
Define

$$a^0=1$$

Then we define the positive powers of $a$ inductively: for integers $k\ge 0$ we define

$$a^{k+1}=a^k\cdot a$$

For integers $l\le -1$ we define

$$a^l=\frac{1}{a^{-l}}$$

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Additive Rule for Exponents lemma

For $a\in \mathbb{R}\setminus \{0\}$

$$a^m{a}^n=a^{m+n}\text{ for }m,n\in \mathbb{Z}$$

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