Denominator

The meaning of the denominator is the divisor of a fraction, that lies below the line in a fraction. It is just the opposite of a numerator, that lies above the line in a fraction. Now you may wonder, what a fraction is? The fraction is a part of a whole. Mathematically, we can represent a fraction by the division of two numbers. For example, 1/5 is one part out of the five equal parts created from that one whole thing. So here, 1 is the numerator and 5 is the denominator. In this article, we will define denominator, understand what is numerator and denominator, the comparison between the numerator denominator, how to find the least common denominator and rationalize the denominator math with more solved examples.

What is a Denominator?

When we express an integer in the fraction form, there appear two numbers that are separated by a horizontal line. The denominator is the number that appears at the bottom of the horizontal line, whereas the number at the top of the line is the numerator. It is not necessary that only numerical values are expressed in the numerator and denominator form, we can also represent the variables in the numerator or denominator form. For example, the fraction  x/y, p/q, m/n, etc., here the variable y, q, and n are the denominators of the fractions respectively.

A fraction consists of two numbers. Symbolically it is represented by two numbers that are separated by a horizontal line (–) or slash line (/). The number above the line is called the numerator, and the number below the line is called the denominator. The term denominator is also used in the ratio and proportion concept.

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In the above picture, 4, 2, and  4 are the denominators of fractions 1/4, 1/2, and 3/4 respectively.

The denominator value can not be zero, because, if the value of the denominator is zero, then the fraction value should be undefined. For example, 2/0 is a fraction, which is undefined because the denominator value is zero. 

Denominator Examples

Some of the examples of the denominator are

What is Numerator and Denominator ?

Definition of Numerator Denominator 

The denominator indicates the number of equal parts from the whole thing that has to be divided.

The numerator indicates the number of divisions that are selected out of the total number from the equal parts.

Let’s understand this with an example.

Given, 3/4 is a fraction. Here, the denominator 4 represents 4 equal divisions that have to be made.

And 3 parts selected out of 4 equal parts created. If we take it in terms of a circle, 3/4 is represented as shown below.

Diagrammatic Representation of 3/4 is as Follows:

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The above diagram clearly shows three equal parts taken out when the whole circle is divided into four equal parts.

What is the Common Denominator?

When two fractions have the same value in the denominator, then they are said to be common denominators. They are like and unlike fractions based on the similarity between the denominators. If we perform the arithmetic operations such as addition and subtraction, then the denominator of the fractions should be common or equal. The fractions can be multiplied or divided even though the denominators of the fractions are not the same or common. Let us see some examples here:

Following are the Examples:

1.\[\frac{1}{2}+\frac{5}{2} = \frac{6}{2} = 3 \]

2.\[ \frac{4}{5} - \frac{2}{5} = \frac{2}{5} \]

3.\[ \frac{5}{4} \times \frac{2}{4} = \frac{10}{16} \]

4.\[ \frac{4}{5} \div \frac{3}{7} = \frac{4}{5} \times \frac{7}{3} = \frac{28}{15}\]

5.\[\frac{8}{9} \times 2 = \frac{8}{9} \]

Least Common Denominator(LCD)

When we have to find the least common denominator of two or more non-zero denominators we can find the smallest whole number which is divisible by each of the denominators' values.

Rationalizing the Denominator

When the denominator of a fraction is an irrational number, then we need to perform rationalization of the denominator. So that further calculations can be made easily on the fraction.

The irrational denominator includes the root numbers.

Example 1. Monomial Denominator

\[\frac{1}{\sqrt{2}}\] has an irrational denominator since it is a square root of 2.

To remove the radical, we have to multiply the numerator and denominator by \[\sqrt{2}\], we get,

\[ \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

We can multiply the top and bottom with the denominator value, to find the smallest possible perfect square in the denominator and remove the radical as per our requirement.

Example 2. Binomial Denominator:

When there are two terms in the denominator.

\[\frac{2}{3+\sqrt{2}} \]

In this case, we have to multiply the numerator and denominator with the conjugate of the denominator.

The meaning of conjugate is the same denominator but with the opposite sign.

The conjugate of  \[ 3+ \sqrt{2}\] is \[ 3- \sqrt{2}\] where the “+ sign “ is replaced with “- sign”

Now, we will rationalize the fraction,

\[ \frac{2}{3+\sqrt{2}} \times \frac{3-\sqrt{2}}{3-\sqrt{2}} = \frac{6-2{\sqrt 2}}{3^2 - {\sqrt{2}}^2} = \frac{6-2\sqrt{2}}{7} \]

Solved Examples:

1.Given fraction \[\frac{1}{3}\] and \[\frac{1}{5}\]. Find the least common denominator of both denominators.

Sol: We have been given two fractions \[\frac{1}{3}\] and \[\frac{1}{5}\].

To find LCD, we need to find multiples of the denominator

3= 3,6,9,12,15,18,21,24,27...

5= 5,10,15,20,25,30,35...

From above, we can see 15 is the common multiple.

Hence 15 is LCD.

2.Rationalize the denominator \[\frac{1}{\sqrt{50}}\].

Sol: To rationalize the denominator, we have to multiply both numerator and denominator of the given rational number by \[\sqrt{50}\]

So \[\frac{1}{\sqrt{50}} \times \frac{\sqrt{50}}{\sqrt{50}} = \frac{\sqrt{50}}{50} = \frac{5\sqrt{2}}{50} = \frac{\sqrt{2}}{10} \]

Hence, rationalization of \[\frac{1}{\sqrt{50}} \] is \[\frac{\sqrt{2}}{10}\]

Conclusion

From the above discussion, we can conclude that the larger the denominator the smaller the pieces, and if the numerator is kept fixed, the larger the denominator the smaller the fraction. If the denominators are bigger, we need to find the least common denominator by factoring in the denominator. If we do not perform prime factorization, then we have to multiply the two denominators together. When the numerator is the same, and the denominator increases, the value of the fraction decreases. When the denominator value is the same, and the numerator value increases, then the value of the fraction increases.