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Basic Mathematics

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					                                 Basic Mathematics


 The following is essentially intended for mature students who are just back in college
and who haven’t had to deal with maths in a long while.
 A few simple, basic rules are reminded below.

    (I)      Priorities
When confronted with something like

2(7  6)  (3  9) / 4 ,

it is essential to get the priorities right.

The safest way to do it is to calculate whatever is inside the brackets first:

2(13)  (12) / 4 .

Then perform the multiplications and the divisions:

26  3

and finally the addition

29 .

The general rule is: brackets first, then multiplications and divisions and finally
additions and subtractions.


    (II)     Signs

How to deal with “subtract  3 from 4 ”?

Another way to put it is: calculate 4  (3) .

In fact,

 (3)  3

so that subtracting  3 amounts to adding 3 .

4  (3)  4  3  7 .
The same kind of rules applies when you multiply two negative numbers:

(4)  (3)  4  3  12 .

However, if you multiply a positive number by a negative number, the result is a
negative number:

4  (3)  (4  3)  12 .



    (III) Simplifying Fractions
Sometimes a fraction can be simplified. For example

3 3 1
 
6 3 2

and the “ 3 ” at the top and the bottom cancel:

3 1
  .
6 2

 The idea is that you decompose the number at the top and the number at the bottom
and cancel the bits they have in common:

60 2  2  3  5 3
                .
75 2  2  5  5 5



    (IV) Adding and Multiplying Fractions

-   Multiplying two fractions is, as a rule, easier than adding them up. For example

3 5 3  5 15
          .
4 7 4  7 28

You just multiply the numbers at the top and the numbers at the bottom. Et voila.


-   Adding two fractions usually requires a bit of work. For example performing

1 4
 
3 3
is easy because the two fractions have the same denominator (denominator = number
at the bottom of the fraction):

1 4 1 4 5
       .
3 3  3   3

If two fractions have a different denominator, the situation is a tad more complicated.
Say you want to calculate

1 1
  .
2 3

You need to find a common denominator, i.e. a denominator that is common to the
two fractions. Multiplying the denominators does the trick, even though it is not
always the most efficient way to do it.

A common denominator in this case is:

  23  6 .

You then have to transform the two fractions so that they both have 6 has a
denominator:

               1 3 1  3   3                       1 2 1  2  2
                       ,                             .
               2 3 2 3 2 6                       3 2 3 23 6

We are now able to add the fractions:

1 1 3 2 3 2 5
         .
2 3 6 6  6   6

				
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