6.2 Approximation

6.12   Definition ($b$ approximates $a$.) Let $\epsilon$ be a positive number, and let $a$ and $b$ be arbitrary numbers. I will say that $b$ approximates $a$ with an error smaller than $\epsilon$ if and only if

$$\vert b-a\vert<\epsilon.$$

Remark: If $b$ approximates $a$ with an error smaller than $\epsilon$, then $a$ approximates $b$ with an error smaller than $\epsilon$, since $\vert a-b\vert=\vert b-a\vert$.

6.13   Definition (Approximation to $n$ decimals.) Let $n\in\mbox{${\mbox{{\bf Z}}}^{+}$}$, and let $a,b$ be real numbers. I will say that $b$ approximates $a$ with $n$ decimal accuracy if and only if $b$ approximates $a$ with an error smaller than $${ {1\over 2}\cdot 10^{-n}}$$; i.e., if and only if

$$\vert b-a\vert<{1\over 2}10^{-n}.$$

6.14   Notation. If I write three dots ($\cdots$) at the end of a number written in decimal notation, I assume that all of the digits before the three dots are correct. Thus since $\pi = 3.141592653589\cdots,$ I have $\pi = 3.1415\cdots$, and $\pi = 3.1416$ with 4 decimal accuracy.

6.15   Example.

$$\pi=3.141592653589793\cdots$$

and

$${{22}\over 7}=3.142857142857142\cdots.$$

Hence

$$3.1415 <\pi<{{22}\over 7} <3.1429,$$

and

$$\left\vert {{22}\over 7} - \pi\right\vert <3.1429-3.1415=.0014<.005={1\over 2}\cdot 10^{-2}.$$

Hence $${ {{22}\over 7}}$$ approximates $\pi$ with an error smaller than $.0014$, and $${ {{22}\over 7}}$$ approximates $\pi$ with 2 decimal accuracy.

6.16   Example. We see that

$$.49 \mbox{ approximates } .494999 \mbox{ with 2 decimal accuracy, }$$

and

$$.50 \mbox{ approximates } .495001 \mbox{ with 2 decimal accuracy, }$$

but there is no two digit decimal that approximates $.495000$ with 2 decimal accuracy.

6.17   Example. Since

$$\vert .49996 -.5 \vert = .00004 < .00005 = {1\over 2}\cdot 10^{-4},$$

we see that $.5$ approximates $.49996$ with 4 decimal accuracy, even though the two numbers have no decimal digits in common. Since

$$\vert .49996 - .4999 \vert = .00006 > {1\over 2} \cdot 10^{-4},$$

we see that $.4999$ does not approximate $.49996$ with 4 decimal accuracy, even though the two numbers have four decimal digits in common.

6.18   Theorem (Strong approximation theorem.) Let $a$ and $b$ be real numbers. Suppose that for every positive number $\epsilon$, $b$ approximates $a$ with an error smaller than $\epsilon$. Then $b=a$.

Proof: Suppose that $b$ approximates $a$ with an error smaller than $\epsilon$ for every positive number $\epsilon$. Then

$$\vert b-a\vert<\epsilon \mbox{ for every } \epsilon \mbox{ in } \mbox{${\mbox{{\bf R}}}^{+}$}.$$

Hence

$$\vert b-a\vert\not=\epsilon \mbox{ for every } \epsilon \mbox{ in } \mbox{${\mbox{{\bf R}}}^{+}$},$$

i.e., $\vert b-a\vert\notin\mbox{${\mbox{{\bf R}}}^{+}$}$. But $\vert b-a\vert\in\mbox{{\bf R}}_{\geq 0}$, so it follows that $\vert b-a\vert=0$, and consequently $b-a=0$; i.e., $b=a$. $\diamondsuit$