.04 X 5 = .2

.2 X 3 = .6

I get .6 so I must be wrong.

Sorry, I tried.

Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of arithmetic for integers, where numbers “wrap around” after they reach a certain value — the modulus. Modular arithmetic was introduced by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

A familiar use of modular arithmetic is its use in the 24-hour clock: the arithmetic of time-keeping in which the day runs from midnight to midnight and is divided into 24 hours, numbered from 0 to 23. If the time is noted at 7 o’clock in the evening — 19:00 in the 24-hour system — and then again 8 hours later, then rather than the time being 27:00 (as in usual addition: 19 + 8 = 27), the time will actually be denoted as 03:00, albeit in the next day. Likewise, if the clock starts at noon (12:00) and 21 hours elapse, then the time will be 09:00 the next day, rather than 33:00 (as in usual addition). Since the hour number starts over at 00 hours after passing 23 hours, this is arithmetic modulo 24 — the hours “wrap around” upon reaching the modulus 24.

Just to confuse you some more. lol …….

Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations of the ring of integers: addition, subtraction, and multiplication. For a fixed modulus n, it is defined as follows.

Two integers a and b are said to be congruent modulo n, if their difference (a−b) is an integer multiple of n. If this is the case, it is expressed as:

The above mathematical statement is read: “a is congruent to b modulo n”.

For example,

because 38 − 14 = 24, which is a multiple of 12. For positive n and non-negative a and b, congruence of a and b can also be thought of as asserting that these two numbers have the same remainder after dividing by the modulus n. So,

because, when divided by 12, both numbers give 2 as remainder.

The same rule holds for negative values of a:

A remark on the notation: Because it is common to consider several congruence relations for different moduli at the same time, the modulus is incorporated in the notation. In spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been clearer if the notation a ≡n b had been used, instead of the common traditional notation.

The properties that make this relation a congruence relation (respecting addition, subtraction, and multiplication) are the following.

If and , then:

[edit] The ring of congruence classes

Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by , is the set . This set, consisting of the integers congruent to a modulo n, is called the congruence class or residue class of a modulo n. Another notation for this congruence class, which requires that in the context the modulus is known, is .

The set of congruence classes modulo n is denoted as and defined by:

When n ≠ 0, has n elements, and can be written as:

When n = 0, does not have zero elements; rather, it is isomorphic to , since .

We can define addition, subtraction, and multiplication on by the following rules:

The verification that this is a proper definition uses the properties given before.

In this way, becomes a commutative ring. For example, in the ring , we have

as in the arithmetic for the 24-hour clock.

The notation is used, because it is the factor ring of by the ideal containing all integers divisible by n, where is the singleton set .

In terms of groups, the residue class is the coset of a in the quotient group , a cyclic group.

The set has a number of important mathematical properties that are foundational to various branches of mathematics.

Rather than excluding the special case n = 0, it is more useful to include (which, as mentioned before, is isomorphic to the ring of integers), for example when discussing the characteristic of a ring.

[edit] Remainders

The notion of modular arithmetic is related to that of the remainder in division. The operation of finding the remainder is known as the modulo operation and is sometimes written as “mod”, so we write “14 mod 12 = 2”. This meaning of “mod” is subtly but significantly different from that introduced in this article; it is true to say “38 ≡ 14 (mod 12)” , but it is not true to say “38 = 14 mod 12” — 38 is congruent to 14 modulo 12, but the remainder of 14 divided by 12 is 2, not 38. To avoid this confusion, the congruence relation is sometimes expressed by using modulo instead of mod like “38 ≡ 14 (modulo 12)” in computer science.

When working with modular arithmetic, each equivalence class is usually represented with its least non-negative member, which is called the common residue. This can be found using long division.