Notes on Rational Numbers

The Set of Rational Numbers

Introduction

The word "ratio" comes from Latin and it can be found in dictionaries, like American Heritage, with the meanings of: (1) Relation in degree or number between two similar things. (2) The relative value of silver and gold in a currency system that is bimetallic. (3)Mathematics. The relation between two quantities expressed as the quotient of one divided by the other. The ratio of 7 to 4 is written 7:4 or 7/4.

Also the word "rational" can be found in dictionaries with the meanings of: (1) Having or exercising the ability to reason. (2) Of sound mind; sane. (3) Manifesting or based upon reason; logical. (4) Mathematics. Designating an algebraic expresion no variable of which appears in an irreducible radical or with a fractional exponent.

The above (4) meaning does not reflect  the concept to be developed in this article. A rational number  will be understood as a number  which can be expressed as a quotient between two integers a, b,  that is a/b, with b different from zero. Examples: 2/3, 8/5, -3/7.

A quotient between two integers is not the only way to represent a rational number. It can be represented as a decimal. Example: 12/5 can be expressed as 2.4

Fractions and rational numbers

Examples of fractions are

Those symbols have the same meaning as 2/3 and 8/5 and correspond to concrete situations like the following:

A plane rectangular region  has been taken as a unit. There has been shaded two thirds of it. (A proper fraction of a unit)

A plane rectangular region U has been taken as a unit. This unit has been divided in 5 equal parts. It has been constructed a second plane rectangular unit with eight rectangular regions, each one  of 1/5 of the original unit. (An improper fraction).

These examples correspond to an early approach to rational numbers.

Rational numbers and number line

Integers can be represented on a number line, by means of equally spaced points.

It is possible too the representation of rational numbers as points on a line.

In the expression a/b the numerator is a and the denominator is b. The denominator indicates the number of equal parts in the unit segment [0, 1]. The numerator indicates the number of parts taken in consideration.

Quotient representation

The representation 2/3 means the same as 4/6. since in the latter case it means that the unit has been divided in 6 equal parts and it has been taken, or considered 3 of them.

In general, a/b means the same as ka/kb,with k ≠ 0

When numerator and denominator are multiplied by the same number k, different from zero, the fraction is said to have been amplified by k.

When numerator and denominator are divided by the same number k, the fraction is said to have been simplified by k.

A necessary and sufficient condition for a/b an c/d represent the same rational number is:

a ∗ d = b ∗ c

Example: 3/7 = 12/28 (3/7 has been amplified by 4)

If a and b have common factors, different than one. the representation a/b can be expressed as a'/b', where a' = ad and b' = bd and  d is the greatest common factor between a and b

Example: 135/765 = 3/17 . In this case, 45 is the GCF of 135 and 765. The fraction 135/765 has been simplified by 45.  Also, it can be verified that 17 times 135 = 3 times 765.

Introduction to decimal notation

It consists in two sequences of digits separated by a point called decimal point.

The decimal point separates the integer part of the number, at the left of that point, and the decimal part of the number to the right of that point.

The digits to the right of the decimal point are  decimal fractions of a unit. They can be read, from left to right as times 1/10, times 1/100, times 1/1000, ...

Examples:
1.5 means 1 + (5/10)
3.25  means 3 + (2 /10) + (5/100)
0.625 means 0 + (6/10) + (2/100) + (5/1000)

Calculators give directly  decimal notations for  fractions given in quotient notation.

Examples: 

3/2 = 1.5 

1/16 = 0.0625 

25/8 = 3.125

5/3 = 1.6666... (cases like this one will be discussed later)

Terminating decimal notation

If the denominator of a fraction has 2 or 5 as their only  factors (the prime factors of ten) then the fraction can be expressed  directly with a denominator which is is a power of 10.

Examples: 3/2 = 15/10 (amplifying by 5) , 7/4 = 175/100  (amplifying by 25), 8/25 = 32/100 (amplifying by 4), 5/8 = 625/1000 (amplifying by 125).

When the denominator is a multiple of 10 there are rules to represent the fraction in decimal notation.

If the denominator is 10, the decimal point will be placed one step to the left in the numerator. Example: for 15/10 the result is 1.5

If the denominator is 100, the decimal point will be placed two steps to the left in the numerator. Example: for 175/100 the result is 1.75

If the denominator is 1000, the decimal point will be placed three steps to the left in the numerator. Example: for 625/1000 the result is 0.625

If the number of steps to the left is greater than the number of figures of the integer, complete with zeros.
Example: ( Remember that 45 is the same as ...000000045)

For 45/10000, the result is 0.0045

It can be seen that multiplicaion by 10 corresponds to one step displacement to the right of the decimal point. Division by 10 corresponds to  one step displacement to the left of the decimal point.
Examples:
 1.56 ∗ 10 = 15.6
1.56/10 = 0.156

Left zeros before of the first digit different from zero in an integer maybe ignored. For example, 000103 means the same as 103. Right zeros after the lasd digit differente from zero in a decimal representation may be ignored. For instance, 0.05 means the same as 0.050 or 0,0500 or 0.05000, or...

0.05 = 5/100; 0.050 = 50/100

This examle shows that the  addition of a zero after the last digit in a decimal representation means the same as amplifying by 10 the corresponding qoutient representation. It does not change its value.

Non terminating decimal notation

If the denominator of a fraction has at least one factor different from 2 or 5 the corresponding decimal notation does not terminate.To decide cut it at a given point depends of the considered problem or situation.

Example: 5/3 = (3/3) + (2/3) = 1 + (2 /3)

2/3 = (20/3) ∗ (1/10)  

20/3 = (18/3) + (2/3) = 6 + (2 /3). 

