Forms of Real 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 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.

The $n^{\text{th}}$ 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 $\sqrt[4]{16}$ is the fourth root of $16.$ Notice that $\sqrt[4]{16}$ simplifies to $2$ because $2$ multiplied by itself $4$ times equals $16.$ The general expression $\sqrt[n]{a}$ represents a number which equals $a$ when multiplied by itself $n$ times. For any real number $a$ and natural number $n,$ the expression $a^{\frac{1}{n}}$ is defined as the $n^{\text{th}}$ 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.