ON RATIONAL NUMBERS, DIVISION ALGORITHMS, AND GREATEST COMMON DIVISORS

An interesting property about the set of integers is the “Division Algorithm” which states that “for any two integers a and b where b is greater than 0, there exist unique integers q and r such that 

 a = bq + r 

where r is greater than or equal to 0 but less than b.” 

The integer q is called quotient, and the integer r is called a remainder while b is the divisor and a is the dividend.

For example, with integers a = 11 and b = 4, we have two unique integers q = 2 and r = 3 also such that 

11 = 4 x 2  + 3.

Note that there is no other integers s and t such that 

11 =  4s + t 

where t is greater than or equal to 0 and t < 4.

Suppose a = -15 and b = 6. 

Although, -15 = 6(-2) + (-3), the remainder cannot be -3 since it is a negative integer. Thus, the quotient cannot be -2. 

Also, although we have -15 = 6(-4) + 9, the remainder cannot be 9 since it is larger than the divisor 6. Thus, the quotient cannot be -4.

The correct combination must be -15 = 6(-3) + 3, and tnhe remainder is equal to 3 it is not a negative integer and it is less than the divisor 6. Thus, the quotient is -3.

The Python programming has built-in commands and operations to find the quotient and the remainder when an integer is divided by a positive integer.To find the remainder between two integers a and b, we can use a % b.

For example, 11 % 4 produces the number 3 as the remainder, The quotient that is equal to 2 can be obtained by (11 - 3) / 4 or simply 11 / 4, if the Python 2.7 version or earlier is used.

A built-in command in Python that produces the quotient and the remainder as a pair is divmod( ... , ... ). 

This divmod(11,4)  will output the pair (2, 3), where q = 2 and r = 3.

A rational number is defined to be a ratio of two integers s and t where t > 0. Thus, a rational number is of the form s / t. 

An integer a is also a rational number since a / 1 is a rational number and it is equal to a.

We can always find a rational number between any two integers 

For example, (3 + 4) / 2 = 7/2  is a rational number between 3 and 4.

Since 7 = 2(3) + 1, we have 7 / 2 = 3 + 1 / 2. The first term in the right-hand side of the equation is an integer and the second-term is a positive rational number less than 1. The second-term is a proper fraction.

Since 34 = 6(5) + 4, we have 34 / 6 = 5 + 4 / 6. The first term is an integer and the second term is a proper fraction which can be reduced to its simplest form 2 / 3. 

Hence, 34 / 6 = 5 + 2 / 3. 

Also,  -2 = 5(-1) + 3. Thus, we have -2 / 5 = -1 + 3 / 5.The first term is also an integer and the second term is a positive rational number less than 1.

Furthermore, 20 = 10(2) + 0 and 20 / 10 = 2 + 0 / 10 = 2 + 0 = 2. The rational number 20 / 10 reduces to an integer 2.

If we have a rational number s / t, then by the Division Algorithm, we have unique integers a and r such that 

s = t x a + r 

where r is greater than or equal to 0 or less than t. 

Thus, the rational number s / t is equal to s / t = a + r / t.

If r = 0, then s / t = a, which is an integer.

The positive integer d is the GCD of  two integers s and t  if d is a common divisor of s  and t and if c is a common divisor of s and t then c is a divisor of d. Thus, d is the largest of all the common divisors of s and t.

If GCD( s,  t ) = 1, then we say that s and t  are relatively prime.

Suppose s = ta +r and r > 0. 

Let d = GCD(r,  t) and let p = r / d and q = t / d. It follows that the GCD(p, q) = 1.

Thus,  s / t = a  +  r / d. Since we can simplify the fraction r / d into a proper fraction p / q where GCD(p, q) = 1.

Hence,   s / t = a + p / q 

where a, p and q are integers with 0 < p < q,  p and q are relatively prime.

Therefore, any rational number can be expressed as an integer or as the sum of an integer and a proper fraction.

by Felix P. Muga II
Associate Professor, Mathematics Department, Ateneo de Manila University
Senior Fellow, Center for People Empowerment in Governance
Chair, Mathematical Sciences Division, National Research Council of the Philippines

The Mathematics of Kits

Next topic:
Immediate - "The Rational Number as the Sum of an Integer and a Proper Fraction";
Main Objective: "Digitizing a Rational Number"