The real numbers include all of the measuring numbers. Real numbers are usually written using decimal numerals, in which a decimal point is placed to the right of the digit with place value one. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus
123.456\,
represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six ". In the US and UK and a number of other countries, the decimal point is represented by a period, whereas in continental Europe and certain other countries the decimal point is represented by a comma. Zero is often written as 0.0 when necessary to indicate that it is to be treated as a real number rather than as an integer. Negative real numbers are written with a preceding minus sign:
-123.456.\,
Every rational number is also a real number. To write a fraction as a decimal, divide the numerator by the denominator. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called irrational. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as 1/2 and the real number 0.333... (forever repeating threes) can be written as 1/3. On the other hand, the real number π (pi), the ratio of the circumference of any circle to its diameter, is
\pi = 3.14159265358979....\,
Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include
\sqrt{2} = 1.41421356237 ...\,
(the square root of 2, that is, the positive number whose square is 2).
Thus 1.0 and 0.999... are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example 2/2, 3/3, 1.00, 1.000, and so on.
Every real number is either rational or irrational. Every real number corresponds to a point on the number line. The real numbers also have an important but highly technical property called the least upper bound property. The symbol for the real numbers is R or \{R}.
Negative numbers are numbers that are less than zero. They are the opposite of positive numbers. For example, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal of the same amount. Negative numbers are usually written by a negative sign (also called a minus sign) in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the set of negative numbers is combined with the natural numbers and zero, the result is the set of integer numbers, also called integers, Z also written \{Z}. Here the letter Z comes from the German word Zahl, (plural Zahlen).
A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. The fraction m/n or
m \over n \,
m represents equal parts, where n equal parts of that size make up one whole. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is:
{1 \over 2} = {2 \over 4}.\,
If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written −7/1. The symbol for the rational numbers is Q (for quotient), also written \mathbb{Q}.
In mathematics, an irrational number is any real number which cannot be expressed as a fraction p/q, where p and q are integers, with q non-zero and is therefore not a rational number. Informally, this means that an irrational number cannot be represented as a simple fraction. It can be proven that irrational numbers are precisely those real numbers that cannot be represented as terminating or repeating decimals, although mathematicians do not take that to be the definition. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational.[1] Perhaps the best-known irrational numbers are π, e and √2.[2][3][4] When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common. A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as I.