A number that can be rational and irrational is called real, therefore this set of numbers is the union of the set of rational numbers (fractions) and the set of irrational numbers (they cannot be expressed as a fraction). Real numbers cover the real line and any point on this line is a real number, and they are designated by the symbol R.
Characteristics of real numbers:
- The set of real numbers is the set of all the numbers that correspond to the points on the line.
- The set of real numbers is the set of all numbers that can be expressed with periodic or non-periodic infinite or finite decimals.
Irrational numbers are distinguished from rational numbers by having infinite decimal places that never repeat themselves, that is, they are not periodic. Therefore, they cannot be exposed as a fraction of two integers. Some irrational numbers are distinguished from other numbers by symbols. For example: ℮ = 2.7182, π = 3.1415926535914039.
In the real line the real numbers are symbolized, each point of the line has a real number and each real number has a point on the line, as a consequence it is not possible to speak of the next in a real number as in the case of natural numbers. Rational numbers are placed on the number line in such a way that in each section, no matter how small, there are infinities. However, and strangely enough, there are infinite gaps that are filled by irrational numbers. Therefore between any two real numbers, X and Y there are rational infinities and irrational infinities, between all of them they fill the line.
Operations with real numbers:
The way to do the operations with real numbers depends on how the numbers are represented. If all the operands are rational numbers, the operations are performed using fractions. If you have to operationalize with irrationals the only way to handle exact values is to leave them as is. If it is necessary to operationalize numerically, it will be necessary to use its decimal representations and since they are infinite decimals, the result can only be given in a close way.
Approximation by default or by excess:
The approximation of irrational numbers in their decimal representation can be:
- By default: if the value to be approximated is less than the number.
- By excess: if the value to be approximated is greater
For example, for the number π, the default approximations are 3 <3.1 <3.14 <3.141 and by excess 3.1416 <3.142 <3.15 <3.2. Rounding or truncation approximation:
Significant figures are all those that are used to express an approximate number, there are two ways to approximate numbers:
By rounding: if the first non-significant figure is 0,1,2,3,4 the previous one remains the same, on the other hand if it is 5,6,7,8,9 the previous figure is increased by one unit, for example: 3, 74281≈ 3.74 and 4.29612 ≈ 4.30.
Truncation approximation: non-significant figures are eliminated, for example: 3.74281≈3.74 and 4.29612 ≈ 4.29.
Scientific notation:
When you want to express very large or very small real numbers, use the scientific notation:
- The integer part made up of a single digit, which cannot be 0.
- All other significant figures are written as a decimal part.
- A power of base ten that gives the order of magnitude of the number.
It is important to emphasize that in scientific notation if the exponent is positive the number is large and if it is negative the number is small, for example: 6.25 x 1011 = 625,000,000,000.
