Rational Numbers

Study Sheet

Rational Numbers

Fractions and decimals, positive and negative, all in one family

What a Rational Number Is

Concept
The Big Idea
rational numbers−2.63/51/3integers−4−17whole07
Each family sits inside the next.

A rational number is any number that can be written as a fraction ab\dfrac{a}{b}, where aa and bb are integers and b≠0b \neq 0. The word “rational” comes from ratio — a comparison of two whole numbers. This one idea gathers up almost every number you have met so far into a single family.

Rational numbers are all the fractions, positive and negative, including the integers and every terminating or repeating decimal. They are closed under the four operations (except division by zero), and the sign rules for integers carry over unchanged. Between any two of them lies another — the number line has no gaps you can point to, though irrationals still fill in more.

The name comes from "ratio": a rational number is any number that can be written as a ratio of two integers. Every integer qualifies because n=n1n = \tfrac{n}{1}, so the integers sit inside the rationals, just as the whole numbers sit inside the integers. Each new family was built so that one more operation always has an answer: integers make subtraction work, rationals make division (by anything but zero) work.

Classifying Numbers

Every one of these is rational, because each can be written as a fraction:

  • Integers: 7=717 = \dfrac{7}{1},   −4=−41-4 = \dfrac{-4}{1},   0=010 = \dfrac{0}{1}.
  • Fractions: 35\dfrac{3}{5},   −29-\dfrac{2}{9},   114\dfrac{11}{4}.
  • Terminating decimals: 0.25=140.25 = \dfrac{1}{4},   −1.5=−32-1.5 = -\dfrac{3}{2}.
  • Repeating decimals: 0.3‾=130.\overline{3} = \dfrac{1}{3}.
Example
Is It Rational?

Show that −2.6-2.6 is rational. Write it as a fraction: −2.6=−2610=−135-2.6 = -\dfrac{26}{10} = -\dfrac{13}{5}. Since −13-13 and 55 are integers and the bottom is not 00, the number is rational.

Side note
Where it goes wrong

Deciding that 0.3‾0.\overline{3} is not rational "because it never ends." Repeating decimals are rational: 0.3‾=130.\overline{3} = \tfrac13. What makes a decimal irrational is never ending AND never repeating, like 2=1.41421…\sqrt2 = 1.41421\ldots

Tip

Tip: Whole numbers and integers are just rational numbers with a denominator of 11. Every integer is rational, but not every rational number is an integer.

Try it
Try it: show it is rational

Write each as a ratio of integers: −5-5, 3.253.25, −0.08-0.08, 2132\tfrac13.

Show the answer

−5=−51-5 = \tfrac{-5}{1}. 3.25=325100=1343.25 = \tfrac{325}{100} = \tfrac{13}{4}. −0.08=−8100=−225-0.08 = -\tfrac{8}{100} = -\tfrac{2}{25}. 213=732\tfrac13 = \tfrac{7}{3}. Each is an integer over a nonzero integer, so each is rational.

Rational Numbers on the Number Line

Concept
Placing and Ordering

Rational numbers fill in the gaps between the integers. To place a fraction, split the space between two whole numbers into equal parts. Negative rationals sit to the left of 00; positive rationals to the right. As always, farther right means greater.

Every rational number has an exact spot: ab\tfrac{a}{b} is reached by cutting each unit into bb equal pieces and counting aa of them from 00 (to the left if negative). Because the spot is exact, comparing two rationals is comparing positions, and the rule is the same as for integers: farther right is greater.

Example
Comparing Two Fractions

Which is greater, 34\dfrac{3}{4} or 58\dfrac{5}{8}? Rewrite with a common denominator of 88:

34=68,58=58.\frac{3}{4} = \frac{6}{8}, \qquad \frac{5}{8} = \frac{5}{8}.

Since 68>58\dfrac{6}{8} > \dfrac{5}{8}, we have 34>58\dfrac{3}{4} > \dfrac{5}{8}.

Side note
Where it goes wrong

Placing −12-\tfrac12 to the left of −1-1 "because it has a fraction." −12-\tfrac12 is half a unit left of 00; −1-1 is a whole unit left. So −1<−12-1 < -\tfrac12.

Example
Ordering with Negatives
−1−1/20.33/4−1.501.5
Left to right: −1<−12<0.3<34-1 < -\tfrac12 < 0.3 < \tfrac34.