Then, 2/3 = [6 + 2/3)]∗(1/10) = 6∗(1/10) + (2/3)∗(1/10) = (6/10) + (2/3)∗(1/10)

That is, 2/3 = (6/10) + (2/3)∗(1/10)

Then, 2/3 = (6/10) + [(6/10) + (2/3)∗(1/10)]∗(1/10) = (6/10) + (6/100) + (2/3)∗(1/100)

Now, 2/3 = (6/10) + (6/100) + [(6/10) + (2/3)∗(1/10)]∗(1/100)

That is, 2/3 = (6/10) + (6/100) + (6/1000) + (2/3)∗(1/1000)

Until now the result is, 2/3 = (666/1000) + (2/3)∗(1/1000) = 0.666 + (2/3)∗(1/1000)

This process can be iterated without end. So,  1.666666... is the decimal notation for 5/3

Alternatively it can be used the division algorithm, taken the dividend as 5.000000... and the divisor as 3.

Periodic decimal notation

In above example one repeated numeric figure apears in the decimal notation of 5/3. The repetition can occur for two or more numeric figures as in the following examples. Every repeated sequence of digits is called a period.

103/33 = 3.12121212... The period is 12 and it begins next to decimal point.

1253/495 = 2.531313131...The period is 31 and it does not begin next to decimal point.

In what follows, the repeating sequence of digits, the period, will be underlined.

Examples:
 1.66666... = 1.6
2.12121212... = 3.12
2.53131313131... = 2.531

Decimal to quotient notation

In the following examples it is described how to express in the form a/b a rational number written in decimal representation.

Example (1) 2.345

This means 2 + 345/1000 , that is (2000 + 345)/1000 = 2345/1000 = 469/200

Example (2) 3.214

To find the fraction F corresponding to this non terminating periodic decimal, a suitable process is the following:

3.214 = F

3214.214 = 1000 ∗ F 

Subtracting in former equations, 3211 = 999 ∗ F

F = 3211/999

Example (3) 12.345

F = 12.345

100 ∗ F = 1234.5

1000 ∗ F = 12345.5

Subtracting in former equations, 900 ∗ F = 11111

F = 11111/900

These processes can be turned systematic by means of a computer program with instructions like the following:

Write the integer part and the decimal part,before the first period, without the decimal point. Call it N1.

Write the same digits, as before, and add the first period. Call it N2.

Numerator = N2 - N1

Write a series of nines a number of times equal to the number of digits of the period and a series of zeros a number of times equal to the number of digits after the decimal point and before the first period. Call it Denominator.

F = Numerator/Denominator.

Example (4): F = 5.3417

N1 = 534

N2 = 53417

Numerator = 52883

Denominator = 9900

F = 52883/9900

Example (5): F = 145.72

N1 =145

N2= 14572

Numerator = 14427

Denominator = 99

F = 14427/99 = 1603/11 

Density of rational numbers

Given an integer number n, it can be determined its next, n + 1. If the number considered is a rational this cannot be possible. Example: If the number is 4.5, somebody could think that 4.6 is its next. However, 4.53, for instance, could be placed between 4.5 and 4.6 on the number line.

In general, given two rational numbers f, g, there is at least a rational number between them. One of those rational numbers is (f + g)/2. Example: 4.55 is in the middle between 4.5 and 4.6. And 4.525 is in the middle between 4.5 and 4.55. This can be done for ever.

This property of rational numbers is called density. The set of rational numbers is dense everywhere. This means that between any two rational numbers there is at least one rational number. Indeed there are an infinity of them.

Above discourse could suggest that every point of a number line corresponds to a rational number. This is not the case. There are points of a number line with no correspondent in the set of rational numbers. This will be discussed in an article about irrational numbers.

Algebraic properties of rational numbers

Addition and multiplication are operations that can be performed in the set of rational numbers. 

The set of integers is a notable subset of rational numbers. In the quotient form, an integer is a fraction which can be represented with 1 as denominator. Examples: 6/1, -13/1.

In decimal form, an integer is represented with one or more zeros after the decimal point. Examples: 6.00, -13.0000

An equation of the form a ∗ x + b = c , with a ≠ 0, has always a unique solution for x. The solution is 

x = (c - b) ∗ a', where a' is the multiplicative inverse of a. It satisfies the equality a´∗ a = 1. Zero is the only number that has not multiplicative inverse.

Example: (2/3) ∗ x + 5/7 = -4/9. Solution: x = (-4/9 - 5/7) ∗ (3/2) = -73/42

Using decimal representation, 0,6 ∗ x + 0.714285 = -0.4

The multiplicative inverse of 0.6 is 1.5

x = (-0.4 - 0.714285) ∗ 1.5 = -1.738095238

Order in rational numbers

Let's consider non negative rationals.

A criterion to know if a/b ≤ c/d  is the following: a/b ≤ c/d iff  a ∗  d ≤ b ∗ c.

Example: To establish an order relation between 13/7 and 12/5, a method could consist in expressing those rationals with the same denominator: 65/35 and 84/35. 
Now it is easy to establish that 65/35 ≤ 84/35, that is, 13/7 ≤ 12/5

Above general criterion can be aplied too: 13 ∗ 5 ≤ 7 ∗ 12.

If the rationals are given in decimal representation, 13/7 = 1.857142  and 12/5= 2.4

For rationals  represented in decimal form, compare the integer parts. If the integer parts are equal, compare the digits after the decimal point from left to right.

Example: f = 2.5689, g = 2.5791. It can be seen that f ≤ g.

If a/b ≤ c/d then d/c ≤ b/a

Example: 5/3 ≤ 7/4; 4/7 ≤ 3/5

1.6 ≤ 1.75; 0.571428 ≤ 0.6

For negative rationals analogous rules can be deduced.