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Reference

Concept

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

$part→whole→ ba ←numerator←denominator $

$a$ 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.

Fractions are also another way to write a division of the numerator by the denominator.

$918 =18÷9 $

A fraction like $918 $ can be simplified to $12 ,$ 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

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.

Concept

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

$acb whereais an integer,b<c,andc =0 $

$-397 ,-261 ,-1125 ,143 ,272 ,383 $

Consider the graphic representation of different mixed numbers.
Mixed numbers represent the rational numbers between any two integers.

Any mixed number can be written as an improper fraction using the following formula.

$acb =ca⋅c+b and-acb =-ca⋅c+b $

Concept

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 $n.$
For any real number $a$ and natural number $n,$ the expression $a_{n1}$ 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.

The resulting number is commonly called a radical. For example, the radical expression $416 $ is the

fourth rootof $16.$ Notice that $416 $ simplifies to $2$ because $2$ multiplied by itself $4$ times equals $16.$

$416 =42_{4} =2 $

The general expression $na $ represents a number which equals $a$ when multiplied by itself $n$ times. $ntimesna ⋅na ⋅⋯⋅na =aor(na )_{n}=a$

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

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