1. Decimal numbers

Decimal numbers

When a fraction is converted into a decimal number, the result is either a finite decimal number or an infinite periodic decimal number. In the latter, the same series of decimals is repeated indefinitely. Infinite periodic decimal numbers are represented by writing the period of the decimal number at least twice and putting three dots at the end of the number. Alternatively, the period can be written only once and an an overline dash can be placed over it: either [[$ 2,1313 ... $]] or [[$ 2,\overline{13} $]].

Each finite decimal number or infinite periodic decimal can be converted into a fraction. However, non-periodic infinite decimal numbers are irrational numbers that cannot be written in fractional form.

When performing calculations with decimal numbers, special attention must be paid to the accuracy of the answers. All significant zeros except the zeros at the beginning of a decimal number and at the end of an integer are considered to be significant numbers. In some cases, the zeros at the end of the integer may be significant, as evidenced by the context.

Addition and subtraction

The answer is given at the accuracy of as many decimal places as is in the most inaccurate initial value.
If the answer is required with a certain accuracy, all output values must be calculated at least one unit more accurately for certainty.

Multiplication and division

The answer must have at most as many significant figures as there are in the most inaccurate initial value.
If the answer is required with a certain accuracy, all output values must be calculated at least one unit more accurately than at least one significant number for certainty.

Example 1

Mark the decimals with an overline.

a) [[$ 0,88... = 0,\overline{8} $]]

b) [[$ 0,5454... = 0,\overline{54} $]]

c) [[$ 1,123123... = 1,\overline{123} \;\;\;\; {\color {red} {\text {The first number one in the number does not belong to the sequence.}}} $]]

Example 2

Convert the decimals to fractions.

a) [[$ 0,7 = \displaystyle\frac {7} {10} $]]

b) [[$ 0,25 = \displaystyle\frac {25} {100}^{(25} = \displaystyle\frac{1}{4} $]]

c) [[$ 8,05 = 8 \displaystyle\frac {5} {100} ^{(5}= 8 \displaystyle\frac {1} {20} $]]

d) [[$ 1,414213562... \;\;\;\; {\color {red} {\text {An unperiodic infinite decimal cannot be represented as a fraction.}}} $]]

Example 3

Convert the number [[$ x = 1,123123… $]] into a fraction.

Because the sequence of decimals is repeated indefinitely, the decimals cannot be viewed as is. There are three numbers in the decimal number section [[$ 123 $]]. If the decimal number is multiplied by [[$ 1000 $]], the decimal part of the new number is the same as the decimal part of the original number.

[[$ 1000x = 1123,123123… $]]

Subtracting the numbers [[$ 1000x $]] and [[$ x $]] completely eliminates the decimal parts of the numbers.

[[$ \begin{equation} \begin{cases} 1000x = 1123,123123... \ \\ x = 1,123123... \end{cases} \end{equation} \\ \\ \color {red}- {\color {red} {\text {Both sides of the equation are subtracted from each other.}}} $]]

[[$ \begin{align} 999x &= 1122 \ \\ x &= \frac {1122} {999} \; \; \; \; \; \; {\color {blue} {\text {Both sides of the equation are reduced by three.}}} \ \\ x &= \frac {374} {333} \end{align} $]]

Answer: The number [[$ 1,123123... $]] gives the fraction [[$ \displaystyle\frac {374} {333} $]].

Example 4

Let us examine the number of significant figures in different numbers.

a) The integer [[$ 40 \:000 $]] has one significant number.

b) The decimal number [[$ 0,140 $]] has three significant numbers.

c) The decimal number [[$ 0,02 $]] has one significant number.

d) The decimal number [[$ 79,10 $]] has four significant numbers.

e) The integer [[$ 7001 $]] has four significant numbers.

f) The integer [[$ 310 $]] has two or three significant numbers, depending on whether the number is rounded.