# decimal system

The decimal system originated in India and reached Europe via Arabic mathematicians. In the decimal system the numbers are written using 10 signs:

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

These numbers have different values according to the position they occupy in the written number. On the furthest right they have unit value, immediately to the left tenfold value, to the left of that one hundredfold, and so on, e.g.

20,349 = 9 units + 4 tens + 3 hundreds + 0 thousands + 2 ten-thousands.

The decimal method writing numbers is possible because every number can be written as a sum of the form

*x* = *a*_{0} × 10^{n} + *a*_{1} × 10^{n–1} + *a*_{2} × 10^{n–2} + ... + *a*_{n} × 10^{0},

where 0 ≤ *a _{i}* < 10, for all

*i*.

E.g. 3059 = 3 × 10

^{3}+ 0 × 10

^{3}+ 5 × 10

^{1}+ 9 × 10

^{0}

3000 + 50 + 9.

The exponent *n*, and the coefficients which appear, are uniquely determined.

The decimal method of writing numbers can be extended to all the real number, e.g.

3.01 = 3 units + 0 tenths + 1 hundredth.

## Decimal fractions

Fractions with denominator 10, 100, 1000, ... are called decimal fractions, 573/1000, 3578/1000.

These fractions can be briefly written as **decimal numbers**: we divide the decimal fraction into a whole number and a 'proper' decimal fraction; we write the whole number first, put a **decimal point** after it, and set down to the right the tenths, hundredths, thousandths, etc.; e.g.

573/1000 = 0.573; 3578/1000 = 3.578.

When we have a proper decimal fraction for clarity we set a 0 to the left of the point. 0.573 is read: 'nought (or zero)-point-five-seven-three'.

## Decimal place

The decimal place is the number of places to the right of a decimal point required to specify a real number to a certain accuracy. For example, the number 4.893302 to three decimal places is written 4.893 (3 d.p.). To five decimal places the same number is 4.89330 (5 d.p.). The convention is to round up the last decimal place if the digit after that place is five or greater, and to round down if it is four or less. Thus the numbers 239.705 and 239.706 are both 239.71 (2 d.p.) to two decimal places, whereas the number 239.704 to two decimal places is 239.70 (2 d.p.). See also significant figures.