Decimal numbers

What are decimal numbers?

Rational numbers that can be expressed as a fraction whose denominator is a power of ten (10, 100, 1000, etc.) are called decimal numbers or, less correctly, decimal fractions. Strictly speaking, a decimal fraction is the explicit fraction (with a power of ten denominator), and is therefore one possible way of expressing a decimal number. Other ways to express decimal numbers are to use decimal notation or percentages, which we will discuss later. Some decimal fractions:

$$\frac{30}{10},\frac{3}{10},\frac{23}{100},\frac{12345}{1000}$$

Reducing a decimal fraction (if possible) always results in a fraction whose denominator is a product of a power of 2 and a power of 5. Therefore, all fractions whose denominator is a product of a power of 2 and a power of 5 represent a decimal number. And vice versa: a decimal number can only be expressed as a fraction if its denominator is a product of a power of 2 and a power of 5. This is because powers of 10 (10, 100, 1000, etc.) have only twos and fives as prime factors.

Decimal notation

Any decimal number can also be expressed in decimal notation, and vice versa:

In which:

In 230 the 23 is in the tens position. In $\frac{23}{100}=0.23$ the 23 is in the hundredths position: "zero point two three" or "twenty-three one-hundredths". In $\frac{12345}{1000}=12.345$ the 12345 is in the thousandths position.

Between the ones position and the tenths position or, in other words, between the integer part and the fractional part of a number, a decimal mark or decimal separator is inserted. In English-speaking countries (and parts of Asia) a period is used as decimal separator: 2.063, pronounced as "two point zero six three". In most European countries a comma is used as decimal separator. International standard (SI/ISO 31-0) is the use of a period as decimal separator.

Sometimes a zero integer part is omitted: 0.001 ("zero point zero zero one") is the same as .001 ("point zero zero one").

Thus, a decimal number can be expressed as a ratio of integers (decimal fraction) or in decimal notation. And a decimal notation always expresses a decimal number. An expression in decimal notation is also referred to as a decimal numeral (or incorrectly a decimal number) or simply a decimal. However, in some sources "decimal" refers to the individual digits after the decimal separator. For example, 23.001 has 3 "decimals" and the third "decimal" is 1.

Some examples:

A decimal fraction can also be converted to a decimal notation by applying long division:

In the next section (non-terminating decimal expansions) we will see what happens if we apply long division to a non-decimal fraction.

Learn the next decimal numbers by heart:

$$\frac{1}{2}=0.5,\frac{1}{4}=0.25,\frac{1}{5}=0.2,\frac{1}{8}=0.125,\frac{1}{25}=0.04,\frac{1}{125}=0.008$$

$$\frac{1}{20}=0.05,\frac{1}{40}=0.025,\frac{1}{50}=0.02,\frac{1}{80}=0.0125$$

Now write $\frac{3}{4}$ , $\frac{4}{5}$ , $\frac{3}{8}$ and $\frac{1}{16}$ in decimal notation, without using long division or a calculator. And how would you calculate $0.4+\frac{3}{5}$ if you had no calculator?

Non-terminating decimal expansions

Non-decimal numbers cannot be precisely expressed in decimal notation. However, they can be approximated to an arbitrary precision by a decimal. As an example we use long division to determine the decimal expansion of 1/3:

The example above shows that 1/3 requires an endless sequence of threes after the decimal point in order to be precisely represented by a decimal notation. The long division never ends, and the same step repeats over and over again, adding more and more threes to the result. The longer the sequence of threes in the decimal expansion, the more accurately 1/3 is approximated.

In the examples above, 1/25 is a decimal number and thus can be expressed precisely using a decimal representation (0.04). The decimal expansion terminates eventually. Of course the long division could be continued, resulting in endlessly repeating trailing zeros in the decimal expansion (0.04000...). But trailing zeros do not affect the value of the number and are generally omitted, except when they express significant figures, indicating "precision".

In the examples above, 41/333 is non-terminating. The long division does not end with remainder zero. The long division can be continued as long as you wish, resulting in an arbitrary long sequence of trailing digits. Because it is impossible to write infinitely many digits, the decimal notation never precisely represents the rational number. However, we see that a same sequence of digits eventually starts repeating endlessly: the decimal expansion eventually becomes periodic. With 1/3 this recurring sequence is 3, with 41/333 the recurring sequence is 123.

There are several notations to indicate an infinitely repeating or recurring part in a decimal expansion. Common use is a vinculum, a horizontal line, over the first repeating digit(s).

In a long division, every subtraction step results in a "remainder" that must be smaller than the denominator. So, there is a finite number of those possible "remainders" and thus a same "remainder" must eventually appear for the second time. That is when the repeating must start because the digit brought down is always zero and a step identical to a step that appeared before, starts a new sequence identical to a sequence that appeared before. This proves that every rational number can be expressed in decimal representation with either a terminating decimal expansion (repeating sequence is zero), or with a non-terminating decimal expansion that eventually starts to endlessly repeat a finite sequence of digits.

