Rational Numbers

Rational numbers and number sets

Every number you have met so far lives in a "family". Each family fits inside a bigger one.

Whole numbers and integers are rational too: $5 = \frac{5}{1}$ and $-3 = \frac{-3}{1}$. So $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$, where $\mathbb{N} = \{1, 2, 3, \ldots\}$ are the natural (counting) numbers and $\mathbb{Z}$ are the integers.

Decimals fit in too. $2.5 = \frac{5}{2}$ and $0.75 = \frac{3}{4}$, so terminating decimals are rational. Repeating decimals are rational as well, and you'll learn how to prove it below. A decimal that never ends and never repeats, like $\sqrt{7}$ or $\pi$, is irrational. We write $\mathbb{I}$ for the irrational numbers, and they get their own page.

Each number belongs to many sets, so "classify" means: name the smallest set that contains it.

Fractions as decimals: terminating or repeating?

Divide the top of a fraction by the bottom and you always get a decimal. It either stops (terminating) or repeats forever. A bar marks the repeating block: $0.\overline{27} = 0.272727\ldots$ Some problems use a hat instead, as in $2.\widehat{7}$, which means the same thing as $2.\overline{7}$.

Repeating decimals come in two flavors:

• Purely repeating: the block repeats right after the decimal point, like $0.\overline{27}$.

• Mixed repeating: some non-repeating digits come first, like $0.8\overline{3}$.

You can tell which one you'll get without dividing. Look at the denominator.

The same idea tells you which numerators work.

From a repeating decimal back to a fraction

Every terminating or repeating decimal is a fraction. Some problems call that fraction the irreducible generating fraction, which just means the fraction in simplest form that equals the decimal.

Terminating decimals are easy: read the place value. $1.35 = \frac{135}{100} = \frac{27}{20}$.

Repeating decimals need a trick. Multiply by a power of $10$ so the endless tails line up, then subtract and the tails cancel.

When digits come before the repeating block (mixed), use two multipliers: one that puts the decimal point just before the repeating block, and one that puts it just after the first block.

Is $0.999\ldots$ really $1$?

Now we can settle Emma's calculator puzzle. First, use the method on $0.\overline{3}$. Let $x = 0.333\ldots$, so $10x = 3.333\ldots$ Subtract: $9x = 3$, so $x = \frac{1}{3}$. That confirms what the calculator suggests: $0.\overline{3} = \frac{1}{3}$ exactly.

Now multiply both sides by $3$:

So $0.\overline{9} = 1$.

Back to the start: Emma's calculator only shows seven $9$s, so it shows a rounded-off version. The full, never-ending $0.999\ldots$ is not "almost" $1$. It is another name for $1$, just as $\frac{1}{2}$ and $\frac{2}{4}$ are two names for one number. The same thing happens with every terminating decimal: $0.5 = 0.4\overline{9}$.

Your turn: prove $0.\overline{9} = 1$ directly with the subtract-the-tails method.

Operating with rational numbers

When an expression mixes fractions, decimals, powers, and roots, convert everything to fractions first (write repeating decimals as fractions, $1.5 = \frac{3}{2}$, $\sqrt{0.25} = \frac{1}{2}$). Then the answer is exact, with no rounding.

Ordering, the number line, and density

Comparing rational numbers

To compare, put the numbers in the same form. Either rewrite the fractions over a common denominator, or turn everything into decimals and line up the digits. For a repeating decimal, write out several digits.

Placing them on the number line

To plot $\frac{a}{b}$ exactly, split each unit into $b$ equal pieces and count $a$ of them from $0$ (to the right if positive, to the left if negative). For an improper fraction it helps to write a mixed number first. $\frac{11}{4} = 2\frac{3}{4}$: go $2$ whole units right, then $3$ fourths more. $-\frac{5}{3} = -1\frac{2}{3}$: go $1$ unit left, then $2$ thirds more.

Density

Between two rational numbers there is always another one, and then another, and so on forever.

Fractions of a quantity

Fractions are everywhere in word problems. Three ideas cover most of them.

Ratios and repeated fractions

"$5$ out of every $8$" is the fraction $\frac{5}{8}$. The rest is $1 - \frac{5}{8} = \frac{3}{8}$. If you know how many are in one group, divide by its fraction to get the total.

When one quantity depends on another by a fraction, and that depends on a third, multiply the fractions along the chain.

If the same fraction is applied again and again, you get a power. This is exactly how a bouncing ball behaves.

A ball dropped from $27$ ft that rebounds to $\frac{2}{3}$ of its previous height reaches $27 \cdot \left(\frac{2}{3}\right)^3 = 27 \cdot \frac{8}{27} = 8$ ft on the third bounce. After $n$ bounces the height is $h \cdot r^n$, a geometric sequence with common ratio $r$. When $r$ is a fraction whose denominator is a power of $2$ (or $5$), the result is a terminating decimal, as you saw earlier.

Working together and sharing

Rates. If a tap fills a tank in $t$ hours, then in $1$ hour it fills $\frac{1}{t}$ of the tank. Rates add when machines work together, and a drain subtracts. Once you know the combined fraction per hour, the time to fill the whole tank is its reciprocal.

Shares. When several people put in fractions of the total, the leftover fraction is $1$ minus the others. If profits are shared in proportion to what each person invested, each person gets the same fraction of the profit as of the investment.

In the widget, the parts $2 : 3 : 5$ are the fractions $\frac{1}{5}, \frac{3}{10}, \frac{1}{2}$ of the total.

Measurement error

When you measure something, the result is usually a decimal that is a little off from the exact value. Rational numbers let us say exactly how far off.

Why two kinds? Missing by $1$ inch matters a lot on a $10$-inch board and almost nothing on a $10$-mile road. The relative error compares the miss with the size of the thing measured.

Puzzles with reciprocals and integer results

The reciprocal (multiplicative inverse) of $\frac{a}{b}$ is $\frac{b}{a}$. Their product is $1$. Puzzles that involve a number and its reciprocal usually turn into an equation you can solve with Algebra 1 tools.

The graph shows $y = x + \frac{1}{x}$ and the horizontal line $y = \frac{10}{3}$. They cross at two points, which are the two solutions of the example below.

Another type of puzzle asks when a fraction with a variable is an integer. The trick is to split off the whole-number part so that only a simple fraction is left.

Summary