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. 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 12.346 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 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.

It is important to note that these decimals can have very different values, depending on their place value.

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

The following are examples of mixed numbers. - 3 79, - 2 16, - 1 512, 1 34, 2 27, 3 38 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