fractions on a number line

To compare two natural numbers I can count their digits or if they have the same digit-count, I scan the digits from left to right. To compare two fractions I have to think harder. For example, to compare $\frac{23}{29}$ and $\frac{25}{31}$ , I end up comparing $23 \cdot 31$ and $25 \cdot 29$ .

Every natural number $n \in \mathbb{N}$ is a multiple of 1, like $n = n \cdot 1$ . So, $n$ represents the point on the number line that is $n$ steps (each step has length 1) to the right of 0. When we compare two natural numbers, we can ignore the step-length (= 1, always); we only need to consider the count of steps. Fractions, on the other hand, can have different step-lengths; so we have to consider two things at the same time: step-count and step-length. For example, $\frac{23}{29} = 23 \cdot \frac{1}{29}$ and it represents the point on the number line that is 23 steps (with step-length = $\frac{1}{29}$ ) to the right of 0. Likewise $\frac{25}{31} = 25 \cdot \frac{1}{31}$ and it represents the point that is 25 steps (with step-length = $\frac{1}{31}$ ) to the right of 0. No wonder comparing fractions is harder. However, if the two fractions have the same denominator or the same numerator, they are easier to compare. Because, only step-count or step-length is changing now. For example, $\frac{23}{29}$ and $\frac{23}{31}$ are easier to compare. We can ignore the step-count (23) and since the second fraction has a smaller step-length ( $\frac{1}{31}$ ), the first fraction is bigger than the second fraction. Likewise, between $\frac{23}{29}$ and $\frac{25}{29}$ , while comparing them we can ignore the step-length. Since the second fraction is taking more (25) steps, the second fraction is bigger than the first fraction.

We can compare $23 \cdot 31$ and $25 \cdot 29$ in place of the two fractions $\frac{23}{29}$ and $\frac{25}{31}$ because, we can manipulate the fractions in an order-preserving manner to reach the two products. Both products have 1 as their step-length, so comparing step-counts is all we need. To compare $\frac{a}{b}$ and $\frac{c}{d}$ , we first note that we are comparing $a$ of $\frac{1}{b}$ -length steps against $c$ of $\frac{1}{d}$ -length steps. If we take $b \cdot d$ times more steps, the points represented by the fractions both move rightwards, but their relative position do not change. So, instead of the two fractions we began with, we can compare $bd \cdot a \cdot \frac{1}{b}$ and $bd \cdot c \cdot \frac{1}{d}$ as proxies, they have the same order between them as the original fractions. This is same as comparing $ad \cdot \left( b \cdot \frac{1}{b} \right)$ against $cb \cdot \left( d \cdot \frac{1}{d} \right)$ . Since, (from 0) taking $b$ number of $\frac{1}{b}$ -length steps (or $d$ number of $\frac{1}{d}$ -length steps) coincide with the point represented by 1 on the number line, we have transformed the original fractions into two fractions whose step-lengths are equal: $b \cdot \frac{1}{b} = 1 = d \cdot \frac{1}{d}$ . As a result, we can ignore the step-length and just compare the step-counts: $a \cdot d$ against $c \cdot b$ . For example, to compare $\frac{23}{29}$ and $\frac{25}{31}$ , I ended up comparing the step-counts $23 \cdot 31$ and $25 \cdot 29$ of the transformed fractions.

Thinking fractions as inhabitants of a number line clarifies why it is alright to write $1.23000$ as $1.23$ . The decimal $1.23000$ represents the fraction $\frac{123000}{100000}$ and $1.23$ represents the fraction $\frac{123}{100}$ . These two fractions represent the same point on the number line. The fraction $\frac{123000}{100000}$ is taking more ( $123000$ ) but smaller $\left( \frac{1}{100000} \right)$ steps and the fraction $\frac{123}{100}$ is taking less ( $123$ ) but bigger $\left ( \frac{1}{100}\right)$ steps. As a result, they both end up on the same point to the right of 0.