Operations on Rational Numbers
Just like with integers, we can perform various operations like addition, subtraction, and multiplication on rational numbers too. Interestingly, when we add, subtract, or multiply two rational numbers, the result obtained is also a rational number. They can be expressed as follows:
Addition of rational numbers: \((\frac{4}{5})+(\frac{1}{3})=\frac{17}{15}\)
Subtraction of rational numbers: \((\frac{4}{5})-(\frac{1}{3})=\frac{7}{15}\)
Multiplication of rational numbers: \((\frac{4}{5})\times (\frac{1}{3})=\frac{4}{15}\)
Rational numbers would not necessarily give another rational number upon division. This is so because the result of a number divided by zero is undefined. With the exception of zero, all other numbers give us rational numbers upon division.
What are the Properties of Rational Numbers?
Rational numbers are those that can be represented in the form of \(\frac{p}{q}\), where both \(p\) and \(q\) are integers, and \(q \neq 0\). The word “rational” comes from the Latin word “ratio,” as all the rational numbers are the result of a ratio. When it comes to the properties of rational numbers, they are categorized as follows:
- The Commutative Property
- The Associative Property
- The Distributive Property
- The Identity Property
- The Inverse Property
Since we have name-dropped the properties of the rational numbers, let us see what each property means.
The Commutative Property
When two rational numbers are added or multiplied, the result remains unchanged irrespective of the way the numbers are arranged. However, the result will change in the case of subtraction and division if the numbers are arranged differently. It can be expressed as:
The commutative property of addition:
a + b = b + a
The commutative property of multiplication:
a × b = b × a
The commutative property of subtraction:
a – b ≠ b – a
The commutative property of division:
a \(\div\) b ≠ b \(\div\) a
Let us see how it works with the help of an example:
For, \(a = \frac{4}{5}\) and \(b = \frac{3}{2}\) | LHS | RHS | Result |
|---|---|---|---|
Check \(a\ +\ b\ =\ b\ +\ a\) | \(a\ +\ b=\ \frac{4}{5}+\frac{3}{2}=\ \frac{23}{10}\) | \(b\ +a =\ \frac{3}{2}+\frac{4}{5}=\ \frac{23}{10}\) | \(LHS = RHS\) |
Check \(a\ \times\ b\ =\ b\ \times\ a\) | \(a\ \times\ b=\ \frac{4}{5}\times\frac{3}{2}=\ \frac{6}{5}\) | \(b\ \times a =\ \frac{3}{2}\times\ \frac{4}{5}=\ \frac{6}{5}\) | \(LHS = RHS\) |
Check \(a\ -\ b\ =\ b\ -\ a\) | \(a\ -\ b=\ \frac{4}{5}-\frac{3}{2}=\ \frac{-7}{10}\) | \(b\ -a =\ \frac{3}{2}-\ \frac{4}{5}=\ \frac{7}{10}\) | \(LHS\neq RHS\) |
Check \(a\ \div\ b\ =\ b\ \div\ a\) | \(a\ \div\ b=\ \frac{4}{5}\div\frac{3}{2}=\ \frac{8}{15}\) | \(b\ \div a =\ \frac{3}{2}\div\ \frac{4}{5}=\ \frac{15}{8}\) | \(LHS\neq RHS\) |
Associative Property
When three rational numbers are added or multiplied, the result remains unchanged irrespective of the way the numbers are arranged. However, the result will be different in the case of subtraction and division if the numbers are arranged differently. It can be expressed as:
The associative properties of addition:
\((a\ +\ b)\ +\ c\ =\ a\ +\ (b\ +\ c).\)
The associative properties of multiplication:
\((a\ \times b)\ \times\ c\ =\ a\ \times\ (b\ \times\ c).\)
The associative properties of subtraction:
\((a\ -\ b)\ -\ c\ \neq\ a\ -\ (b\ -\ c).\)
The associative properties of division:
\((a\ \div\ b)\ \div\ c\ \neq\ a\ \div(b\ \div\ c).\)
For, \(a=\frac{1}{5}, b=\frac{1}{4},\) and \(c=\frac{1}{3}\) | LHS | RHS | Result |
|---|---|---|---|
Check \((a+b)+c=a+(b+c)\) | \((a+b)+c=(\frac{1}{5}+\frac{1}{4})+\frac{1}{3}=\frac{47}{60}\) | \(a+(b+c)= \frac{1}{5}+(\frac{1}{4}+\frac{1}{3})=\frac{47}{60}\) | \(LHS = RHS\) |
Check \((a\ \times\ b)\ \times\ c\ =a\ \times\ (b\ \times\ c)\) | \((a\ \times\ b)\ \times\ c\ =\ (\frac{1}{5}\times\frac{1}{4})\times\frac{1}{3}=\frac{1}{60}\) | \(a\ \times\ (b\ \times\ c)\ =\ \frac{1}{5}\times(\frac{1}{4}\times\ \frac{1}{3})=\frac{1}{60}\) | \(LHS = RHS\) |
Check \((a-\ b)\ -\ c\ =a\ -\ (b\ -\ c)\) | \((a\ -\ b)\ -\ c\ =\ (\frac{1}{5}-\frac{1}{4})-\frac{1}{3}=\ \frac{-23}{60}\) | \(a\ -\ (b\ -\ c)\ =\ \frac{1}{5}-(\frac{1}{4}-\ \frac{1}{3}) =\ \frac{17}{60}\) | \(LHS\neq RHS \) |
Check \((a\div\ b)\ \div\ c\ =a\ \div\ (b\ \div\ c)\) | \((a\ \div\ b)\ \div\ c\ =\ (\frac{1}{5}\div\frac{1}{4})\div\frac{1}{3}=\ \frac{12}{5}\) | \(a\ \div(b\ \div c)\ =\ \frac{1}{5}\div(\frac{1}{4}\div\ \frac{1}{3})=\ \frac{4}{15}\) | \(LHS\neq RHS \) |
The Distributive Property
The distributive property states that when an expression is in the form of \(a\ \times(b\ +\ c)\) , it can be written as \(a\ \times\ (b\ +\ c)\ =\ (a\ \times\ b)\ +\ (a\ \times c)\) for the addition of two rational numbers. Similarly, the expression \(a\ \times\ (b\ -\ c)\) can be written as \(a\ \times\ (b\ -\ c)\ =\ (a\ \times b)\ -\ (a\ \times c)\) for subtracting two rational numbers.