The period of a repeating sequence, that is, the number of digits in a repeating sequence, can never be more than the denominator minus one. In the first of two examples above the possible "remainders" are 1,2,3,4,5,6 and they all appeared in the long division.

$$\frac{1}{29}=0.\overline{0344827586206896551724137931}$$ The repeating sequence of $\frac{1}{97}$ contains 96 digits!

Next some examples. Can you check them using long division?

$$\frac{2}{3}=0.\overline{6}$$ $$\frac{1}{6}=0.1\overline{6}$$ $$\frac{1}{9}=0.\overline{1}$$ $$\frac{252455}{33300}=7.58\overline{123}$$

Conversion

We can use long division to convert a fraction to a (non-terminating) decimal expansion. But how to convert a non-terminating decimal representation to a fraction? We can try to reason to a conversion, for instance with 0.333...:

$$\begin{array}{l}0.3=\frac{3}{10}=\frac{9}{30}=\frac{10-<!--\; -\; -->1}{30}=\frac{1}{3}-<!--\; -\; -->\frac{1}{30}\\ 0.33=\frac{33}{100}=\frac{99}{300}=\frac{100-<!--\; -\; -->1}{300}=\frac{1}{3}-<!--\; -\; -->\frac{1}{300}\\ 0.333=\frac{333}{1000}=\frac{999}{3000}=\frac{1000-<!--\; -\; -->1}{3000}=\frac{1}{3}-<!--\; -\; -->\frac{1}{3000}\end{array}$$

The more trailing threes you add to the decimal expansion, the closer you get to 1/3. The fraction that is subtracted from 1/3 becomes smaller and smaller, approaching "infinitely small" as the number of trailing threes approaches "infinitely many".

Let's try a different approach and see if we can find a more general method:

$$\begin{array}{rl}0.333\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\times <!--\; \times \; -->9=& 0.333\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\times <!--\; \times \; -->\left(10-<!--\; -\; -->1\right)\\ \phantom{0.333\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\times <!--\; \times \; -->9}=& 3.333\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}-<!--\; -\; -->0.333\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\\ \phantom{0.333\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\times <!--\; \times \; -->9}=& 3\\ \frac{0.333\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\times <!--\; \times \; -->9}{9}=& \frac{3}{9}\\ 0.333\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}=& \frac{3}{9}=\frac{1}{3}\end{array}$$

$$\begin{array}{rl}0.123123\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\times <!--\; \times \; -->999=& 0.123123\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\times <!--\; \times \; -->\left(1000-<!--\; -\; -->1\right)\\ \phantom{0.123123\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\times <!--\; \times \; -->999}=& 123.123123\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}-<!--\; -\; -->0.123123\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\\ \phantom{0.123123\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}\times <!--\; \times \; -->999}=& 123\\ 0.123123\mathrm{\dots <!--HORIZONTAL\; ELLIPSIS\; ...\; -->}=& \frac{123}{999}=\frac{41}{333}\end{array}$$

The above examples show that a decimal expansion with integer part zero and the repeating sequence first occurring right after the decimal point, can be expressed as a fraction with the repeating sequence as numerator and a denominator that consists only of the digits 9, as many as there are digits in the repeating sequence:

$$0.\overline{6}=\frac{6}{9}=\frac{2}{3}$$ $$0.\overline{18}=\frac{18}{99}=\frac{2}{11}$$ $$0.\overline{012}=\frac{012}{999}=\frac{12}{999}=\frac{4}{333}$$

We can use this method to convert any decimal expansion to a fraction:

$$\begin{array}{l}0.00\overline{4}=\frac{1}{100}\times <!--\; \times \; -->0.\overline{4}\\ \phantom{0.00\overline{4}}=\frac{1}{100}\times <!--\; \times \; -->\frac{4}{9}=\frac{4}{900}=\frac{1}{225}\end{array}$$ $$\begin{array}{l}1.23\overline{4}=\frac{1}{100}\times <!--\; \times \; -->123.\overline{4}=\frac{1}{100}\times <!--\; \times \; -->\left(123+0.\overline{4}\right)\\ \phantom{1.23\overline{4}}=\frac{1}{100}\times <!--\; \times \; -->\left(123+\frac{4}{9}\right)=\frac{123}{100}+\frac{4}{900}\\ \phantom{1.23\overline{4}}=\frac{1107}{900}+\frac{4}{900}=\frac{1111}{900}=1⁤<!--\; INVISIBLE\; PLUS\; -->\frac{211}{900}\end{array}$$

Thus, any rational number can be expressed as a decimal expansion. Decimal numbers can be expressed with a terminating decimal expansion, and all other rational numbers can only be expressed with a non-terminating repetitive decimal expansion.

Rounding

Fractional numbers are often represented in decimal notation, rather than as a fraction expressed with a numerator and denominator. It is often easier to compare fractional numbers in decimal notation than expressed with a numerator and denominator. And computers generally store and work with numbers in decimal (or binary) representation.

It is impossible to involve an infinite number of digits, so non-decimal fractional numbers (with non-terminating decimal expansions) are generally approximated by decimal fractional numbers (with terminating decimal expansions). This is usually not a problem, because in practice no one ever needs infinite precision. Nevertheless, it does introduce errors, and errors may add up or may be multiplied in further calculations.

