Rational Numbers That Are Not Integers – Fractions, Decimals, Ratios
Rational numbers that are not integers can be written as fractions, ratios, terminating decimals, or repeating decimals.
Rational Numbers – Exercises
Instructions: For questions 1 to 5, identify the rational numbers from the lists provided. For questions 6 to 8, express the rational numbers in their simplest form. For questions 9 and 10, rationalize the denominator. You may want to study the examples below first.1) 3.456382 , 176739/892548 , 4p
2) 582 , 1/3 , \(\frac{2}{\sqrt{4}}\)
3) 9: 1 , 0.5876 , \(\sqrt{3}\)
4) 1/6 , 1589572 , \(\sqrt{2}\)
5) 24/72, \(\frac{\pi}{\sqrt{16}}\) , \(\frac{\sqrt{3} – 1}{-2}\)
6) \(\frac{1}{3} \times \frac{3}{5}\)
7) \(\frac{1}{6} \div \frac{4}{9}\)
8) \(\frac{x/y}{v/w}\)
9) \(\frac{3}{\sqrt{2}}\)
10) \(\frac{5}{\sqrt{3}}\)
Rational Numbers – Answers
Answer to question 1
1) The correct answers are:3.456382 and 176739/892548
3.456382 is rational because it is a terminating decimal.
176739/892548 is rational because it is a fraction. When simplified, it is also a terminating decimal.
4p is not rational because it is a non-terminating decimal.
Answer to question 2
2) The correct answers are:582 , 1/3 , and \(\frac{2}{\sqrt{4}}\)
582 is rational because it is an integer.
1/3 is rational because it is a repeating decimal.
\(\frac{2}{\sqrt{4}}\) is rational because it can be simplified to 1.
Answer to question 3
3) The correct answers are:9: 1 and 0.5876
9: 1 is rational because it is a ratio.
0.5876 is rational because it is a repeating decimal.
\(\sqrt{3}\) is not rational because it is a non-terminating decimal.
Answer to question 4
4) The correct answers are:1/6 and 1589572
1/6 is rational because it is a fraction. When simplified, it is also a repeating decimal.
1589572 is rational because it is an integer.
\(\sqrt{2}\) is not rational because it is a non-terminating decimal.
Answer to question 5
5) The correct answers is: 24/7224/72 is rational because is a fraction. When simplified, it is also a repeating decimal.
\(\frac{\pi}{\sqrt{16}}\) is not rational because p is a non-terminating decimal.
\(\frac{\sqrt{3} – 1}{-2}\) is not rational, even through the denominator has been rationalized. That is because the square root of 3 is a non-terminating decimal.
Answer to question 6
6) The correct answer is: 1/5When you multiply fractions, you have to multiply the numerators and then the denominators. Then simplify, if possible.
\(\frac{1}{3} \times \frac{3}{5} =\)
\(\frac{1 \times 3}{5 \times 3} =\)
\(\frac{3}{15} =\)
\(\frac{1}{5}\)
Answer to question 7
7) The correct answer is: 3/8When you divide fractions, you have to invert the second fraction and then multiply.
\(\frac{1}{6} \div \frac{4}{9} =\)
\(\frac{1}{6} \times \frac{9}{4} =\)
\(\frac{1 \times 9}{6 \times 4} =\)
\(\frac{9}{24} =\)
\(\frac{3 \times 3}{3 \times 8} =\)
\(\frac{3}{8} \)
Answer to question 8
8) The correct answer is: xw/yvWhen you divide fractions that are rational expression like this one, you have to invert the second fraction and then multiply.
\(\frac{x/y}{v/w} =\)
\(\frac{x}{y} \div \frac{v}{w} =\)
\(\frac{x}{y} \times \frac{w}{v} =\)
\(\frac{x \times w}{y \times v} =\)
\(\frac{xw}{yv}\)
Answer to question 9
9) The correct answer is:\(\frac{3\sqrt{2}}{2}\)
Multiply both the numerator and the denominator by the square root in order to rationalize the denominator.
\(\frac{3}{\sqrt{2}} = \)
\(\frac{3 \times \sqrt{2}}{\sqrt{2}\times \sqrt{2}} = \)
\(\frac{3\sqrt{2}}{2}\)
Answer to question 10
10) The correct answer is:\(\frac{5\sqrt{3}}{3}\)
Multiply both the numerator and the denominator by the square root in order to rationalize the denominator.
\(\frac{5}{\sqrt{3}}\)
\(\frac{5 \times \sqrt{3}}{\sqrt{3}\times \sqrt{3}} = \)
\(\frac{5\sqrt{3}}{3}\)
Simplifying Rational Numbers
You will need to simplify rational number integers, as well as rational numbers that are not integers, on your algebra exam.
Expressing fractions in their simplest form may involve adding, subtracting, multiplying, or dividing fractions, as well as finding the lowest common denominator.
You may want to view our pages on fractions and decimals if you need to review these skills.
Now look at the following example question.
Simplifying Rational Numbers – Example 1
Identify the rational numbers in the following list.
3 , 1/4 , 0.256733 , π , 1: 5 , \(\sqrt{2}\)
Answer: 3 , 1/4 , 0.256733 , 1: 5
- 3 is a rational number because it is an integer.
- 1/4 is a rational number because it is a fraction.
- 0.256733 is a rational number because it is a repeating decimal.
- π is not a rational number because it is a non-terminating decimal.
- 1: 5 is a rational number because it is a ratio. It can also be expressed as a fraction.
- \(\sqrt{2}\) is not a rational number because it is a non-terminating decimal, when simplified.
What is a Rational Number – with Example
Rational numbers can be expressed in the form a/b where a and b are integers and b is not zero.
Rational Numbers – Integers
Integers are rational numbers because they can be written in the form a/b.
For example, the integer 7 can be written as 7/1.
So, integers are rational numbers because they can be written as fractions, with the integer in the numerator and 1 in the denominator.
Rational Numbers That Are Not Integers – Denominators
You may see questions on your algebra test that ask you to rationalize the denominator in a fraction.
This means that you have to express the denominator as a rational number.
The denominator may include a radical or polynomial.
So, you may need to know how to carry out the FOIL method in order to rationalize a denominator.
Look at the following example.
Example 2:
Rationalize the denominator in the fraction below.
\(\frac{1}{1 + \sqrt{3}}\)
Answer:
\(\frac{1 – \sqrt{3}}{-2}\)
Now study the steps in the solution below
Use the FOIL method to simplify the denominator.
\(\frac{1}{1 + \sqrt{3}} =\)
\(\frac{1}{1 + \sqrt{3}} \times \frac{1 – \sqrt{3}}{1 – \sqrt{3}} =\)
\(\frac{1 \times (1 – \sqrt{3})}{(1 + \sqrt{3})(1 – \sqrt{3})} =\)
\(\frac{1 – \sqrt{3}}{1^2 + \sqrt{3} – \sqrt{3} – \sqrt{3}^2} \)
Then perform the subtraction and simplify the square root to simplify further.
\(\frac{1 – \sqrt{3}}{1^2 + \sqrt{3} – \sqrt{3} – \sqrt{3}^2} =\)
\(\frac{1 – \sqrt{3}}{1 – 3} =\)
\(\frac{1 – \sqrt{3}}{-2}\)
More Algebra Practice
Numbers that are not rational numbers are called irrational numbers.
You may want to see our posts on irrational numbers and radicals.
