Numbers: Real numbers
The notion of real number
Real numbers can be described by approximating them with decimal numbers. When we start with a decimal number and subsequently add an appropriate decimal to it, we get an increasingly accurate description of the number. This is the principle behind the following definition.
Decimal developments and real numbers
A decimal development is an infinite sequence of decimal numbers, wherein each number results from the previous one by the addition of a single digit at the right side.
The general form of a decimal development is \[\epsilon r_{-n}r_{1-n}\cdots r_{-1}r_0.r_1r_2r_3\cdots\tiny,\] where #\epsilon# is the sign #+# or #-#, #n# a non-negative integer, and each #r_i# a digit. The point between #r_0# and #r_1# is called the decimal point.
The real number represented by the above decimal development is \[\epsilon\left(r_{-n}\cdot 10^n +r_{1-n}\cdot 10^{n-1}+\cdots +r_0+r_1\cdot 10^{-1}+r_2\cdot 10^{-2}+\cdots\right)\] The row of digits starting at a certain place in the decimal development, is called the tail of the number. The part of the number before that is called the head. The head is a decimal number. A head with #i# decimals has exactly #i# numbers after the decimal point.
If the tail of a decimal development has a pattern #www\cdots # in which #w# is a finite set of numbers constantly being repeated, then the tail is called repetitive or #w#-repetitive.
The set of all real numbers is denoted by #\mathbb{R}#.
The sum defining the real number has infinitely many terms. If you start with the integer and next adding the contributions from the digits behind the decimal point, you get an increasingly accurate approximation of the real number. This is the essence of real numbers: they can be approximated pretty good but are often impossible to write exactly. These kind of infinite sums are covered extensively in the chapter Sequences and series in the Calculus Course.
On the number line the decimal development can be interpreted as follows: let #r# be a positive real number. The biggest integer to the left of #r# is the head #r_{-n}r_{1-n}\cdots r_0# with zero decimal places. Next up is the decimal point. Then we search for the largest part with length #\frac{r_1}{10}# which we can add to the head, in which #r_1# is a digit, without the result being bigger than #r#. Then we search for the largest part with length #\frac{r_2}{100}# which we add to #r_{-n}r_{1-n}\cdots r_0.r_1#, in which #r_2# is a digit, without the result being bigger than #r#, and so on.
The area of a circle with radius #1# is #\pi=3.141592\cdots# Of this decimal development only the head with 6 decimal places is given. This page is not long enough to completely an infinite series. Neither our entire course material.
The zeros on the far right of a #0#-repetitive tail of a real number are often omitted: \[-3.25000\cdots =-3.25\]
If behind the decimal point only zeros occur, then together with the zeros also the decimal point itself is omitted. The result is an integer. Integers are so conceived as decimal numbers: \[2015 = 2015. = 2015.0 = +2015.0 = +2015.00000\cdots\]
In the decimal development of #\pi# the head #3.14159# occurs. The rounding #3.1416# is a better approximation to #\pi# then #3.1415#. But in the decimal development, the head #3.1415# occurs and not #3.1416#. The best rounding may thus differ from the digit in the decimal development.
The decimal development of a real number
Any real number unequal to #0# has a unique decimal development without #9#-repetitive tail.
This is called the decimal development of that number.
Hence every real number can be randomly exact approximated with the help of decimal numbers. Moreover, this approximation defines this in total. More exact proof can be provided after the theory in the chapter Sequences and series.
We shed more light on two exceptions:
The number #0# is excluded because the zero can be notated in many ways: \[0=0.0= 0.000000\cdots=+0.000000=+00000.000000\cdots\]
Decimal representations with a #9#-repetitive tail are excluded because \[+0.9999\cdots = 1\tiny.\]
Here are two proofs of this equality.
The first evidence used Linear equations with one unknown. We write #x=+0.9999\cdots# and determine #10\cdot x = 9+x#. This holds since \[\begin{array}{rcl}10\cdot x &=& 10\cdot 0.9999\cdots\\ &&\phantom{xx}\color{blue}{\text{definition of }x}\\ &=& 10\cdot\left(\frac{9}{10}+\frac{9}{10^2}+\frac{9}{10^3}+\frac{9}{10^4}+\cdots\right)\\ &&\phantom{xx}\color{blue}{\text{definition of }0.9999\cdots}\\&=& 9+\frac{9}{10}+\frac{9}{10^2}+\frac{9}{10^3}+\cdots\\&&\phantom{xx}\color{blue}{\text{brackets expanded}}\\&=&9+0.9999\cdots\\ &&\phantom{xx}\color{blue}{\text{definition of }0.9999\cdots}\\ &=&9+x\\ &&\phantom{xx}\color{blue}{\text{definition of }x}\end{array}\] These steps are known in the case of a finite number of terms, but here an infinite number of terms are added. Later we will learn that this is allowed here. The solution of the linear equation #10\cdot x = 9+x# with #x# is equal to #x=1#.
The second proof uses geometrical series. There, we use the fact that #0.9999\cdots# is the infinite sum #\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+\cdots# of which the theory says that it is equal to #\frac{9}{10}\cdot\frac{1}{1-\frac{1}{10}}#, which can be simplified as #1# .
The statement claims two things simultaneously:
- every real number has a decimal development without a #9#-repetitive tail,
- two different decimal developments without #9#-repetitive tails, in which a digit unequal to #0# occurs, belong to different real numbers.
- The first claim can be substantiated by executing the above described process, which will be discussed extensively later on.
The second claim, about different decimal developments is less difficult to show: if two decimal developments differ than they also differ in their positions on the number line, and the approximation with parts #\frac{r_n}{10^n}# for suitable numbers #r_n# will also be different. From this it can be reduced that the one real number is greater than the other one. Again, this will be discussed later.
Or visit omptest.org if jou are taking an OMPT exam.
