Numbers: Rational numbers
Decimal numbers
The picture of the real numbers as points on a line is geometrically appealing. On the other hand, an algebraic definition is not easy. Before we look into calculating with real numbers, we elaborate on the following special rational numbers.
Decimals
A digit is one of the numbers #0#, #1#, #2#, #3#, #4#, #5#, #6#, #7#, #8#, #9#.
A number of the form #\frac{a}{10^n}#, wherein #a# and #n# are integers with #n\ge0# is called a decimal number. This number is often written as #a# (possibly complemented with zeros from the left side) with a decimal point before the #n#-th digit from the right.
The digits behind the decimal point are called the decimals.
The digits are those numbers that can be written with exactly one symbol. The fact that there are #10#, is due to our habit of working with the decimal system (decima comes from "a tenth" in Latin).
Decimals are also called the decimal digits after the decimal point.
| #\frac{3}{10}= .3 =0.3# | # \frac{1}{2}=\frac{5}{10} = 0.5# | |
| #\frac{192}{10^0}= 192# | #\phantom{xyzabcd}# | #\frac{31 51 49 2}{10^6}=3.141592# |
| #\frac{-12}{10^7}=-0.00 00 012# | # \frac{-1234567890}{10^{5}}= -12345.6789# |
Arithmetic operations of decimal numbers
Addition, subtraction and multiplication of decimal numbers results in another decimal number.
Division of a decimal number by #2# or #5# as well.
But in general, dividing a decimal number by an other decimal number does not result in a decimal number.
Let #a#, #b#, #m#, #n# be integers with #m\ge0# and #n\ge0#, in such a way that #\frac{a}{10^n}# and #\frac{b}{10^m}# are any two decimal numbers.
Given the theory Addition and subtraction of fractions, the addition of two decimal numbers is given by \[\frac{a}{10^n}+\frac{b}{10^m}=\frac{10^ma+10^nb}{10^{m+n}}\tiny.\] This shows that the sum of two decimal numbers is another decimal number.
The subtraction of two decimal numbers works in the same manner.
The product of two decimal numbers is \[\frac{a}{10^n}\cdot\frac{b}{10^m}=\frac{a\cdot b}{10^{m+n}}\tiny,\] hence again another decimal number.
Dividing #\frac{a}{10^n}# by #10# results in #\frac{a}{10^{n+1}}#, which clearly is another decimal number.
Division of a decimal number by #2# produces the same result as multiplication by #5# (a decimal number) and dividing by #10#. We have just seen that this results in another decimal number.
For #5# instead of #2# the reasoning is the same (with #2# instead of #5# ).
In order to show that a division does not always work like this, we take #a=1#, #n=0#, #b=3# and #m=0#, is such a way that we are dealing with the decimal numbers #1# and #3#. Suppose the division of #1# by #3# results in a decimal number. Then there are integers #c# and #p# with #p\ge0# such that #\frac{1}{3} = \frac{c}{10^p}#. We rewrite this to #c=\frac{10^p}{3}#. The left-hand side is an integer, but the right-hand side is the result of the division of #10^p# by #3#. But this division always has remainder #1# and is therefore not an integer, which contradicts the left-hand side. This contradiction means that the division of #1# by#3# does not result in a decimal number.
For, division by #20# is the same as multiplying by #5# and dividing by #100#.
- Multiplying #753.469# by #5# results in the decimal number #3767#.
- Division of #3767# by #100# corresponds with the digits moving #2# places to the left of the decimal point. (If there are not enough decimal places you can of course add zeros) Hence, the answer is #37.67345#.
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