Properties of Rational Numbers

1. Closure Property

This shows that the operation of any two same types of numbers is also the same type or not.

a. Whole Numbers

If p and q are two whole numbers then

OperationAdditionSubtractionMultiplication Division
Whole numberp + q will also be the whole number.p – q will not always be a whole number.pq will also be the whole number.p ÷ q will not always be a whole number.
Example6 + 0 = 68 – 10 = – 23 × 5 = 153 ÷ 5 = 3/5
Closed or NotClosedNot ClosedClosedNot Closed

b. Integers

If p and q are two integers then

OperationAdditionSubtractionMultiplication Division
Integersp + q will also be an integer then.p – q will also be an integer.pq will also be an integer.p ÷ q will not always be an integer.
Example-3 + 2 = – 15 – 7 = – 25 × 8 = – 405 ÷ 7 = – 5/7
Closed or NotClosedClosedClosedNot Closed

c. Rational Numbers

If p and q are two rational numbers then

OperationAdditionSubtractionMultiplication Division
Rational Numbersp + q will also be a rational number.p – q will also be a rational number.pq will also be a rational number.p ÷ q will not always be a rational number
Example \dfrac{-4}{7}+\dfrac{6}{11}\\ \\=\dfrac{-44+42}{77}\\ \\=-\dfrac{2}{77}. \dfrac{3}{7}-\dfrac{8}{5}\\ \\=\dfrac{15+56}{35}\\ \\=\dfrac{71}{35}. \left(-\dfrac{4}{5}\right)\times\left(-\dfrac{6}{11}\right)\\  \\=\dfrac{24}{55}.p ÷ 0
= not defined
Closed or NotClosedClosedClosedNot Closed

2. Commutative Property

This shows that the position of numbers does not matter.

a. Whole Numbers

If p and q are two whole numbers then 

OperationAdditionSubtractionMultiplication Division
Whole Numberp + q = q + pp – q ≠ q – pp × q = q × pp ÷ q ≠ q ÷ p
Example3 + 2 = 2 + 38 –10 ≠ 10 – 8 – 2 ≠ 23 × 5 = 5 × 33 ÷ 5 ≠ 5 ÷ 3
CommutativeYesNoYesNo

b. Integers

If p and q are two integers then

OperationAdditionSubtractionMultiplication Division
Integerp + q = q + pp – q ≠ q – pp × q = q × pp ÷ q ≠ q ÷ p
ExampleTrue5 – 7 = – 7 – (5)-5 × 8 = 8 × (–5)-5 ÷ 7 ≠ 7 ÷ (-5)
CommutativeYesNoYesNo

c. Rational Numbers

If p and q are two rational numbers then

OperationAdditionSubtractionMultiplication Division
Rational numberp + q = q + pp – q ≠ q – pp × q = q × pp ÷ q ≠ q ÷ p
Example \dfrac{-4}{7}+\dfrac{6}{11}\\ \\=\dfrac{6}{11}+\left(\dfrac{-4}{7}\right) \dfrac{3}{7}-\dfrac{8}{5}\\ \\ \ne\left(-\dfrac{8}{5}\right)-\dfrac{3}{7} \left(-\dfrac{4}{5}\right)\times\left(-\dfrac{6}{11}\right)\\ \\=\left(-\dfrac{6}{11}\right)\times\left(-\dfrac{4}{5}\right) \left(-\dfrac{4}{5}\right)\div\left(-\dfrac{6}{11}\right)\\ \\ \ne\left(-\dfrac{6}{11}\right)\div\left(-\dfrac{4}{5}\right)
Commutativeyes NoyesNo

3. Associative Property

This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.

a. Whole Numbers

If p, q and r are three whole numbers then

OperationAdditionSubtractionMultiplication Division
Whole Numberp + (q + r) = (p + q) + rp – (q – r) = (p – q) – rp × (q × r) = (p × q) × rp ÷ (q ÷ r) ≠ (p ÷ q) ÷ r
Example3 + (2 + 5) = (3 + 2) + 58 – (10 – 2) ≠ (8 -10) – 23 × (5 × 2) = (3 × 5) × 210 ÷ (5 ÷ 1) ≠ (10 ÷ 5) ÷ 1
AssociativeYesNoYesNo

b. Integers

If p, q and r are three integers then

OperationIntegersExampleAssociative
Additionp + (q + r) = (p + q) + r(– 6) + [(– 4)+(–5)] = [(– 6) +(– 4)] + (–5)Yes
Subtractionp – (q – r) = (p – q) – r5 – (7 – 3) ≠ (5 – 7) – 3No
Multiplicationp × (q × r) = (p × q) × r(– 4) × [(– 8) ×(–5)] = [(– 4) × (– 8)] × (–5)Yes
Divisionp ÷ (q ÷ r) ≠ (p ÷ q) ÷ r[(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)]No

