Real Number Theorems

Quick revision for theorems

Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b,

there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

Theorem 1.2 (Fundamental Theorem of Arithmetic) : Every composite number

can be expressed ( factorised) as a product of primes, and this factorisation is

unique, apart from the order in which the prime factors occur.

Theorem 1.3: Let $p$ be a prime number. If $p$ divides $a^{2},$ then $p$ divides a, where $a$ is a positive integer.

Theorem 1.4: $\sqrt{2}$ is irrational.

Theorem 1.5: Let $x$ be a rational number whose decimal expansion terminates.

Then $x$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are coprime, and the

prime factorisation of $q$ is of the form $2^{\circ} 5^{m},$ where $n, m$ are non-negative integers.

Theorem 1.6: Let $x=\frac{p}{q}$ be a rational mamber, such that the prime factorisation of $q$ is of the form $2^{n} 5^{\text {" }},$ where $n, m$ are non-negative integers. Then $x$ has a

decimal expansion which terminates.

Theorem 1.7: Let $x=\frac{p}{q}$ be a rational number, such that the prime factorisation of $q$ is not of the form $2^{n} 5$ ", where $n, m$ are non-negative integers. Then, $x$ has a decimal expansion which is non-terminating repeating (recurring).

So we can conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating.