Real Numbers The set of real numbers Rational numbers

Recurring decimals (Infinite decimals, period)
Purely recurring decimals

Converting decimal number to a fraction
Converting terminating decimal to a fraction
Converting the purely recurring decimal to a fraction
Converting the mixed recurring decimal to a fraction
Exponential or scientific notation of decimal numbers
Converting from a number to scientific notation
Converting from scientific notation to a decimal number
The set of real numbers
 The set of real numbers, denoted R, is the set of all rational and irrational numbers,  R = Q U I. The real numbers or the reals are either rational or irrational and are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line, the number line or the real line. The set of natural numbers
N = {1, 2, 3,  . . . , n, n + 1,  . . . }, the positive integers used for counting.
The set of whole numbers
N0 = { 0, 1, 2, 3,  . . . }, is just like the set of natural numbers except that it also includes zero.
The set of integers
Z = { . . . , 3, 2, 1, 0, 1, 2, 3,  . . . }, consists of all natural numbers, negative whole numbers and  zero. This means that the set of natural numbers is a subset of integers, i.e., N is a subset of  Z.
The set of rational numbers
Q = { a/b | a, b Î Z, b is not 0 }, is the set of all proper and improper fractions. That is, a ratio or quotient of two integers a and b, where b is not zero.
All integers are in this set since every integer a can be expressed as the fraction a/1 = a. Thus, the set of all natural numbers N is proper subset of integers Z and the set of integers is proper subset of the set of rational numbers, N is proper subset of Z is proper subset of  Q. Rational numbers can be represented as integers, fractions, terminating decimals and recurring or repeating decimals.
The set of irrational numbers, denoted I, is the set of numbers that cannot be written as ratio of two
integers. An irrational number expressed as a decimal never repeat or terminate. The irrational numbers are precisely those numbers whose decimal expansion never ends and never enters a periodic pattern, such as 0.1020030004..., p, Ö2, Ö3, or any root of any natural number that is not a perfect root is an irrational number.
Rational numbers
Decimal representation of rational numbers or fractions
In order to convert a rational number represented as a fraction into decimal form, one may use long division.
By dividing the numerator by the denominator we get a terminating or a recurring decimal.
If the final remainder is 0 the quotient is a whole number or a finite or terminating decimal, i.e. a decimal with a limited number of digits after the decimal point.
 Examples: Sometimes when dividing, the division will never stop as there is always a remainder.
These fractions convert to a recurring, (periodic or infinite) decimals, and they have an unlimited number of digits after the decimal point.
 Examples: A rational number is either a terminating decimal or recurring decimal.
We can determine which fraction will convert to terminating decimal and which to recurring decimal only if the given fraction is expressed in its lowest terms, that is, when its numerator and the denominator have no common factor other than 1.
Expanded form of decimal number, decimal fractions
The integer and fractional parts of a decimal number are separated by a decimal point.
Converting decimals to fractions involves counting the number of places to the right of the decimal point.
This will give the corresponding place value, which then determines the number of zeros that will be used in forming the denominator.
The numerator of the fraction equivalent is the number without the decimal point, and the denominator is 1 followed by the number of zeros corresponding to the number of decimal places.
 Example: Terminating decimals
The irreducible fractions, i.e., the vulgar fractions in lowest terms whose the only prime factors in a denominator are 2 and/or 5 can be converted to terminating decimals.
That is, the terminating decimals represent rational numbers whose fractions in the lowest terms are of the form a/(2n · 5m).
 Examples:  Recurring decimals (Infinite decimals, period)
Purely recurring decimals
The irreducible fractions, i.e., the vulgar fractions in lowest terms whose prime factors in the denominator are other than 2 or 5, that is, the prime numbers from the sequence (3, 7, 11, 13, 17, 19, ...) convert to the purely recurring decimals, i.e., the decimals which start their recurring cycle immediately after the decimal point.
 Examples:  Mixed recurring decimals
The irreducible fractions, i.e., the vulgar fractions in lowest terms whose denominator is a product of 2's and/or 5's besides the prime numbers from the sequence (3, 7, 11, 13, 17, 19, ...) convert to the mixed recurring decimals, i.e., the decimals that have some extra digits before the repeating sequence of digits.
The repeating sequence may consist of just one digit or of any finite number of digits. The number of digits in the repeating pattern is called the period. All recurring decimals are infinite decimals.
 Examples:  All fractions can be written either as terminating decimals or as recurring/repeating decimals.
Converting decimal number to a fraction
Converting terminating decimal to a fraction
Every terminating decimal and recurring decimal can be converted to a fraction a/b, a Î Zb Î N.
 Examples:  Converting the purely recurring decimal to a fraction
When converting the purely recurring decimal less than one to fraction, write the group of repeating digits to the numerator, and to the denominator of the equivalent fraction write as much 9’s as is the number of digits in the repeating pattern..
 Examples: Converting the mixed recurring decimal to a fraction
When converting the mixed recurring decimal less than one to fraction, write the difference between the number formed by the entire sequence of digits, including the digits of the recurring part, and the number formed only by the digits of the non-recurring pattern to its numerator.
To the denominator of the equivalent fraction write as much 9’s as is the number of digits in the repeating pattern and add as much 0’s as is the number of digits in the non-recurring pattern.
 Examples:  Exponential or scientific notation of decimal numbers
Exponential or scientific notation is used to express very large or very small numbers.
A number in scientific notation is written as the product of a number (the coefficient) and a power of 10 (the exponent), i.e.,
coefficient ´ 10exponent
The coefficient should have exactly one non-zero digit to the left of the decimal point. The exponent indicates, how many places the decimal point was moved to the left or to the right. If the decimal point was moved to the left, the exponent is positive, if moved to the right the exponent is negative.
Converting from a number to scientific notation
 Examples: a)   302,567,908 = 3.02567908 · 108      b)   0.000040635 = 4.0635 · 10−5.
Converting from scientific notation to a decimal number
 Examples: a)   2.09085 · 107 = 20908500               b)   7.81 · 10−5 = 0.0000781   College algebra contents 