Rational And Irrational Numbers With Examples[pdf]

Rational and Irrational numbers are numbers that can be expressed in the form of a ratios.

Rational Numbers and Irrational Numbers

A rational number is any number that can be written or expressed as the quotient (or fraction) of any two integers (or whole numbers). The denominator of such a fraction cannot be zero.

Any whole number can be called a rational number since it can be expressed as a fraction with 1 as the denominator. They can be represented on a number line.

Examples of rational numbers are :

3/4 , -5/6, 18/11, -9/5, 21/3, -12/3, 14, .29, 0.26, -0.798156, √81, √6.25

√81 = 9 and √6.25 = 2.5 so both are rational numbers

Irrational Numbers:

8/0, 53/0, 𝜋 (𝜋 = 3.1415926535…) , e (e = 2.718281828459…), √7 (approx. 2.64575131106…)

Number lines are another way to visualize rational numbers.

What is the difference between rational and irrational numbers?

Any number that can be written as the ratio of ab where a and b are both integers.

Definition of a Rational Number:

Any number that can be written as the ratio of ab where a and b are both integers

and b≠0

Definition of a Irrational Number:

Any number that cannot be written as the ratio of ab where a and b are both integers and b≠0.

Comparing Rational Numbers:

With integers, it is easy to observe that 17 is larger than 11 just by comparing the values of the numbers. But comparing rational numbers in fraction form will often require rewriting the fractions with a common denominator.

Compare Rational Number Having Same Denominator:

If denominators are the same then simply compare numerators. There is no need to think about denominators. Based on the value of numerator numbers can be compared to whether it is smaller or greater.

Example 1: 4/6 is greater than 1/6.

Here denominator is common which is 6 and the numerators are 1 and 4. Comparing 1 and 4, 4 is greater.

Therefore, 4 is greater than 1

Example 2: 3/10 is less than 5/10

10 is the denominator that is common to both fractions.

3 and 5 are numerators and 3 is less than 5 so 3/10 is less than 5/10

Comparing rational numbers with different denominators:

This is a very easy task. Similar to fractions with different denominators, we will make denominator common. Once done, we can simply compare numerators.

Example 1: Suppose we want to determine which is larger\? 2/3 or 5/6

Since we have different denominators 3 and 6 let’s convert them to a fraction with the same denominator.

L.C.M of 3 and 6 is 6

so 2/3 is equivalent to (2 x 2)/(3 x 2) = 4/6

Now we have a common base that is 6. Comparing 4 and 5 is now easy.

4 is less than 5 so, 4/6 is less than 5/6

Writing a Rational Number in Decimal Form

To write the rational numbers in decimal form do actual division.

simply resort to doing the operation by hand using long division.

Terminating Versus Repeating Decimals:

There are two kinds of decimals

1. Terminating
2. Repeating

Terminating decimals are those that occur when you change a rational number into a decimal and the string of numbers contained in the decimal answer STOPS.

Example of terminating rational numbers are:

0.25, 0.333, 0.56, 0.89, 0.112

In case of Repeating rational numbers string of numbers in the decimal answer doesn’t stop.

Examples are: 0.2333333…, 0.7777…, 0.89999…, 0.565656…, 0.1324132413241324….

Writing Terminating Decimals as Fractions:

If you want to convert decimal numbers back into fractions, use the place of the last digit to determine the denominator of the fraction.
For example, given the decimal number 0.9, since 9 is the last digit of the decimal and it is in the tenth place, the denominator is 10. Thus, we have the fraction 9/10

Writing Repeating Decimals as Fractions:

Example: lets try converting 0.455555555…

First, we set the variable x equal to the repeating decimal, so let

x = 0.4555555555…

Next, because there is only a single digit (the “5”) that repeats, we multiply both sides of the above equation by 10, so that we obtain

10x = 4.5555555555…

Now, here is where a “trick” comes into play. We next do something that we have never done before—we actually subtract the two equations as follows:

Solving for x gives, x= 4.1/9

Converting to whole number gives: 41/90

Comparing Fractions Using Decimals:

When comparing two or more numbers to determine which number is the greatest or smallest, you can change the fractions (or rational numbers) to decimal form as an alternative to trying to convert the fractions so that they all have a common denominator. Let’s do a sample problem. Which is greater: 79/86 or 92/100? The decimal equivalent of 79/86 = 0.918604651163…; whereas, the decimal equivalent of 92/100 = 0.92, therefore we see that 92/100 is larger since the “2” in the hundredths place of 0.92 is greater than the “1” in the hundredths place of 0.9186….

Conclusion:

1. if the numbers in a decimal end at some point, we call that kind of decimal terminating; however, if a number or sequence of numbers in a decimal continue to repeat on into infinity, we call that a repeating decimal.