Definition of a Rational Number

Rational numbers are of the form

\frac{a}{b}  with the special case that  b\neq0

because division by 0 is meaningless (undefined)

where a is any integer and b is any integer other than 0

a, b\in\mathbb{Z}  with  b\neq0

Recall that the Integers \mathbb{Z} are the positive and negative whole numbers along with 0.

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Definition of Addition of 2 Rational Numbers

Here is a definition of the addition of Rational numbers. This is just a formula and it works.

\frac{a}{b}+\frac{c}{d}=\frac{a\times d}{b\times d}+\frac{b\times c}{b\times d}=\frac{ac}{bd}

Example

\frac{3}{4}+\frac{2}{3}=\frac{3\times3}{4\times3}+\frac{2\times4}{4\times3}=\frac{9+8}{12}=\frac{17}{12}

The best way to add 2 Rational numbers is to find the Least Common Denominator LCD between the 2 fractions. The formula does not always give the LCD, but it will reduce to the right answer. If the 2 denominators have a common factor, then their product will not be the LCD.

Commutative Property

The order in which we add 2 Rational numbers does not matter. It will always give the same sum no matter which Rational number we add to the other.

This is called the Commutative Property of Addition

\frac{a}{b}+\frac{c}{d}=\frac{c}{d}+\frac{a}{b}

Example

\frac{3}{5}+\frac{2}{3}=\frac{2}{3}+\frac{3}{5}

Addition of 3 or more Rational Numbers and the Associative Property

We add 3 or more Rational numbers in the same way, we just must account for each denominator while trying to find a common denominator.

The order in which we add the numbers together also does not matter.

This is called the Associative property of Addition

\frac{a}{b}+(\frac{c}{d}+\frac{e}{f})=(\frac{a}{b}+\frac{c}{d})+\frac{e}{f}=\frac{c}{d}+(\frac{a}{b}+\frac{e}{f})

Example

\frac{3}{4}+(\frac{1}{3}+\frac{5}{6})=\frac{3}{4}+(\frac{2}{6}+\frac{5}{6})=\frac{3}{4}+\frac{7}{6}=\frac{9}{12}+\frac{14}{12}=\frac{23}{12}

(\frac{3}{4}+\frac{1}{3})+\frac{5}{6}=(\frac{9}{12}+\frac{4}{12})+\frac{5}{6}=\frac{13}{12}+\frac{7}{6}=\frac{13}{12}+\frac{10}{12}=\frac{23}{12}

\frac{1}{3}+(\frac{3}{4}+\frac{5}{6})=\frac{1}{3}+(\frac{9}{12}+\frac{10}{12})=\frac{1}{3}+\frac{19}{12}=\frac{4}{12}+\frac{19}{12}=\frac{23}{12}  <h2>Additive Inverse and Identity</h2> <section></section> \frac{a}{b}+(-\frac{a}{b})=0 \frac{a}{b}+0=\frac{a}{b}$

Closure:

The result of adding two rational numbers is another rational number.

a + b

Associative:

The way in which the summands are grouped does not change the result.

(a + b) + c = a + (b + c)

The opposite of the opposite of a number is equal to the same number.

As a result of these properties, the subtraction of two rational numbers is defined as the addition of the minuend plus the opposite of the subtrahend.

a − b = a + (−b)

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.