Modular Arithmetic - An introduction
Modular Arithmetic is a special base arithmetic in which only the remainder is reported. That is, it can be considered as remainder arithmetic.
For example,
considering the number of days of the week on a circular dial or circular clock
face as shown below.
If
we take Sunday as the starting point (0), Monday (1) and so on clockwise around
the dial, we can find any day we want. For example, starting at Sunday, what
day will be in 18 days?
Solution:
When
we start on Sunday, we will return to Sunday every seven days. So 18 days from
Sunday will be 2 (7) + 4; this is Thursday.
We
could also solve this problem by using the idea of remainder. That is dividing
18 by 7 then we consider the remainder.
Now
the remainder (4) shows there is four days after Sunday and that is Thursday.
From
the above the example, the number of days in the cycle is called its modulus
and each day is called a cyclic variable. This type of
circular arithmetic is called Modular Arithmetic.
NB:
The modular arithmetic base 7 is written as mod7, base 9 is written
as mod9 and so on. In general, in any modn arithmetic, the remainder
will form a set with elements {0, 1, 2 ...
n – 1}
So
that if n = 7, mod7 = {0, 1, 2, 3, 4, 5, 6}
n = 10, mod10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and so on.
The
equivalent sign (≡) is used in modular arithmetic.
Equivalent Values in
Modular Arithmetic:
Two
or more number are said to be equivalent if they give the same remainder when
divided by a particular modulo. For numbers such as 3, 8, 13, 18, ... are equivalent
in arithmetic modulo 5.
Example:
simplify the following
1. 17mod3
2. 28mod9
3. 125mod10
Solution:
1. 17mod3
≡ 2mod3. This is because (17 divided by 3 = 5 remainder 2.)
2. 28mod9
≡ 1mod9. This is because (28 divided by 9 = 3 remainder 1.)
3. 125mod10
≡ 5mod10. This is because (125 divided by 10 = 12 remainder 5.)
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The Venn diagram – Two-Set Problem. - New!
Labels: Algebra, All Topics
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