The document discusses rational and irrational numbers. It defines rational numbers as numbers that can be represented as fractions with integer numerators and non-zero denominators. Irrational numbers are numbers that cannot be expressed as fractions and have infinite, non-repeating decimal representations. The set of all rational and irrational numbers is called the set of real numbers. The document also discusses converting between decimals and fractions, as well as different types of intervals on the real number line including open, closed, and half-closed intervals.

1. BILINGUAL SECTION – MARÍA ESTHER DE LA ROSA
REAL NUMBERS
1. FRACTIONS. RATIONAL NUMBERS
A rational number is a number that can be represented as the quotient of two integers, with a non-zero denominator. It
is represented by .
In practice, the terms rational number and fraction are used as synonyms.
2. OPERATIONS WITH RATIONAL NUMBERS
This is explained in the theory of the past year
3. DECIMAL TO FRACTION
There are three different types of decimal numbers: Exact decimal, periodicly pure an periodicly mixed.
• Exact Decimal to Fraction
The numerator of the fraction has the number given without the decimal point and the denominator has the
number 1 followed by as many zeros as there are decimal places in the original decimal number.

2. • Repeating Decimal to Fraction
1. Periodicly pure:
The numerator of the fraction has the number given without the decimal point, minus the integer part (the
number before the decimal), and denominator has the number formed by as many digits of nine as there are
decimal points (quantity of numbers after the decimal point).
2. Periodicly mixed
The numerator of the fraction has the number given without the decimal point, minus the integer
part followed by the non-recurrent decimal places and the denominator has a number formed by as
many digits of nine as the number has recurrent decimal places, followed by so many zeros as the digits have
the decimal non-recurrent.
ON LINE ACTIVITIES:
http://www.topmarks.co.uk/maths-games/7-11-years/fractions-and-decimals
http://www.abcya.com/fractions_to_decimals.htm
http://www.sheppardsoftware.com/mathgames/fractions/FractionsToDecimals.htm
4. IRRATIONAL NUMBERS
A number is irrational if there are infinite decimal places non-recurrent, therefore it can not be expressed in the form of
fraction.
The best known irrational number is , which is defined as the relationship between the perimeter of the circle and its
diameter.
= 3.141592653589...

3. Other irrational numbers are:
• The number e e = 2.718281828459...
• The golden ratio, , used by artists from all periods (Phidias, Leonardo da Vinci, Alberto Dürer, Dalí,..) in
the proportions of their works.
5. REAL NUMBERS: The set formed by the rational numbers and irrational numbers is the set of real numbers,
and is designated by .
6. INTERVALS
An interval is a to set of real numbers between two given points: a and b, which are called ends of the interval. There
are three different types of intervals:
1. Open Interval (a, b): is the set of all real numbers greater than a and smaller than b.
(a, b) = {x / a < x < b}
2. Closed Interval [a, b]: is the set of all real numbers greater than or equal to a and less than or equal to
b.
[a, b] = {x / a ≤ x ≤ b}
3. Half-Closed Intervals (They are also called half-open intervals)
(a, b] = {x / a < x ≤ b}
[a, b) = {x / a ≤ x < b}

4. 8. HALF LINE
The half-line is determined by a number.
(a, +∞) = {x / a < x < +∞}
[a, +∞) = {x / a ≤ x < +∞}
(-∞, a) = {x / -∞ < x < a}
(-∞, a] = {x / -∞ < x ≤ a}