What Number Theory is all about?

Number Theory has a simple meaning it is the Theory of Numbers.

But why the development of Number Theory is in such a way in which it stands now. Thus this post is a bird eye view of Number Theory.

In Number Theory, we focus on Integers and their properties.

Now, \(\sqrt{2}\) is not an integer. Number Theory is also not interested in that number. So in Number Theory, we study the properties of the set of integers only.

The specialty of the set of Integers?

Now take the set of real numbers. Pick two numbers 3.2 and 4.5. Now 3.2 divides 4.5 and the quotient is 1.40625. Well, the point is that if you take the set of all real numbers (R), then any non-zero number always divides another real number.

But now you take the set of Integers (Z). Pick your numbers to say 32 and 45. Now 32 doesn’t divide 45 to give another integer.

This is the idea that Integers (Z) are not closed under Division, but the Real Numbers (R) and Rational Numbers (Q) are closed under Multiplication.

What do you mean by closed under some ‘operation’?

We say that a Collection of Numbers S is closed under some operation (*) if we take \(x, y \in\) S, then \(x*y \in ) S.

Now, check the above quoted statement.

This gives a huge range of opportunities for exploration among the integers (Z).

Thus we introduce some ideas as a natural consequence of the inquisitiveness of Human Beings.

  • The idea of Divisibility and Division Algorithm – The technology to find out exactly when an integer divides another.
  • The idea of Primes – The atoms of the numbers when we see through the lens of multiplication.
  • Unique Prime Factorization Theorem The idea of a unique representation of integers as a product of prime powers.
  • GCD, Bezout’s Lemma and Euclidean Algorithm – The commonness of two integers by the string of primes.

All the above properties of Numbers is with respect to the Multiplicative Property of Integers.

The ideas of Mathematical Induction and Well-Ordering Principle are at the heart of the specialty of Natural Numbers (N).

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