A fraction is an expression where and are integers, and . On this page we review some basic facts about fractions. We will also see why these facts are true. The way we'll approach the issues will be typical for approaching the kind of mathematics you will learn in great depth in this class.

You are familiar with these **facts**:

- The number on top of the fraction bar is the
**numerator**, and the number at the bottom is the**denominator**of the fraction. - An integer can be considered a fraction with a denominator equal
to 1.
- Multiplying numerator and denominator of a fraction with the
same non-zero factor does not change the fraction. (In elementary
school the term
*equivalent*fractions is used, but a more mature view is that equivalent fractions denote the same real number, and hence need not be distinguished other than for clarity and simplicity.) - Fractions with the same denominator are added (or subtracted) by
adding (or subtracting) the
numerators (and keeping the denominator the same).
- Fractions with
*different*denominators are first turned into fractions with the same denominators, by multiplying numerators and denominators with suitable factors, and then added using the rule above. - Fractions are multiplied by multiplying numerators and
multiplying denominators.
- The
**reciprocal**of a fraction is obtained by switching numerators and denominators. - Dividing a fraction by another fraction is equivalent to
multiplying with the reciprocal of the second fraction.

Note how we explain the addition of fractions with distinct denominators in terms of the addition of fractions with the same denominator, and the division of fractions in terms of the multiplication of fractions.

These provide instances of a principle whose statement is deceptively simple, but whose applications and implications are far reaching:

** Reduce your problem to one you have solved before.**

A fraction such as is simply a way to denote the result of dividing 3 by 4. In general, a fraction denotes the result of dividing by . (This is true even if and aren't integers, an illustration of the principle that we make definitions in simple contexts and then generalize them so that all relevant rules remain true. However, if and aren't integer then the expression is called a ratio or quotient.)

Division is defined as the solution of a multiplication problem. Thus the fraction

.

It's important to understand this basic definition, since all of the above rules can be derived from it.

We do, however, require another principle

** Doing the same on both sides of an equation creates another valid equation.**:

The solution of the equation does not change if we multiply on the left and right with the same factor to obtain the equivalent equation . We have as an immediate consequence the above identity:

Let's suppose we are given two fractions

As we discussed, these fractions are defined by the equations

We now add on both sides of the first equation. This will give
another valid equation. However, on the left we will not use the
number , but instead the expression . We can do this because
the second equation asserts that equals . Henceforth we will
simply say that * we add the second equation to the first*, but the
underlying reasoning is that we add the same thing on both sides of
the first equation, we just give it different names on the two sides
of the equation. This operation gives the new equation

It is clear how this argument applies to general fractions with
the same denominator, and to the * subtraction* (as opposed to the
* addition*) of fractions.

When adding two fractions like
and
we
first convert them to fractions with the same denominator by applying
rule number 3 above. Thus we have to find a ** common
denominator**. As a practical matter, the smaller the denominator the
easier it is to manipulate the numbers, and therefore we like to use
the ** least common denominator**. However, the product of the two
denominators always works, and it is often, like in this case, also the
least common denominator.
So in this example we use

Thus

Suppose we are given two fractions

This means and satisfy the equations

Proceeding as above, and multiplying on both sides of the first equation with , but calling it on the left, we obtain

This can be rewritten as

Suppose we want to divide the same fractions as above. So we ask what is

Division is the inverse process of multiplication, so satisfies the equation

The relevance of all this is that a **common
denominator** of two fractions is a common multiple of the two
denominators, and the
**least common denominator** is the least
common multiple of the two denominators.

The least common multiple *LCM* and the greatest common
factor *GCF* of two numbers
*m* and *n* are related by the fact

LCM = m*n/GCFFor example, the GCF of

12 = 4*6/2

Think about this and send me your explanation of this fact!

** Mixed Numbers** are fractions written as a natural number plus a
fraction where the denominator is greater than the numerator. For
example,

Mixed numbers are popular because the integer part gives an indication of their size, but otherwise they have little to recommend them. They form an exception (the only exception) to the rule that a missing operator means multiplication, and they make the arithmetic operations harder to carry out. We will not use them in this class and I recommend you ignore their existence.

For large numerators and denominators the most practical way of finding common factors is the Euclidean Algorithm described elsewhere, but for many small factors there are simple rules available. They are listed in the following Table. It's a good exercise to think about why these rules hold. If you can't figured it out drop me a note!

Factor | Rule | Examples |

2 | last digit is 0, 2, 4, 6, or 8 | 2, or 127174 |

3 | sum of digits is divisible by 3 | 111 (s.o.d. = 1+1+1 = 3.) 111= 3*37, or 212,319,231 (s.o.d. = 24), 212,319,231 = 3*70,773,077. |

4 | the last two digits form a number that is divisible by 4. | 1,232, or 12,135,432,196 |

5 | last digit is 0, or 5 | 58,213,475 |

6 | Apply tests for 2 and 3 | 228, or 5,832 |

7 | there is no good test, divide by 7 | 2,443 |

8 | the last three digits form a number that is divisible by 8. | 25,432, or 2,942,600 |

9 | sum of digits is divisible by 9 | 111 (s.o.d. = 3+3+3 =9.) 333= 9*37. 242,319,231 (s.o.d. = 27), or 242,319,231 = 9*26,924,359. |

10 | last digit is 0 | 20, or 123,456,780 |

There is also a well known rule for divisibility by 11. You form one sum by adding the first, third, fifth, etc. digit, and another by adding the second, fourth, sixth, etc. The number is divisible by 11 is the difference of the sums is. For example, suppose we want to check

*m = 5,123,456,789.*

The sum of the digits in the odd numbered positions is
9+7+5+3+1 = 25. The sum of the digits in the even numbered positions
is 8+6+4+2+5 = 25. The difference is 25-25 = 0. 0 is divisible by 11, and so is *m*. Indeed,

*5,123,456,789 = 11*465,768,799.*

Check it out!