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MAT081 :: Lecture Note :: Week 03
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### Introduction to Fractions

Instructor to class: How do we define fraction?

Instructor to class: Have you ever defined fraction as follows? A fraction is an unsolved division problem (i.e. it is a quotient of two numbers).

`[humor]` Five out of every four people have difficulty understanding fractions.

Fractions are used to represent part-of-a-whole of something. Examples: Zelmo ate 1/2 of the whole pie. Edith spent 3/4 of a full hour sending an email message to Zelmo. Truman completed only 2/9 of the nine part exercise.

```                                 x
x ÷ y  is also the fraction  ---  (also written  x/y)
y

'x' (dividend) is the numerator
'y' (divisor) is the denominator
```

Since the divisor divides into the dividend, the denominator divides into the numerator.
[terminology exercise: No or Yes: Dividing the denominator into the numerator results in a terminator.]

A fraction can never have a denominator of zero because division by zero is undefined.

Ignoring negative numbers... A proper fraction is a fraction having a numerator that is less than the denominator. An improper fraction is a fraction having a numerator greater than or equal to the denominator.

```   2/3 -- proper fraction (2 < 3)
7/4 -- improper fraction (7 > 4)
5/5 -- improper fraction; equals 1
0/8 -- proper fraction (0 < 8)
9/0 -- undefined fraction (denominator is 0)
```

Although improper fractions are called improper, there is nothing improper about them (i.e. they are not "bad" fractions).

A unit fraction is a fraction that has one as its numerator.

Every whole number 'n' can be written as a fraction by using 'n' in the numerator and one in the denominator.

A mixed number is a whole number plus a fraction.

```   1/8    -- unit fraction
5/1    -- the value 5
7 1/4  -- mixed number; 7 is a whole number;  1/4 is a fraction
7 1/4 is 7 + 1/4; most of the time the fractional part
of mixed number is a proper fraction
```

A fraction consisting of two integers is called a rational number. [Observe how the word rational contains the word ratio. Fractions are ratios.]

A fraction is negative if either the numerator is negative or the denominator is negative. If both the numerator and denominator are negative, then the fraction is positive.

GDT::BAB:: About the Fraction Five Divided-By Eight

##### Multiplication

Multiply the numerators to get the numerator of the product. Multiply the denominators to get the denominator of the product.

```   2   4     8      2 * 4 is 8
- * - =  --
3   5    15      3 * 5 is 15
```
##### Division

To divide one fraction by another one, multiply the top fraction by the reciprocal of the bottom fraction.

If dividing mixed numbers, then convert the mixed numbers to improper fractions and then perform the division.

Reciprocal is the multiplicative inverse of a number. For a fraction, it's obtained by "flipping the fraction."

```   2   1
- ÷ -
5   7

rewritten

2         2   7       14       4
-         - * -   =   --  =  2 -
5         5   1        5       5
---
1
-
7
```
##### Equivalent Fractions

Given a fraction, both the numerator and denominator can be multiplied or divided by the same number and the result is an equivalent fraction.

```   3   5   15
- * - = --
4   5   20
```

Fractions can be added or subtracted only if they are like fractions (i.e. have common denominators). Typically we want the common denominator to be the smallest common denominator, but to help get started we will obtain a common denominator by using the product of the two denominators. Once a common denominator has been determined and the numerators adjusted (i.e. equivalent fractions calculated), then the numerators are added or subtracted.

##### Example
```   1   1
- + -
5   4

multiply the two denominators to get a common denominator
5 * 4 equals 20

1(n)    1(n)
---- +  ----
20      20

adjust the numerators finding equivalent fractions
given the common denominator

1(4)    1(5)       9
---- +  ----   =  ---
20      20        20

A  +   B

_ _ _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ _ _     0/20

A A A A B B B B B _ | _ _ _ _ _ _ _ _ _ _     9/20
```