Fractions: equivalence and comparison

Every fraction greater than \(1\) can be written as the sum of a whole number and a fraction less than \(1\); conversely, the sum of a non-zero whole number and a non-zero fraction less than \(1\) is a fraction greater than \(1\): Such a writing, like \(2 + \dfrac{1}{3}\), is called a mixed number.

To write \(\dfrac{n}{b}\) as a whole number plus a fraction less than \(1\):

1. Count the whole units: each whole unit is made of \(b\) parts, so divide with remainder: \(n = q \times b + r\), with a remainder \(r\) smaller than \(b\).
2. The \(r\) remaining parts form the fraction: \(\dfrac{n}{b} = q + \dfrac{r}{b}\), and \(\dfrac{r}{b} < 1\) because \(r < b\).

Conversely, to recompose a single fraction, each whole unit becomes \(b\) parts. On the number line, the decomposition also places the fraction between two consecutive whole numbers: \(q < \dfrac{n}{b} < q+1\) (when the remainder is not zero — if \(r = 0\), the fraction is exactly the whole number \(q\)).

Once a fraction greater than \(1\) is written as a whole number plus a fraction less than \(1\), its place on the number line is settled: the whole number is the one just below, and the next whole number is just above. $$\dfrac{7}{3} = 2 + \dfrac{1}{3} \qquad\text{so}\qquad 2 < \dfrac{7}{3} < 3$$ The two neighbours are read off the decomposition: never compute anything else. If the fractional part is zero, the fraction is that whole number itself, like \(\dfrac{6}{3} = 2\).

Two fractions are equivalent when they are the same amount of the same unit: on two identical bars, they cover the same length.

• Say it. “If, for the same unit, I make parts twice as small and I take twice as many parts, then I take the same amount.” Quarters cut in two give eighths: \(\dfrac{3}{4} = \dfrac{6}{8}\).
• See it on two identical bars. The first bar is cut into quarters, the second into eighths: \(\dfrac{3}{4}\) and \(\dfrac{6}{8}\) cover the same length.
• Find a missing numerator. \(\dfrac{?}{8} = \dfrac{1}{2}\): eighths are \(4\) times smaller than halves, so take \(4\) times more parts: \(\dfrac{4}{8} = \dfrac{1}{2}\).
• Find a missing denominator. \(\dfrac{2}{3} = \dfrac{4}{?}\): twice as many parts are taken, so the parts are twice as small: sixths, \(\dfrac{2}{3} = \dfrac{4}{6}\).

Some equalities come back so often that they are known, not recomputed. Cutting each part into the same number of equal smaller parts multiplies the number of parts and the number taken by that same number — the value does not change. Cut each half into two equal parts and you get quarters; cut it into five equal parts and you get tenths.

	\(=\)		\(=\)
\(\dfrac{1}{2}\)		\(\dfrac{2}{4}\)		\(\dfrac{5}{10}\)

$$\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{6} = \dfrac{5}{10} = \dfrac{50}{100} \qquad\qquad \dfrac{1}{4} = \dfrac{2}{8} = \dfrac{25}{100} \qquad\qquad \dfrac{1}{10} = \dfrac{10}{100}$$ The last ones are worth knowing on their own: tenths and hundredths are the fractions the decimal numbers are built from.

To simplify a fraction is to find an equivalent fraction that uses the smallest possible whole numbers for its numerator and denominator.
The procedure is to divide both the numerator and the denominator by the same number (a common factor). This process may need to be repeated until no whole number greater than \(1\) divides both the numerator and the denominator.

For two fractions with the same denominator, the fraction with the larger numerator is larger.

The standard procedure for comparing two fractions with different denominators is as follows:

1. Find a common denominator: Identify a common multiple of both denominators. A simple method is to multiply the denominators together.
2. Create equivalent fractions: Convert each fraction into an equivalent fraction with the chosen common denominator.
3. Compare the numerators: Once the denominators are the same, the fraction with the larger numerator is the greater fraction.

A fraction is compared to \(1\) by comparing its numerator to its denominator:

• if the numerator is smaller than the denominator, the fraction is smaller than \(1\): \(\dfrac{13}{21} < 1\);
• if they are equal, the fraction equals \(1\): \(\dfrac{21}{21} = 1\);
• if the numerator is larger, the fraction is larger than \(1\): \(\dfrac{16}{12} > 1\).

The same idea compares a fraction to \(\dfrac{1}{2}\): \(\dfrac{5}{12} < \dfrac{6}{12} = \dfrac{1}{2}\), so \(\dfrac{5}{12} < \dfrac{1}{2}\).

Two fractions with the same non-zero numerator are ordered the other way round from their denominators: the larger the denominator, the smaller the parts, so the smaller the fraction. $$\dfrac{3}{5} > \dfrac{3}{8} \qquad \text{because a fifth is larger than an eighth, and there are three of each.}$$

1. Write all the fractions with the same denominator (a common multiple of the denominators).
2. Order the numerators: with equal denominators, the larger the numerator, the larger the fraction.
3. Write the chain with the original fractions, in increasing or decreasing order.