UPSKILL MATH PLUS

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Learn more### Theory:

If a set of numbers is closed for a particular operation, then it is said to possess the closure property for that operation.

**Rational Numbers:**

**i) Addition:**

Adding two rational numbers result in another rational number. Hence, rational numbers under addition is closed.

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

Example:

$\frac{8}{5}+\phantom{\rule{0.147em}{0ex}}\frac{(-2)}{5}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{8+\left(-2\right)}{5}=\frac{6}{5}\phantom{\rule{0.147em}{0ex}}$

**ii) Subtraction:**

Subtracting two rational numbers result in another rational number. Hence, rational numbers under subtraction are closed.

$\frac{a}{b}-\frac{c}{d}$

Example:

$\frac{8}{5}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\frac{(-2)}{5}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{8-\left(-2\right)}{5}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{10}{5}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}2\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}2/1$

**iii) Multiplication:**

Multiplying two rational numbers result in another rational number. Hence, rational numbers under multiplication is closed.

$\frac{a}{b}\times \frac{c}{d}$

Example:

$\frac{8}{5}\phantom{\rule{0.147em}{0ex}}\times \frac{(-2)}{5}=\phantom{\rule{0.147em}{0ex}}\frac{\phantom{\rule{0.147em}{0ex}}8\times (-2)}{25}=\frac{-16}{25}$

**iv) Division:**

Dividing two rational numbers may result in an undefined number with which is not a rational number. Hence, rational numbers under division is not closed.

$\frac{a}{b}\xf7\frac{c}{d}$

Example:

**i)**$\frac{8\phantom{\rule{0.147em}{0ex}}}{2}\xf7\frac{(-2)}{2}\phantom{\rule{0.147em}{0ex}}=\frac{-16}{4}\phantom{\rule{0.147em}{0ex}}=\frac{-8}{2}$______ Closed.

**ii)**$\frac{8}{0}\xf7\frac{\phantom{\rule{0.147em}{0ex}}(-2)}{0}$_____Not closed because it is undefined.

Important!

Since not all the rational numbers on division operation satisfy the closure property, we can say that the division on the rational number does not satisfy closure property.