Elementary arithmetic is the simplified portion of arithmetic which includes the operations of addition, subtraction, multiplication, and division.
Elementary arithmetic starts with the natural numbers and the written symbols (digits) which represent them. The process for combining a pair of these numbers with the four basic operations traditionally relies on memorized results for small values of numbers, including the contents of a multiplication table to assist with multiplication and division.
In Swami Bharati Krishna Tirtha's Vedic mathematics, the auxiliary fraction method is used to convert a fraction to its equivalent decimal representation. The "auxiliary fraction" is not a true fraction, but is simply a mnemonic aid used in the calculation. The method is essentially the long division algorithm adapted for mental calculation. It is simplest when the fraction's denominator is one less than a multiple of 10, when it uses the identity
Variants of the method used when the denominator is not one less than a multiple of 10 become progressively more complex but still in the realm of mental math or with one line of notation.
An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered). In other words, a fraction a⁄b is irreducible if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to irreducible rational fractions.
An equivalent definition is sometimes useful: if a, b are integers, then the fraction a⁄b is irreducible if and only if there is no other equal fraction c⁄d such that |c| < |a| or |d| < |b|, where |a| means the absolute value of a. (Let us recall that to fractions a⁄b and c⁄d are equal or equivalent if and only if ad = bc.)