## Algebra Rules for Arithmetic

### Multiplication can be distributed across addition.

If you're multiplying something with a sum of two or more other values, you can distribute the multiplication to each of the values, then sum the result.

### Multiplying the numerator is the same as multiplying the fraction.

A multiplication or division of the numerator of a fraction affect the fraction as a whole (and vice versa). If you need to multiply a fraction, multiplying the numerator does the job.

### Dividing the numerator equals multiplying the denominator.

If you *divide* the numerator by a particular number, it has the same effect on the fraction's overall value as if you *multiply* the denominator by that same number. A division above equals a multiplication below.

### Dividing the denominator equals multiplying the numerator.

In the same fashion as the above rule, *dividing* the denominator of a fraction has the same effect as *multiplying* the numerator. A fraction below equals a multiplication above.

### Fractions can be summed by multiplying across between numerators and denominators, and multiplying denominators for a common denominator.

If top and bottom of a fraction are both multiplied by the same number, the fraction remains unchanged. So, we can sum two fractions by first multiplying each fraction's numerator and denominator with the other fraction's denominator. Then, since both the fractions now have the same denominator (the product of the two denominators), we can combine them into one fraction, with the sum in the numerator.

### How to turn a subtraction of two fractions into one fraction.

This is another version of rule 5, but for *subtraction* of two fractions, rather than addition.

### Reversing a subtraction in both the numerator and the denominator of a fraction leaves the fraction's value unchanged.

Reversing a subtraction gives the inverse result: ``5-3 = 2; 3-5 = -2``. In a fraction, if both the numerator and the denominator are inverted, the value of the fraction stays the same. So, if we reverse a subtraction in both the numerator and denominator, the value of the fraction is unchanged.

### Fractions with common denominators can be combined.

Two fractions with common denominators can be added by adding the numerators and leaving the denominator unchanged. Going the other direction, we can also break apart a fraction with an addition in the numerator into two fractions (each with the common denominator).

### Multiplication and division (by the same number) cancel each other out.

Division is the inverse of multiplication: if ``{a \over b} = c`` then ``b*c = a``. This means that the fraction ``{ac \over c}`` is equal to ``a``, since we are multiplying ``a`` by ``c`` and then immediately dividing it by ``c`` again, which puts us right back where we started. Since we know that ``{ac+bc \over c} = {ac \over c} + {bc \over c}`` (see rule 8), and based on the above we can see that ``{ac \over c} = a`` and ``{bc \over c} = b``, we have our result: ``a+b``.

### If the numerator and denominator of a fraction are both fractions, it can be converted into a fraction of two multiplications.

Combining rules 3 and 4, we can multiply the denominator of the bottom fraction with the numerator of the upper fraction, which gives the combined numerator, and cancels the denominator of the lower fraction; we can then multiply the denominator of the upper fraction with the numerator of the lower fraction, to give the combined denominator and cancel the denominator of the upper fraction.