Let us see how it works with the help of an example:
For, \(a = \frac{1}{4}, b = \frac{3}{4},\) and \(c = \frac{7}{4}\) | LHS | RHS | Result |
|---|---|---|---|
Check \(a\ \times(\ b\ +\ c)\ =(a\ \times b)\ +(a\times\ c)\) | \(a\ \times(\ b\ +\ c)\ =\ \frac{1}{4}\times(\frac{3}{4}+\frac{7}{4})\) \(=\frac{1}{4}\times(\frac{10}{4})\) | \((a\ \times b)\ +(a\times\ c)\)\(=\ (\frac{1}{4}\times\frac{3}{4})+(\frac{1}{4}\times\frac{7}{4})\) \(=\ (\frac{3}{16})+(\frac{7}{16})\) | \(LHS = RHS\) |
Check \(a\ \times(\ b\ -\ c)\ =(a\ \times b)\ -(a\times\ c)\) | \(a\ \times(\ b\ -\ c)\ =\ \frac{1}{4}\times(\frac{3}{4}-\frac{7}{4})\) \(=\frac{1}{4}\times(\frac{-4}{4})\) | \((a\ \times b)\ -(a\times\ c)\)\(=\ (\frac{1}{4}\times\frac{3}{4})-(\frac{1}{4}\times\frac{7}{4})\) | \(LHS = RHS\) |
The distributive property does not apply to rational numbers for performing division or multiplication.
The Identity Property
In the rational number system, 0 is known as an additive identity and 1 is known as a multiplicative identity.
The Inverse Property
For any rational number \(\frac{a}{b}\), the additive inverse is \(\frac{-\ a}{b}\) and the multiplicative inverse is \(\frac{b}{a}\).
Rapid Recall
Solved Examples
Example 1:
If \(2\ \times\ (4\ \times\ 5)\ =\ 40\), what would be the product of \((2\ \times\ 4)\ \times\ 5\)?
Solution:
As we know, the multiplication of an associative property gives the same result, irrespective of the order.
So, the result for the given equation will be:
\(2\ \times\ (4\ \times\ 5)\ =\ (2\ \times\ 4)\ \times\ 5\)
\(=\ 40\)
Example 2:
Solve the given equation using the distributive property of multiplication over addition: \(10\times(7+14).\)
Solution:
According to the distributive property of multiplication over addition, the given equation can be written as:
\(10\times(7+14) = (10\times7)+(10\times14)\) [First, multiply the given numbers]
\(=\ 70+140.\) [Add the products obtained]
\(=\ 210\) [Result]
Example 3:
Reese arranged four jelly beans vertically and seven jelly beans horizontally. How can she get the same number of jelly beans by rearranging the rows and columns of jelly beans?
Solution:
Given, Reese arranged a total of 4 x 7 = 28 jelly beans in 4 vertical columns and 7 horizontal rows.
According to the commutative property of multiplication, the multiplication of two numbers gives the same result.
So, we can say that \(4\times7=7\times4=28\).
Hence, Reese can build 7 vertical columns and 4 horizontal rows to make a new row with the same number of jelly beans.
As a result, the new arrangement will have 7 vertical columns and 4 horizontal rows.
Example 4:
If \(\frac{-15}{17}\) is an additive inverse of \(\frac{15}{17}\), what will be the multiplicative inverse of \(\frac{15}{17}\)?
Solution:
According to the inverse property, the multiplicative inverse of \(\frac{15}{17}\) is \(\frac{17}{15}.\)
Example 5:
What would be the additive identity and multiplicative identity for the rational number \(\frac{29}{30}\)?
Solution:
According to identity property, the multiplicative identity for the number \(\frac{29}{30}\) can be determined by: \(\frac{29}{30}\ \times1=\frac{29}{30}.\) [Multiply by 1]
Also, the additive identity: \(\frac{29}{30}+0=\frac{29}{30}.\) [Sum with 0]
The most common use of rational numbers in the physical world is to measure quantities like length, mass, time, or something else that cannot be accurately described by integers alone.
Pythagoras, an ancient Greek mathematician, created rational numbers.
If a number is written as a fraction, where both the denominator and the numerator are integers, and the denominator is a non-zero value, we can call it a rational number.
The major properties of rational numbers are commutative, associative, and distributive properties.