Difference between a Common Fraction and a Decimal Number

To identify what is the difference between a common fraction and a decimal number it is enough to observe both elements: one represents a rational number, and the other includes in its constitution an integer part and a decimal part.

A “common fraction” is the expression of one quantity divided by another, without carrying out said division. Mathematically, a common fraction is a rational number, which is defined as the quotient of two whole numbers “a / b”, where b ≠ 0.

A "decimal number" is a number that consists of two parts: an integer part and a decimal part..

To separate the integer part from the decimal part, a comma is placed, called a decimal point, although depending on the bibliography a period is also used..

Decimal numbers

A decimal number can have a finite or infinite number of numbers in its decimal part. Also, the infinite number of decimal places can be decomposed into two types:

Periodic

That is, it has a repeating pattern. For example, 2.454545454545…

Not periodic

They have no repeating pattern. For example, 1.7845265397219…

Numbers that have a periodic infinite or finite number of decimal places are called rational numbers, while those that have a non-periodic infinite number are called irrational.

The union of the set of rational numbers and the set of irrational numbers is known as the set of real numbers.

Differences between common fraction and decimal number

The differences between a common fraction and a decimal number are:

1- Decimal part

Every common fraction has a finite number of numbers in its decimal part or an infinite periodic number, while a decimal number can have an infinite non-periodic number of numbers in its decimal part..

The above says that every rational number (every common fraction) is a decimal number, but not every decimal number is a rational number (a common fraction).

2- Notation

Every common fraction is denoted as the quotient of two whole numbers, while an irrational decimal number cannot be denoted in this way..

The most used irrational decimal numbers in mathematics are denoted by square roots (√ ), cubic (³√ ) and higher grades.

Besides these, there are two very famous numbers, which are the Euler number, denoted by e; and the number pi, denoted by π.

How to go from a common fraction to a decimal number?

To go from a common fraction to a decimal number, just make the corresponding division. For example, if you have 3/4, the corresponding decimal number is 0.75.

How to go from a rational decimal number to a common fraction?

The reverse process to the previous one can also be done. The following example illustrates a technique for moving from a rational decimal number to a common fraction:

- Let x = 1.78

Since x has two decimal places, then the previous equality is multiplied by 10² = 100, with which we obtain that 100x = 178; and solving for x it results that x = 178/100. This last expression is the common fraction that represents the number 1.78.

But can this process be done for numbers with a periodic infinite number of decimal places? The answer is yes, and the following example shows the steps to follow:

- Let x = 2.193193193193…

As the period of this decimal number has 3 digits (193) then the previous expression is multiplied by 10³ = 1000, with which we obtain the expression 1000x = 2193.193193193193 ... .

Now the last expression is subtracted from the first and the entire decimal part is canceled, leaving the expression 999x = 2191, from which we obtain that the common fraction is x = 2191/999.

References

1. Anderson, J. G. (1983). Technical Shop Mathematics (Illustrated ed.). Industrial Press Inc.
2. Avendaño, J. (1884). Complete manual of elementary and higher primary instruction: for the use of aspiring teachers and especially of the students of the Normal Schools of the Province (2 ed., Vol. 1). Printing of D. Dionisio Hidalgo.
3. Coates, G. and. (1833). The Argentine arithmetic: Complete treatise on practical arithmetic. For the use of schools. Print of the state.
4. From sea. (1962). Math for the workshop. Reverte.
5. DeVore, R. (2004). Practical Problems in Mathematics for Heating and Cooling Technicians (Illustrated ed.). Cengage Learning.
6. Jariez, J. (1859). Complete course of physical and mechanical mathematical sciences applied to industrial arts (2 ed.). Railway printing house.
7. Palmer, C. I., & Bibb, S. F. (1979). Practical math: arithmetic, algebra, geometry, trigonometry, and slide rule (reprint ed.). Reverte.