Rational Numbers Between Two Given Numbers

A number is a mathematical object which is used to count, measure, and label the mathematical concepts. Numbers are the basic units of Mathematics. All types of numbers used in Mathematics are grouped under the banner of a Number system. Various kinds of numbers include prime numbers, composite numbers, odd numbers, even numbers, natural numbers, whole numbers, integers, decimal numbers, fractions, rational numbers, irrational numbers, real numbers, and imaginary numbers. All the numbers that exist in reality are called the real numbers. All the numbers that do not exist and are assumed to explain a few mathematical concepts are called imaginary numbers. Real numbers are broadly classified into rational numbers and irrational numbers. Rational numbers are the numbers that can be expressed in the form of a fraction whose denominator is not equal to zero. Irrational numbers are the numbers that cannot be expressed in the form of a fraction such that the denominator is not equal to zero.

Terminology of Rational Numbers

In the context of the set Q, the term rational refers to the fact that a rational number is a ratio of two integers. "Rational" is frequently used as a word in mathematics, abbreviating "rational number." In some cases, the adjective rational denotes that the coefficients are rational numbers. A rational point, for example, is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; and a rational polynomial is a polynomial with rational coefficients, though the term "polynomial over the rationals" is preferred to avoid confusion between "rational expression" and "rational function" (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). A rational curve, on the other hand, is one that may be parameterized by rational functions rather than one that is defined over the rationals.

The Topological Properties of Real Numbers

The rationals are a dense subset of the real numbers, with rational numbers arbitrarily near to every real number. Rational numbers are the only numbers having finite expansions as regular continuous fractions, which is a related feature. 

The rationals have an order topology due to their order. The rational numbers have a subspace topology because they are a subspace of the real numbers. Using the absolute difference metric d(x, y) = |x y|. The rational numbers form a metric space, yielding a third topology on Q. The rationals are transformed into a topological field when all three topologies coincide. The rational numbers are a good example of a space that is not compact locally. Topologically, the rationals are defined as the only countable metrizable space without isolated points. In addition, the space is completely isolated.

Etymology

Despite the fact that rational numbers are now described in terms of ratios, the term rational is not derived from the term ratio. Ratio, on the other hand, is derived from rational: the earliest use of ratio in its contemporary sense was documented in English around 1660, while the use of rational for qualifying numbers was documented over a century earlier, in 1570.  This definition of rational was derived from the mathematical definition of irrational, which was first used in 1551 in "Euclid's translations." The ancient Greeks "avoided heresy by preventing themselves from conceiving of those (irrational) lengths as numerals," according to legend. As a result, such lengths were irrational, in the sense of illogical, and "not to be discussed."

This is comparable to the etymology of imaginary and real numbers.

Rational Numbers Between Two Rational Numbers

Rational numbers are the numbers which can be expressed in the form of p and q where q ≠ 0. Examples for rational numbers are prime and composite numbers, odd and even numbers, decimals, and fractions. A number of rational numbers between two rational numbers can be located. Between any two rational numbers, countless rational numbers can be found. 

How to Find Rational Numbers Between two Rational Numbers With the Same Denominator Value?

• The first step in determining the rational numbers between two rational numbers is to check the value of the denominators.

• If the denominator values are the same, check the value of the numerators.  

• If the numerators differ by a large value, then the rational numbers between the two rational numbers can be written in the increments of one for the numerator without altering the value of the denominator.

• Example: The 5 rational numbers between 1/9 and 7/9 are 2/9,3/9,4/9,5/9,6/9

• If the values of the numerators differ by a lesser value than the number of rational numbers to be found, then the numerators and denominators of both the rational numbers are multiplied by multiples of 10. 

Example:  If 10 rational numbers are to be found between 2/7and 5/7, both the rational numbers are to be multiplied with 10/10.

2/7  x 10/10=20/70

5/7  x 10/10=50/70

The 10 rational numbers between 2/7 and 5/7 can be written as the rational numbers between 20/70 and 50/70. The 10 rational numbers are 21/70, 22/70,23/70, 24/70, 25/70, 26/70, 27/70, 28/70, 29/70 and 30/70.

How to find Rational Numbers Between two Rational Numbers with Different Values of Denominators?

• To find the rational numbers between two rational numbers with different denominators, the denominators should be equated. 

• Equating the denominators can be done either by finding their LCM or by multiplying the denominators of one to both the numerator and denominator of the other.

Example

If the rational numbers between ⅔ and ¾ are to be found. 

LCM of 3 and 4 is 12. When the denominators are equated by the LCM method, the equivalent rational numbers are 8/12 and 9/12.

The same rational numbers will be obtained when the denominator of one rational number is multiplied to the numerator and denominator of the other. 

2/3 x 4/4 = 8/12 and 3/4 x 3/3 = 9/12.

Once the denominators are equated, the same rules of finding the rational numbers between two rational numbers having the same denominator is used.

Rational Numbers Between two Rational Numbers Example:

1. Find rational numbers between ¼ and ½ (at least 5).

Solution:

The rational numbers ¼ and ½ have different denominators.

Equate the denominator. 

1/4 x 2/2 = 2/8 and 1/2 x 4/4 = 4/8

So the rational numbers are 2/8 and 4/8.

5 rational numbers between these two rational numbers cannot be written. So, both the numerator and denominator of the two rational numbers are multiplied by 10.

2/8 x 10/10 = 20/80 and 4/8 x 10/10 = 40/80

The 5 rational numbers between ¼ and ½, therefore, are 21/80, 22/80, 23/80, 24/80, and 25/80.

2. How to find the rational numbers between two rational numbers ¼ and ¾. (At Least 10)

Solution:

The rational numbers ¼ and ¾ have the same denominators. 

It is not possible to write 10 rational numbers between ¼ and ¾ .

So, the numerator and denominator of both the rational numbers are multiplied by 10.

1/4 x 10/10 = 10/40 and 3/4 x 10/10 = 30/40

Therefore, the 10 rational numbers between ¼ and ¾ are:

11/40, 12/40, 13/40, 14/40, 15/40, 16/40, 17/40, 18/40, 19/40 and 20/40

Fun Facts

• Rational numbers can always be expressed in the form of ratios.

• The denominator of a rational number cannot be an irrational number because any rational number divided by an irrational number is an irrational number.