Rational Numbers

The set $\mathbb{Q}$

The natural numbers and the integers are also rational: $5 = \frac{5}{1}$ and $-3 = \frac{-3}{1}$. That is why $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$. Numbers that cannot be written as a fraction, such as $\pi$ or $\sqrt{7}$, are irrational ($\mathbb{I}$); we deal with them in the next section.

To classify a number, simplify first: $\frac{8}{2} = 4$ is a natural number, even though it "looks like" a fraction.

From fraction to decimal: exact or repeating?

Dividing the numerator by the denominator always gives a decimal. Whether it is exact or repeating depends only on the prime factors of the denominator of the irreducible fraction.

The repeating part is marked with a bar: $0.\overline{3} = 0.333\ldots$ and $1.2\overline{3} = 1.2333\ldots$ In some exercises you will see a hat ($\widehat{3}$): it means the same thing.

The generating fraction

Every exact or repeating decimal can be written as a fraction: its generating fraction. The idea is to multiply by powers of 10 so that the repeating part cancels when you subtract.

From this comes a short rule:

Combined operations with rational numbers

When an expression mixes repeating decimals, powers, roots, and parentheses, convert everything to fractions before operating. The result comes out exact, with no rounding.

For powers of fractions, a negative exponent flips the base: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$. Dividing by a fraction is equivalent to multiplying by its reciprocal.

Fractions in problems: part of a whole

"$\frac{2}{5}$ of something" means multiply: $\frac{2}{5} \cdot$ total. And "of the rest" means the fraction applies to what is left, not to the original total.

Ratios and work rates

A ratio like "5 out of every 8" is a fraction: $\frac{5}{8}$. To convert it to a percent, turn it into a decimal and multiply by $100$. If you know how many failed, find what fraction of the total they represent: $1 - \frac{5}{8} = \frac{3}{8}$ failures.

When several ratios are chained together ("it makes $\frac{5}{2}$ turns for every turn of the previous one"), the fractions are multiplied.

For joint work, you add the fractions of work done per hour. If a faucet fills a tank in $4$ h, in $1$ h it fills $\frac{1}{4}$ of the tank. A drain that empties it in $8$ h contributes $-\frac{1}{8}$ per hour.

Repeating fractions and sharing

If at each step something is multiplied by the same fraction, the result after $n$ steps is the initial value times the fraction raised to the $n$th power: $\text{initial} \cdot \left(\frac{p}{q}\right)^{n}$.

If an amount is shared proportionally to what each person contributed, each person gets the same fraction of the share as the fraction they contributed to the total.

Order, number line, and density

To order rational numbers, put them in the same form: all as decimals (with the repeating ones written with several digits) or all as fractions with a common denominator.

To plot a fraction exactly on the number line, separate the whole part and divide the unit into as many parts as the denominator says: $\frac{5}{3} = 1 + \frac{2}{3}$ is two thirds past $1$; $-\frac{7}{4} = -1 - \frac{3}{4}$ is three quarters to the left of $-1$.

Density

Between any two distinct rational numbers, however close they are, there are always infinitely many more rational numbers. This property is called density.

The rational numbers, together with this order and the operations of addition and multiplication, form what is called an ordered field. But note: even though they are dense, the rational numbers do not fill the whole number line; points like $\sqrt{2}$ are not rational.

Measurement error with fractions

If the exact value is $V$ and the measured or approximate value is $v$:

• Absolute error: $E_a = |V - v|$.

• Relative error: $E_r = \dfrac{E_a}{|V|}$. It compares the error with the size of what was measured; it can be expressed as a fraction or a percent.

With fractions, convert the decimal to a fraction and subtract using a common denominator.

Challenges with reciprocals and algebraic fractions

A number and its reciprocal. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$. Adding them gives $\dfrac{a}{b} + \dfrac{b}{a} = \dfrac{a^2 + b^2}{ab}$. You look for that fraction to equal the given one and try small integers. For example, $\frac{a^2+b^2}{ab} = \frac{10}{3}$ holds with $a=3,\ b=1$, since $3 + \frac{1}{3} = \frac{10}{3}$.

Algebraic fractions with integer value. To find out when $\frac{\text{something with } n}{n+k}$ is an integer, separate the whole part by dividing the numerator by the denominator. Then require that the denominator divide the remainder.

Back to the beginning: $1 \div 3 = 0.\overline{3}$ never ends, and $3 \times 0.\overline{3} = 0.\overline{9} = 1$ exactly. What was missing on Ximena's screen was nothing at all: infinitely many digits simply do not fit.

Summary