Approximations of real numbers

Numbers: Real numbers

Approximations of real numbers

Because it is impossible to completely write down the notation of infinite sequences, almost all of the times we are satisfied with approximations of real numbers by decimal numbers: with a head of the decimal development. The precision is expressed in the number of decimal places.

The approximation of #\pi# to five decimal places is #3.14159#.

The number #\pi# can (in principle) be determined up to any desired precision, with help of the first parts of the decimal development. \[ \begin{array}{rcl} 3. &=& 3\\ 3.1 &=& 3+\frac{1}{10}\\ 3.14 &=& 3+\frac{1}{10}+\frac{4}{100}=\frac{314}{100}\\ 3.141 &=& 3+\frac{1}{10}+\frac{4}{100}+\frac{1}{1000}=\frac{3141}{1000}\\ 3.1415 &=& 3+\frac{1}{10}+\frac{4}{100}+\frac{1}{1000}+\frac{5}{10000}=\frac{31415}{10000}\\ 3.14159 &=& 3+\frac{1}{10}+\frac{4}{100}+\frac{1}{1000}+\frac{5}{10000}+\frac{9}{10000}=\frac{314159}{100000}\\&{\cdots}& \end {array}\]

Note the difference between decimal approximation and the decimal development: #3.1415# occurs in the decimal development of #\pi#, but the best approximation of #\pi# to four decimal places is #3.1416#.

Examples of decimal developments that can be easily be described, are the ones that have a repetitive tail, such as #3.111\cdots# or #2.913913913913913\cdots#. We will later see that all these exceptions are rational numbers.

Here is a method to calculate a head of the decimal development on the basis of equations of the number with decimal numbers.

Entier

When #x# is a real number, then we use #\lfloor x\rfloor# to indicate the largest integer that is less than or equal to #x#. This number is called the entier of #x#.

The word "entier" is French and means "whole". Pronounce it as "anti a".

Examples:

If #x# is an integer, then #\lfloor x\rfloor# holds. Integers are the only numbers with this property.
If #x=\frac{a}{b}# is a rational number, with integers #a#, #b# and #b# unequal to #0#, then #\lfloor x\rfloor = \lfloor a/b\rfloor#, is the quotient of #a# when dividing with remainder by #b#.
#\lfloor \pi\rfloor=3# and #\lfloor \e\rfloor=2#.
#\lfloor \sqrt{5}\rfloor=2# and #\lfloor \sqrt{11}\rfloor=3#.

Determination of the decimal development

Let #r# be a non-negative real number and #n# a natural number. The head of the decimal development of #r# with #n# decimals equals \[\frac{\lfloor{r}\cdot10^n\rfloor}{10^n}\tiny.\]

If #r\lt0#, then the head of the decimal development of #r# with #n# decimals is equal to #-\frac{\lfloor{-r}\cdot10^n\rfloor}{10^n}#.

To see why the formula #h=\frac{\lfloor{r}\cdot10^n\rfloor}{10^n}# for the head #h# of the decimal development of #r# with #n# decimals is correct for #r\ge0#, we reduce: \[ \begin{array}{rcccccl}0&\le&10^n\cdot r - \lfloor10^n\cdot r\rfloor&\lt&1&\phantom{h}&\color{blue}{\text{definition entier}}\\0&\le& r - \frac{\lfloor10^n\cdot r\rfloor}{10^n}&\lt&\frac{1}{10^n}&\phantom{h}&\color{blue}{\text{rule of calculation 5 for inequalites}}\\ 0&\le& r - h&\lt&\frac{1}{10^n}&\phantom{h}&\color{blue}{\text{definition }h}\\ h&\le& r &\lt&h+\frac{1}{10^n}&\phantom{h}&\color{blue}{\text{rule of calculation 3 for inequalites }}\\\end {array}\] Thanks to the last rule and the fact that #h# is a decimal number, we can use the squeezing of a real number between decimal numbers to conclude that #h# is the head of decimal development of #r#.

In order to add, subtract, multiply, and divide real number in terms of decimal development, it is necessary to be able to do this for the head of the decimal developments of the digits. Because those heads are rational numbers, we can perform the mentioned operations on them. The question remains as to how good the approximation is. This can be solved in general, but here we will not dwell on it. Instead, we show how, in a number of practical cases, the approximation can be found. To this end, an upper and a lower bound of the real number are given that are rational and are sufficiently close to each other for rendering the desired approximation.

Euler's number, with notation #\e#, is a real, non-rational, number. Its tail is not repetitive. The decimal development begins as follows: \[\e = 2.718281828459\cdots\] Approximate #10\cdot { \e}# by a decimal number up to 3 digits after the decimal point.

#27.183#

This can be seen by using the approximation #2.718282 = \frac{2718282}{1000000}# of #\e#:
\[10\cdot { \e}\approx 10\cdot { \frac{2718282}{1000000} } = \frac{2718282}{100000} \approx 27.183\tiny.\] The tail of the decimal development, which we omit here, does not affect the three digits after the decimal point. We can show this by using inequalities: from #2.718281 \le \e \le 2.718282# follows \[{ 27.18281 } \le 10\cdot { \e} \le { 27.18282 }\tiny.\] From this it becomes clear that the calculated decimal approximation of #10\cdot { \e}# up to three digits after the decimal point is correct.

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