Decimal numbers

Calculating with numbers: Decimal numbers

Decimal numbers

In natural sciences, the value of a quantity is often indicated in decimal notation. For example, a weight of half a kilogram is noted as 0.5 kg. In the Netherlands and Belgium they are called kommagetallen because the comma is used as a separator in a decimal number; the same holds for Germany and France. But on calculators and in English-speaking countries the decimal point is used and one speaks of floating-point numbers. For this reason we will use the convention of a decimal point.

Decimal number

A decimal number is a number in a decimal system with a decimal point: 2 is not a decimal number, but 2. and 2.0 is. The digits after the decimal point are called the decimals.

Mathematically, 2.1 is the same as 2.10 and 2.100, and so on, because an extra zero at the end does not change the value of the number. From a scientific point of view, the decimal numbers are different: then, numbers are often the results of measurements or calculations and the extra zeros indicate their accuracy.

Examples

\(\frac{3}{2}=1.5\)

\(\frac{1}{8}=0.125\)

1/2 or 0.5? Mathematicians do not consider numbers such as \(\frac{1}{2}\) and \(\frac{1}{4}\) the same as \(0.5\) and \(0.25\), respectively. The first notation belongs to doing mathematically exact calculations and the second notation is more something for applications in which calculations with rounding is used.

Decimal notation Decimal notation was introduced in Europe by Simon Stevin (1548-1620) in his book De Thiende ("The thenth") from 1585.

Construction of a decimal number The decimal number \(123.45\) consists of 1 hundred, 2 tens, 3 units, 4 tenths and 5 hundredths because \[123.45=1\times 100+ 2\times 10 + 3 + 4\times 0.1 + 5\times 0.01\] We also conclude from this that every decimal number is actually a fraction: \(123.45=\frac{12345}{100}=\frac{2469}{20}\).

Moving the decimal point \[\begin{aligned} \text{decimal point to the right}\qquad & \text{decimal point to the left} \\\begin{aligned} 43.15 &= 43.15\times 1 \\ 431.5 &= 43.15\times 10 \\ 4315 &= 43.15\times 100 \\ 43150 &= 43.15\times 1000\end{aligned} \qquad &\begin{aligned} 43.15 &= 43.15\times 1 \\ 4.315 &= 43.15\times 0.1 \\ 0.4315 &= 43.15\times 0.01 \\ 0.04315 &= 43.15\times 0.001\end{aligned} \end{aligned}\] By the way, multiplying by \(0.1\) is the same as dividing by \(10\). So in the right column you can also read \(4.315=43.15\div 10\). And multiplying by \(0.01\) is the same as dividing by \(100\). And so forth.

Rounding

A decimal number can be rounded to a certain number of decimal places. Rounding up or down depends on the value of the next decimal. For example, when rounding to three decimal places, the value of the fourth decimal is decisive: if it is less than 5, it is rounded down, otherwise it is rounded up.

When rounding down, the decimal to which it is rounded remains the same. When rounding up, a 1 is added. If the relevant decimal is 9, then it becomes a 0 and the digit before it is increased by 1.

Rounding is indicated by using the approximate symbol \({}\approx{}\,\).

Examples

\[\begin{aligned}1.39954&\approx 1\text{ rounded to a natural number,}\\ &\phantom{\approx 1}\;\text{ because }3<5\\ &\approx 1.4\text{ rounded to 1 decimal place,}\\ & \phantom{\approx 1.4}\;\;\text{because }9\ge 5\\ &\approx 1.40\text{ rounded to 2 decimal places,}\\ & \phantom{\approx 1.40}\;\text{ because }9\ge 5\\ &\approx 1.400\text{ rounded to 3 decimal places,}\\ & \phantom{\approx 1.400}\;\,\text{because }5\ge 5\\&\approx 1.3995\text{ rounded to 4 decimal places,}\\ & \phantom{\approx 1.3995}\;\,\text{ because }4< 5\end{aligned}\]