# Fraction Frantics

## It's a well known fact that 1/*x* is the reciprocal of *x*. However:

## 1 ÷ (1/*x*) = *x* itself!

### Note: *x* isn't equal to zero since division by zero is technically undefined.

## If you add another 1 & another division sign, (don't forget to use parentheses!) you get:

## 1 ÷ (1 ÷ (1/*x*)) = 1/*x*

## Now you're back at the reciprocal!

## It depends on how many 1's, division signs & parentheses you have what the correct quotient will be! (Yes, the 1 in "(1/*x*)" counts as one of them!) If the number of 1's is __odd__, you get 1/*x*; if the number of 1's is __even__, you get *x*. Don't forget that each opening parenthesis "(" must have a closing parenthesis ")".

## Another thing about this math trick:

## 1/*x* = *x*^{-1}, so (*x*^{-1})^{-1} = *x*^{1} = *x*.

If exponents are negative, then you're supposed to divide the base by itself (**x** **number** of times) instead of multiply. The base is the number that you're multiplying(or dividing) by itself!

## Parentheses can be used to multiply exponents! Remember this rule about exponents below:

## (x^{a})^{b} = x^{ab}

## Multiplying(or dividing) 2 negatives makes a positive product(or quotient), so will multiplying an __even__ number of negatives. An __odd__ number of negatives will make the product negative, so:

## ((((*x*^{-1})^{-1})^{-1})^{-1})^{-1} = *x*^{-1} = 1/*x*

## There are 5 negative 1's as exponents, so since 5 is an __odd__ number, their product is negative. The parentheses are used to multiply them so that you get -1 as the exponential product.

Note: Some of the other math tricks show proof that a number is even or odd! Guess which ones they are!

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© Derek Cumberbatch