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The real numbers are **the set that includes the natural, integer, rational and irrational numbers** . It is represented by the letter ?.

The word *real* is used to distinguish these numbers from the imaginary number *i* , which is equal to the square root of -1, or ?-1. This expression is used to simplify the mathematical interpretation of effects such as electrical phenomena.

In addition to the particular characteristics of each set that makes up the superset of real numbers, we mention the following characteristics.

All real numbers have an order:

In the case of fractions and decimals:

Each real number can be written as a decimal. Irrational numbers have endless and unrepeatable decimal places, for example, the number pi ? is approximately 3.14159265358979 ...

All numbers are represented by the ten symbols: 0, 1, 2, 3, 4, 5, 6. 7, 8, and 9, which are called

A symmetric number is one that added with its corresponding natural number gives zero. That is, the symmetric of

Positive integers are numbers greater than zero, while numbers less than zero are negative integers.

Integers are used to:

- represent positive numbers: gains, degrees above zero, distances to the right;
- represent negative numbers: debts, losses, degrees below zero and distances to the left.

This means that he owes 7,000 coins.

Those quantities that cannot be expressed in whole form or as a fraction that are incommensurable are also irrational. For example, the ratio of the circumference to the diameter the number ? = 3.141592…

The roots that cannot be expressed exactly by any whole or fractional number are irrational numbers:

- The sum of two real numbers is closed, that is, if
*a*and*b*? ?, then a + b ? ?. - The sum of two real numbers is commutative, so a + b = b + a.
- The sum of numbers is associative, that is, (a + b) + c = a + (b + c).
- The sum of a real number and zero is the same number; a + 0 = a.
- For each real number there is another symmetric real number, such that its sum is equal to 0: a + (- a) = 0
- The multiplication of two real numbers is closed: if
*a*and*b*? ?, then a. b ? ?. - The multiplication of two numbers is commutative, so a. b = ba
- The product of real numbers is associative: (ab) .c = a. (B .c)
- In multiplication, the neutral element is 1: so, a. 1 = a.
- For each real number
*to*different from zero, there is another real number called the multiplicative inverse, such that: a. a_{-1}= 1. - If
*a,**b**and**c*? ?, then a (b + c) = (a. B) + (a. C)

The discovery of real numbers is attributed to the Greek mathematician Pythagoras. For him there was no rational number whose square is two:

So the ancient Greeks saw the need to call these **irrational numbers** .