Reference

Forms of Real Numbers

Concept

Fraction

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

l part→ whole→ a/b l←numerator ←denominator

There are many possible ways of reading fractions, but one universal method is saying a over b. Fractions where a is less than b are called proper fractions. Fractions where a is greater than or equal to b 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. 18/9=18÷9

A fraction like 189 can be simplified to 21, or just 2. It is important to keep in mind that the denominator of a fraction can never be equal to 0 because the quotient of division by 0 is always undefined.

Concept

Decimal Numbers

Numbers that lie between integers on the number line can be written as decimal numbers. Decimal numbers 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 decimal number 12.346.

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 12. Since there is a decimal part, 0.346, the number is greater than 12 but less than 13. Therefore, when plotting 12.346 on a number line, the point will lie between 12 and 13.

Concept

Mixed Number

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

a bc [0.5em] whereais an integer,b

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.

Extra

Writing a Mixed Number as an Improper Fraction

Any mixed number can be written as an improper fraction using the following formula. a bc=a* c+b/c and - a bc=- a* c+b/c

Concept

n^(th) Root

The n^(th) root of a real number a expresses another real number that, when multiplied by itself n times, will result in a. In addition to the radical symbol, the notation is made up of the radicand a and the index n.

The resulting number is commonly called a radical. For example, the radical expression sqrt(16) is the fourth root of 16. Notice that sqrt(16) simplifies to 2 because 2 multiplied by itself 4 times equals 16. sqrt(16) = sqrt(2^4) = 2 The general expression sqrt(a) represents a number which equals a when multiplied by itself n times.

sqrt(a) * sqrt(a) * ... * sqrt(a)_(ntimes)=a or ( sqrt(a) )^n=a

For any real number a and natural number n, the expression a^(1n) is defined as the n^(th) root of a. Note that a root with an even index is defined only for non-negative numbers. Therefore, if n is even, then a must be non-negative.

Just as with exponents, the most common roots have special names: square roots and cube roots have an index of 2 and 3, respectively.

Extra

Recommended exercises