By Michal Paszkiewicz.

# Algebra and fractions

### 1. Multiply out and simplify the following: \( (a + b)(a - b) \)

*Worth: half a chocolate.*
### 2. Before western mathematics benefitted from the arrival of Arabic algebra, the majority of mathematics consisted of geometry.

The difference of two squares is proved in Euclid's **Elements** (written in 300 B.C.) by taking two squares of side a and b, and lining up the two rectangles containing the area in a that is not contained by b, i.e. the difference of two squares. This new rectangle is of length a added to b and width b subtracted from a.

The fact algebra comes from Arabic sources means that at first, it would have contained symbols we do not often see today. Simplify the following problem that uses Hebrew characters aleph ( \( \aleph \) ) and beth ( \( \beth \) ):

$$ \frac{ \aleph }{ \beth - 2 } + \frac{ \beth + 2 }{ \aleph } $$
*Worth: one chocolate*
### 3. Simplify the following: \( \frac{1}{(x + 2)} + \frac{1}{(x - 2)} \)

*Worth: half a chocolate*
### 4. Simplify the following: \( \frac{1}{(\aleph + 2)} + \frac{1}{(\aleph - 2)} + \frac{2 (\aleph + \frac{1}{ (\aleph ^2 - 4 )})}{( \aleph ^ 2 + 4 )} \)

*Worth: three chocolates*