Arithmetic » Numbers

Contents

Numbers
Decimal numbers
    Decimals
    Point or comma as decimal separator?
Rounding off
Subdivision in number systems (natural, whole, rational, real and complex) Other numeral systems

Numbers

Let's begin with looking at an example: the number 42 583.
This is a number that consists of five digits.
We pronounce this number as forty-two thousand and five hundred (and) eighty-three.
The 'and' before eighty-three is often left out in American English.
Every digit has a certain value. The position of the digit indicates its value.

Our numbers consist of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Ten parts is 'decima', so we use the decimal numeral system.
Also known as the decimal number system or base-10 number system.
The digits are called Arabic numerals, which is a little confusing as in current day Arabia, other digits are used.
Because the position of the digit indicates its value, the numeral system is a positional numeral system.
Conclusion: We work with a decimal positional numeral system.

Value

In the table underneath you can see what the value of each digit is of the nine digit number 378 946 123.
You might have noticed a space is used every three digits instead of the comma you maybe expected. Further down in this article under the heading 'Point or comma as decimal separator?' you can read why this is.

DigitValueValue as a number
3three hundred million300 000 000
7seventy million70 000 000
8eight million8 000 000
9nine hundred million900 000
4forty thousand40 000
6six thousand6000
5five hundred500
2twenty20
3three3

The number 378 946 123 is pronounced 'three hundred and seventy-eight million nine hundred and forty-six thousand one hundred and twenty-three' or as 'three hundred seventy-eight million nine hundred forty-six thousand one hundred twenty-three' if you are not used to using and in the numbers.

Every position of a digit gives you the value of a power of 10.
In the following table you can see this clearly.

109108107106105104103102101100
5 378 946 1235378946123

Bigger numbers (short scale)

NameValue
Thousand 103 = 1000
Million 106 = 1 000 000
Billion 109 = 1 000 000 000
Trillion 1012 = 1 000 000 000 000
Quadrillion 1015 = 1 000 000 000 000 000
Quintillion 1018 = 1 000 000 000 000 000 000
Sextillion 1021 = 1 000 000 000 000 000 000 000
Septillion   1024 = 1 000 000 000 000 000 000 000 000

The list counts on with the Latinish numbers. See Wikipedia for a couple more.

The so called short scale is used in most English speaking countries. In continental Europe, the long scale is used. A billion in England is therefore different than the 'billion' in France or the 'biljoen' in the Netherlands.
Confusing, right? See this Wikipedia article why this is.

Bigger numbers (long scale)

NameValue
Thousand 103 = 1000
Million 106 = 1 000 000
Milliard 109 = 1 000 000 000
Billion 1012 = 1 000 000 000 000
Billiard 1015 = 1 000 000 000 000 000
Trillion 1018 = 1 000 000 000 000 000 000
Trilliard 1021 = 1 000 000 000 000 000 000 000
Quadrillion   1024 = 1 000 000 000 000 000 000 000 000

With computer memory other names are used. See computer memory.

Decimal number

A decimal number is not a whole number.
Part of the number is behind the (decimal) point.
Examples are 3.5 and 4.375.
These are pronounced three point five and four point three seven five.
You call the (decimal) point the decimal separator.
Sometimes people will call the decimal point a dot, full stop or period.

Decimals

A digit behind the decimal separator is called a decimal.
So 3.5 has one decimal and 4.375 has three decimals.

Point or comma as decimal separator?

In countries where the point is the decimal separator sometimes commas are used to make for ease of reading. The commas are written every three digits in front of the decimal separator. For example 12,457,846.12.

However, in large parts of the world the comma is used as a decimal separator. In those countries the point is used for ease of reading. So in these countries you will get 12.457.846,12.

You can understand that a number like 35.283 can be interpreted in both 'systems'.
This is 352831000 in some countries, but more than 35 thousand in other countries!
That is the reason that internationally the official endorsed notation is to only use the point or comma as decimal separator.
For ease of reading a (small) space is used.
That way there is no danger of any confusion.
In four digit numbers a space is not used.

So never write 12,457,846.12 anymore!
In countries with the dot as decimal separator it will become 12 457 846.12.
In countries with the comma as decimal separator it will become 12 457 846,12.

The space can also be used for very small numbers. One billionth can therefore be written as 0.000 000 001.

Rounding off

Sometimes you will have to round off numbers.
How this works can be found at rounding off.

Subdivision in number systems

Our numbers are divided into natural, whole, rational, real and complex numbers.
Look at Number systems.

Other numeral systems

Look at Roman, binary, octal and Babylonian numerals


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