c. Rational Numbers

If p, q and r are three rational numbers then

OperationIntegers ExampleAssociative
Additionp + (q + r) = (p + q) + r\dfrac{-1}{2}+\left{\dfrac{3}{7}+\left(-\dfrac{4}{3}\right)\right}=\left{\dfrac{-1}{2}+\dfrac{3}{7}\right}+\left(-\dfrac{4}{3}\right)\\ \\ \dfrac{-1}{2}+\left{\dfrac{9-28}{21}\right}=-\dfrac{1}{2}+\dfrac{19}{21}=\dfrac{17}{42}\\  \\ \left{\dfrac{-7+6}{14}\right}+\left(-\dfrac{4}{3}\right)=\dfrac{-1}{14}+\left(-\dfrac{4}{3}\right)=\dfrac{17}{42}
Hence LHS=RHS
Yes
Subtractionp – (q – r) = (p – q) – r\dfrac{-2}{3}-\left(\dfrac{-4}{5}-\dfrac{1}{2}\right)=\left[\dfrac{2}{3}-\left(-\dfrac{4}{5}\right)\right]-\dfrac{1}{2} \\ \\ \dfrac{-2}{3}-\left(\dfrac{-4}{5}-\dfrac{1}{2}\right)=-\dfrac{2}{3}-\left(-\dfrac{13}{10}\right)=\dfrac{19}{30}\\ \\ \left(\dfrac{-2}{3}-\dfrac{4}{5}\right)-\dfrac{1}{2}=\dfrac{22}{15}-\dfrac{1}{2}=\dfrac{29}{30}
Hence LHS  \neq RHS
No
Multiplicationp × (q × r) = (p × q) × r \dfrac{2}{3} \times\left{-\dfrac{6}{7} \times \dfrac{4}{5}\right}=\left{\dfrac{2}{3} \times-\dfrac{6}{7}\right} \times \dfrac{4}{5}\\ \\ \dfrac{2}{3} \times-\dfrac{24}{35}=-\dfrac{48}{105}\\ \\ -\dfrac{12}{21} \times \dfrac{4}{5}=-\dfrac{48}{105}
Hence LHS = RHS
Yes
Divisionp ÷ (q ÷ r) ≠ (p ÷ q) ÷ r \dfrac{1}{2} \div\left{-\dfrac{1}{3} \div \dfrac{2}{5}\right}=\left{\dfrac{1}{2} \div\left(-\dfrac{1}{3}\right)\right} \div \dfrac{2}{5} \\ \\ \dfrac{1}{2} \div\left{-\dfrac{1}{3} \times \dfrac{5}{2}\right}=\dfrac{1}{2} \times-\dfrac{6}{5}=-\dfrac{3}{5}\\ \\ \left{\dfrac{1}{2} \times-\dfrac{3}{1}\right} \div \dfrac{2}{5}=-\dfrac{3}{2} \times \dfrac{5}{2}=-\dfrac{10}{4}
Hence LHS  \neq RHS
No

Additive Identity

Zero is the additive identity for whole numbers, integers and rational numbers.

IdentityExample
Whole Numbera + 0 = 0 + a = aAddition of zero to whole number2 + 0 = 0 + 2 = 2
Integerb + 0 = 0 + b = bAddition of zero to an integerFalse
Rational Numberc + 0 = 0 + c = cAddition of zero to a rational number\frac{2}{5}+0=0 \frac{2}{5}
= \frac{2}{5}

Multiplicative Identity

One is the multiplicative identity for whole numbers, integers and rational numbers.

IdentityExample
Whole Numbera × 1 = aMultiplication of one to the whole number5 × 1 = 5
Integerb × 1= bMultiplication of one to an integer– 5 × 1 = – 5
Rational Numberc × 1= cMultiplication of one to a rational number \frac{2}{5}+1=\frac{2+5}{5}=\frac{7}{5}

Additive Inverse

Additive inverse of a number is the number that when added to yields zero.

IdentityExample
Whole Numbera +(- a) = 0Where a is a whole number5 + (-5) = 0
Integerb +(- b) = 0Where b is an integerTrue
Rational Numberc + (-c) = 0Where c is a rational number \frac{2}{5}+\left(-\frac{2}{5}\right)=0

Multiplicative Inverse

The multiplicative inverse of any rational number

 \mathrm{\frac{a}{b}\ is\ \frac{b}{a}\ as\ \frac{a}{b}\times\frac{b}{a}=1}

Example

The reciprocal of \frac{4}{5} is \frac{5}{4}.

Distributivity of Multiplication over Addition and Subtraction for Rational Numbers

This shows that for all rational numbers p, q and r

1. p(q + r) = pq + pr

2. p(q – r) = pq – pr

Example

Check the distributive property of the three rational numbers \frac{4}{7},-\frac{2}{3} and \frac{1}{2}.

Solution

Let’s find the value of

 \frac{4}{7} \times\left\{\left(\frac{-2}{3}\right)-\frac{1}{2}\right\} \text {and} \left\{\frac{4}{7} \times\left(\frac{-2}{3}\right)\right\}-\left(\frac{4}{7} \times \frac{1}{2}\right)\\ \\ \frac{4}{7} \times\left\{\left(\frac{-2}{3}\right)-\frac{1}{2}\right\}=\frac{4}{7} \times\left\{\frac{(-2)}{3}-\frac{1}{2}\right\}\\ \\ =\frac{4}{7} \times\left\{\frac{(-4)-3}{6}\right\}\\ \\ =\frac{4}{7} \times \frac{(-7)}{6}\\ \\ =\frac{4 \times(-7)}{7 \times 6}\\ \\ =\frac{-4}{6}\\ \\ =\frac{-2}{3}\\ \\ \left\{\frac{4}{7} \times\left(\frac{-2}{3}\right)\right\}-\left\{\frac{4}{7} \times \frac{1}{2}\right\} =\left\{\frac{4}{7} \times \frac{(-2)}{3}\right\}-\left\{\frac{4}{7} \times \frac{1}{2}\right\} \\ \\=\frac{(-8)}{21}-\frac{4}{14} \\ \\ =\frac{(-8)}{21}-\frac{2}{7} \\ \\ =\frac{(-8)-6}{21} \\ \\ =\frac{-14}{21} \\ \\=\frac{-2}{3} \\ \\\text{This shows that}\\ \\ \frac{4}{7} \times\left\{\left(-\frac{2}{3}\right)-\frac{1}{2}\right\}=\left\{\frac{4}{7} \times\left(-\frac{2}{3}\right)\right\}-\left(\frac{4}{7} \times \frac{1}{2}\right)

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