Real numbers/Euclid's
division lemma.
Introduction:
Since our childhood we have been using it for fundamental
operations of addition, subtraction, multiplication and division.We have
applied these operations on natural numbers in teachers. Rational number, an
irrational number.
Euclid's Division lemma tells us about the divisibility of
integers. It is quite easy to state and understand. It takes that any positive
integer A can be divided by another positive integer be.In such a way that it
leaves a remainder are.That is a smaller than be.This is nothing but the usual
long division process.Euclid's Division lemma provides a stepwise procedure to
compute the head chef of two positive integers. This stepwise procedure is
known as Euclid's Algorithm.We can use the same for finding the H.C.F of
positive integers.
The fundamental theorem of arithmetic tells us about
expressing positive integers as a product of prime integers. It states that
every positive integer is either prime or it can be factored as a product of
powers of prime integers. This theorem has many significant applications in
mathematics and in other fields.
To find head, CF, and LCM of a positive integer, by using
the fundamental theorem of arithmetic in numbers, such as 
, 
we know that the decimal representation of a
rational number is either terminating or if it is not terminating then it is
repeating.The prime factorization of a denominator of a rational number
completely reveals the nature of its decimal representation.In fact, by looking
at the prime factorization of the denominator of a rational number, one can
easily tell about its decimal representation, whether it is terminating or non
terminating, repeating.
Divisibility:
A non zero integer ‘a’ is set to divide an integer ‘b’ if
there exists an integer c such that b=ac.
The integer b is called the dividend integer ‘a’ is known as
the divisor and the integer ‘c’ is known as the quotient.
For example, Three divide 36 because there is an integer 12
such that 36=3x12. However, 3 does not divide 35 because there does not exist
an integer c such that 35 = 3xc Is not true for an integer c.
If a non zero integer ‘a’ divides an integer ‘b’ then we
write a|b. This is read as “a divides b”.
When a|b, we say that ‘b is divisible by a’ or ‘a is a
factor of b’ or ‘b is a multiple of a’ or ‘a is a divisor of b’.
We Observe that:
●
-4|20, because their existence integers -5 such that 20
= -4x(-5)
●
4|-20, because their existence integers -5 such that
-20 = 4x(-5)
●
-4|-20, because
their existence integers -5 such that -20=-4x5
Properties of divisibility:
Divides every non zero
integer, That is 
|a For every non zero
integer a.- 0 Is divisible by every
non zero integer a, That is a|0 For every non zero integer a.
- 0 does not divide any
integer.
- If a is a non zero integer
and b is any integer then a|b
a|-b, -a|b and -a|-b - If a and b are nonzero
integer then, a|b and b|a
a=
b - If a is nonzero, integer
and b, c are any two integers then,
a|b and a|c a|b
,a|bc, a|bx , for any
integer x.
- Ifa and c are nonzero
integers and b,c are any two integers then,
●
a|b and c|d ac|bd
●
ac|bc a|b
Euclid's
division lemma is a fundamental mathematical result that was first stated and
proved by the ancient Greek mathematician Euclid in his book "Elements".
The lemma is a statement about the properties of the division of integers, and
it forms the basis for many of the other results in number theory.
In its most general form, Euclid's division lemma states that
for any two positive integers a and b, there exist unique integers q and r such
that a = bq + r, where 0 ≤ r < b. This statement is often called the
"division algorithm" because it provides a systematic way to divide
one integer by another.
The
proof of Euclid's division lemma is relatively straightforward, and it relies
on the well-known principle of mathematical induction. This principle states
that if a statement is true for a particular value of a variable (the
"base case"), and if the statement is also true for any other value
of the variable whenever it is true for the previous value (the "inductive
step"), then the statement must be true for all values of the variable.
In the case of Euclid's division lemma, the base case is when a
and b are both equal to 1. In this case, it is clear that a = bq + r, where q =
0 and r = 0. This satisfies the conditions of the lemma, because 0 ≤ r < b.
For the inductive step, we assume that the statement of the
lemma is true for some arbitrary pair of positive integers a and b. We then
consider the case where a is increased by a multiple of b, so that a + kb = (k
+ 1)b. By the induction hypothesis, we know that there exist unique integers q
and r such that a = bq + r, where 0 ≤ r < b.
We
can then use this equation to express (k + 1)b in terms of a and b:
(k
+ 1)b = (bq + r) + kb = bq + (r + kb)
This equation shows that (k + 1)b can be written as a multiple
of b plus a remainder. Since 0 ≤ r + kb < (k + 1)b, it follows that the
integers q and r + kb must be the unique integers that satisfy the conditions
of the lemma for the case where a = (k + 1)b.
This
completes the inductive step, and it shows that the statement of Euclid's
division lemma is true for all pairs of positive integers a and b. This is a
powerful result, because it allows us to divide any integer by any other
integer using a simple and systematic procedure.
The
uniqueness of the integers q and r that satisfy the conditions of the lemma is
also an important property. This means that there is only one possible value of
q and one possible value of r for any given pair of integers a and b. This
property is useful because it ensures that the division algorithm always
produces the same result, regardless of how it is applied.
In
addition to the division algorithm, Euclid's division lemma has many other
applications in number theory and other areas of mathematics. For example, it
can be used to prove the existence and uniqueness of prime factorizations, to
prove the Euclidean algorithm for finding the greatest common divisor of two
integers, and to prove the well-ordering principle.
Overall, Euclid's division lemma is a fundamental result in
mathematics that has many important applications. It provides a simple and
systematic way to divide one integer by another, and it has played a key role
in the development of many other important mathematical concepts.