Proof 16: Fraction rules of algebra


The denominator in a fraction tells ‘how many portions we split a value into’. Split it into 2 or 3 or 4 portions: $\frac12$, $\frac13$ or $\frac14$ and so on. “Split” it into 1 portion, and we still have the same number of course: $\frac51=5$. But what if you split it into zero portions? That doesn’t make sense – it is not possible. It is a necessity for a division or fraction that the denominator is not zero:

$$\frac1a \text{ requires } a\neq0\quad_\blacksquare$$

Your 1000 € in tax money is put into several different accounts by the state, maybe evenly spread on 4 accounts, so 250 € in each.

  • You could write this as a negative amount lost, so that $-1000\,€$ are divided into 4 accounts: $\frac{-1000\,€}{4\,\text{accounts}}=-250\,€/\text{account}$.
  • Or you could write it as the amount $1000\,€$ divided into 4 “negative” accounts: $\frac{1000\,€}{-4\,\text{accounts}}=-250\,€/\text{account}$.

Either way, this is “negative” money, leaving your private account; it doesn’t matter if the negative sign is at the numerator or denominator:

$$-\frac ab=\frac{-a}b=\frac a{-b} \quad_\blacksquare $$


4 apples divided into 4 portions’ will naturally mean 1 apple per portion: $\frac 44=1$. The same for ‘5 apples divided into 5 portions’ or ‘6 apples into 6 portions’ or any other number. So, generally:

$$\frac aa=1 \quad_\blacksquare$$


If you have 3 eighths, $\frac 38$, two times, then you have six eighths: $2\cdot \frac 38=\frac 68$. This is the same as simply multiplying the numerator with 2: $ \frac {2\cdot 3}8=\frac 68$. Multiplications of the whole fraction correspond to multiplying just the numerator:

$$a\cdot \frac bc=\frac{ab}c\quad_\blacksquare$$

Four ninths times 2’ will double it (there will be double as many ninths): $\frac 49\cdot 2=\frac89$. ‘Four ninths times a half’ will half it (there will be half as many ninths): $\frac49 \cdot \frac 12=\frac29$. This is the same as having the denominator of 2 in the denominator of the fraction itself, which will also divide by 2: $\frac4{2\cdot 9}=\frac29$. When multiplying with a fraction, you can merge the denominators together:

$$\frac ab\cdot \frac 1c=\frac a{bc}\quad_\blacksquare$$

6 apples slices divided by 2’ means ‘how many portions do we have if we put 2 apples in each’. The answer is of course 3 (3 portions with each 2 apples means in total 6 apples). ‘6 apples divided by a half’ in the same way means ‘how many portions do we have if we put half an apple in each’. That will of course be 12 (half an apple in 12 portions is 6 apples in total). Dividing with a half will double the number, as if the fraction was flipped:

$$a /\frac bc=a\cdot \frac cb\quad_\blacksquare$$

A box contains 20 pieces of fruit: 12 bananas and 8 pears. To remove half of the fruit, you can either just take 10 pieces away, or you can take 6 of the bananas and 4 of the pears away. The result is the same, no matter if you divide each part with 2, or if you divide the total with 2. In other words, the division can be joined and split; it doesn’t matter:

$$\frac ac+\frac bc=\frac{a+b}c\quad_\blacksquare$$

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