Approximation by decimal fractions is done by rounding. Non-terminating decimal expansions are rounded depending on the desired precision.

It is up to the user to decide how many digits after the decimal point should be involved. The more digits, the more accurate the approximation is.

Earlier we rounded quantities to their significant figures. The same procedures can be used to round a decimal expansion. In the example above, the first 1/3 is rounded to (whole) tenths or rounded to the nearest tenth: $\frac{1}{3}\approx \; <!--\; \approx \; -->\frac{3}{10}$ . The second is rounded to hundredths, the third is rounded to thousandths.

We have seen that 2.9349999... = 2.935. Hence:

Arithmetic with decimal numbers

Adding and subtracting

Adding and subtracting decimal numbers in decimal notation is pretty straightforward. The same algorithms can be used as before:

Multiplication and division by powers of 10

Multiplying a number by a power of 10 moves the decimal point to the right an equal number of digits as zeros the power of 10 has.
Dividing a number by a power of 10 moves the decimal point to the left an equal number of digits as zeros the power of 10 has.

Thirteen times ten. Ten has one zero. The decimal point moves one position to the right, turning 13 into 130. Thirteen divided by a hundred. A hundred has two zeros. The decimal point moves two position to the left, turning 13 into 0.13.

Moving the decimal point works because we are simply adding or removing zeros to the denominator:

Long multiplication

When we use the multiplication algorithm (long multiplication) we "remove" the decimal points first. In the result we finally insert a decimal point again. For instance 34.14 × 0.5: We use the algorithm to calculate 3414 × 5. This is equal to calculating 34.14 × 100 × 0.5 × 10 = 34.14 × 0.5 × 1000. So the result must be divided by 1000 to get the answer to 34.14 × 0.5. Or in other words:

Before using the algorithm, how much do you think a half of 34.14 is?

So what we can do is simply count the total number of positions after the decimal points in the original factors: 3 in the example above (the position of 1,4 and 5). Then, in the result, move the decimal point from the end this much places to the left: from 17070 to 17.070 in the example above.

An other example: 1.37 × 0.052 = ...

An other example: write ${2.4}^2$ as a decimal:

Long division

When we use the division algorithm (long division) we "remove" the decimal point from the divisor i.e. the denominator first.

Before using long division, how much do you think 100.275/25 approximately is?

Fractions vs. decimals

As we have learned, decimals are not appropriate if you want to keep exact results: 1/3 is an exact number but 0.3333 is only an approximation of that number. In addition, decimals often involve more digits than the corresponding fraction, which, in some cases, makes calculating more complicated. For instance, 0.125 uses more digits than 1/8. A calculation like 1/8 × 32 = 4 is fairly easy, while 0.125 × 32 = 4 takes a lot more work. Nevertheless, decimals are widely used. Why?

The same algorithms for basic arithmetic operations on integers can be used, except you have to keep track of where the decimal point goes. And you only need one format to represent any number: the decimal notation. For these (and other) reasons computers and computer software generally use decimals and not fractions.

One of the advantages of decimal notation is the ability to express numbers in a uniform way. A half in decimal notation is 0.5, while in fractions there are infinite ways to express a half because the denominator can be any even number: 1/2, 11/22, 29/58 etc, etc. With decimals the denominator is always a power of 10. We are comfortable with powers of 10 because 10 is the base of our numeral system. We have a good and immediate sense of where 0.418530 sits between 0 and 1. With the fraction 131/313, this is not so clear. And comparing fractions with different denominators: most people will probably immediately agree with 0.6 > 0.57 but 3/5 > 4/7 is not that obvious.

Using decimals has many advantages. But non-terminating decimals have to be rounded off. And rounding errors may accumulate in calculations. Given fractions in mathematical equations should generally not be converted into decimals. For example, we want to derive a formula for the volume of a shape made up of two identical right circular cones (radius r, height h) that share the same circular base and point in opposite directions. For the volume of the total shape we need to multiply the volume of a single right circular cone ($\frac{1}{3}<!--\; INVISIBLE\; TIMES\; -->\mathrm{\pi }<!--\; INVISIBLE\; TIMES\; -->r^2<!--\; INVISIBLE\; TIMES\; -->h$) by 2:

We maintain the fractions and do not converse to decimals. This way we obtain an exact result.

What if we would have converted 1/3 to 0.33? The decimal in the result would have been 0.66. But 2/3 should be rounded to 0.67.

And what if we want to calculate the volume of an actual piece with this shape rounded to the nearest thousandth or the nearest millionth? Therefore, we maintain the fractions in the general formula and only the result of a calculation of the volume of an actual piece with this shape will be represented as a decimal, rounded to the desired precision.

Decimals are particularity suitable for measured values or values to be measured. Measured values are approximations anyway and also often represented as (terminating) decimals by the measuring instrument. It makes no sense to convert these values into fractions to maintain precision in calculations. And unlike fractions, decimals provide a way of indicating how precise your measurement was (see significant digits).