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Forms of Real Numbers



Fractions are a specific type of ratio that compares a part to a whole. Fractions are rational numbers written in the form where the numerator is the part and the denominator is the whole.

There are many possible ways of reading fractions, but one universal method is saying over Fractions where is less than are called proper fractions. Fractions where is greater than or equal to are called improper fractions.
Applet that shows the names of different fractions and visualizes them using tiles
Fractions are also another way to write a division of the numerator by the denominator.
A fraction like can be simplified to or just It is important to keep in mind that the denominator of a fraction can never be equal to because the quotient of division by is always undefined.


Decimal Numbers

Numbers that lie between integers on the number line can be written as decimal numbers. These consist of an integer part, a decimal point as a separator, and a non-zero decimal part written to the right of the decimal point. Consider the number as an example.
The decimal 12.346 where 12 is an integer part, . is a decimal point, and 346 is a decimal part
The integer part of this number is Since there is a decimal part, the number is greater than but less than Therefore, when plotting on a number line, the point will lie between and
It is important to note that these decimals can have very different values, depending on their place value.


Mixed Number

A mixed number consists of a non-zero integer number and a proper fraction.

The following are examples of mixed numbers.
Consider the graphic representation of different mixed numbers.
An applet that illustrates different mixed numbers
Mixed numbers represent the rational numbers between any two integers.


Writing a Mixed Number as an Improper Fraction
Any mixed number can be written as an improper fraction using the following formula.



The root of a real number expresses another real number that, when multiplied by itself times, will result in In addition to the radical symbol, the notation is made up of the radicand and the index
The resulting number is commonly called a radical. For example, the radical expression is the fourth root of Notice that simplifies to because multiplied by itself times equals
The general expression represents a number which equals when multiplied by itself times.

For any real number and natural number the expression is defined as the root of Note that a root with an even index is defined only for non-negative numbers. Therefore, if is even, then must be non-negative.
power of 1/n as root
Just as with exponents, the most common roots have special names: square roots and cube roots have an index of and respectively.