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    This week we shall continue with our study of the type of numbers which are created when we take some kind of root $ \sqrt{\,\,\,}$ of another \emph{positive} number. These quantities we now know to be called \fillin[surds], and last week we learned the basic arithmetic of these objects.\\

Before we begin our review / brain warm-up let us contextualise today's theory: We will begin by inserting \textbf{surds} into the algebraic structures of our first topic ($(a+b)^2 = a^2 + 2ab + b^2$, etc...) where $a$ and $b$ are now surd terms.\\

Then we will learn about something known as \emph{rationalising the denominator} which is an important simplification method in mathematics, and one of the earlier conventions adopted by the \textit{Babylonians} so that they would not be dividing by incomensurable quantities - i.e. what does it mean to divide $4$ apples into $\sqrt{2}$ quantities? That would be asking to divide the 4 apples into $1.41421356237\ldots$ parts, where the `dot, dot, dot' means the denominator never terminates!\\

After practising the above simplifications, we return to Pythagora's Theorem and Geometry. Specifically we attempt problems in 3 dimensions where the need for visualisation and mathematical stamina increase.\\

The final 2 sections then deal with rationalising \emph{binomial denominators} and converting irrational numbers to fractions respectively. The latter of these makes for a cool party trick: \textsc{What fraction is equal to the repeating decimal $0.81818181\ldots$?} 

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    \fbox{\large\parbox{2in}{\[0.\overline{81} = \]}}
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