Order −12, 0.3, −1, 34-\dfrac{1}{2},\ 0.3,\ -1,\ \dfrac{3}{4} from least to greatest. As decimals: −0.5, 0.3, −1, 0.75-0.5,\ 0.3,\ -1,\ 0.75. Reading the number line left to right:

−1<−12<0.3<34.-1 < -\tfrac{1}{2} < 0.3 < \tfrac{3}{4}.
Tip

Tip: To compare tricky rationals, turn them all into the same form — either common-denominator fractions or decimals. With negatives, remember −12-\frac{1}{2} is greater than −1-1.

Try it
Try it: order

Order −23-\dfrac{2}{3}, −0.7-0.7, 16\dfrac{1}{6}, −54-\dfrac{5}{4} from least to greatest.

Show the answer

As decimals: −0.666…-0.666\ldots, −0.7-0.7, 0.1666…0.1666\ldots, −1.25-1.25. Farthest left is −1.25-1.25, then −0.7-0.7, then −0.666…-0.666\ldots: −54<−0.7<−23<16-\tfrac54 < -0.7 < -\tfrac23 < \tfrac16.

Absolute Value of Rational Numbers

Concept
Distance from Zero

The absolute value of a rational number is its distance from 00 on the number line, so it is never negative. We write it with bars:

∣−23∣=23,∣−1.7∣=1.7,∣0.9∣=0.9.\left|-\tfrac{2}{3}\right| = \tfrac{2}{3}, \qquad |{-1.7}| = 1.7, \qquad |0.9| = 0.9.

Absolute value is distance from 00, and distance does not depend on direction or on whether the number is whole. −75-\tfrac75 is 1251\tfrac25 units left of 00, so ∣−75∣=75\left|-\tfrac75\right| = \tfrac75. The rule "drop the sign" works for fractions and decimals for the same reason it works for integers.

Example
Dropping the Sign
−2−10127/5−7/5
−75-\tfrac75 is 75\tfrac75 units from 00.

∣−75∣=75\left|-\dfrac{7}{5}\right| = \dfrac{7}{5}   because −75-\dfrac{7}{5} is 75\dfrac{7}{5} units from 00. Watch a sign outside the bars: −∣−2.4∣=−(2.4)=−2.4-|{-2.4}| = -(2.4) = -2.4. The bars make the inside positive first; the outside sign then stays.

Side note
Where it goes wrong

Writing ∣−3.2∣=−3.2|{-3.2}| = -3.2 because "the number inside is already negative." The bars report a distance, and distances are never negative: ∣−3.2∣=3.2|{-3.2}| = 3.2.

Tip

Tip: Absolute value works the same for fractions and decimals as for integers: find the distance from 00, which just means make the inside positive.

Try it
Try it: evaluate

Evaluate ∣−94∣−∣12∣\left|-\dfrac{9}{4}\right| - \left|\dfrac{1}{2}\right| and −∣−0.6∣-|{-0.6}|.

Show the answer

∣−94∣=94\left|-\tfrac94\right| = \tfrac94 and ∣12∣=12\left|\tfrac12\right| = \tfrac12, so the difference is 94−24=74\tfrac94 - \tfrac24 = \tfrac74. For the second, the bars give 0.60.6 and the outside sign makes it −0.6-0.6.

Adding and Subtracting Fractions

Concept
Common Denominator First, Then Sign Rules

To add or subtract fractions, first get a common denominator. Then combine the numerators using the integer sign rules:

  • Same signs: add the numerators, keep the sign.
  • Different signs: subtract, keep the sign of the larger.

Remember: subtracting means adding the opposite.

A negative fraction is a negative count of pieces: −26-\tfrac26 is two sixths to the left. Once both fractions count the same-sized pieces, the numerators are ordinary integers, and adding them uses the integer rules: −2+5=3-2 + 5 = 3 sixths. The denominator never takes part in the sign work.

Example
Unlike Denominators with a Negative
−3/6−2/6−1/601/62/63/64/65/61−1/3+5/6
Two sixths left, then five sixths right: 36=12\tfrac36 = \tfrac12.

Compute −13+56-\dfrac{1}{3} + \dfrac{5}{6}. Common denominator is 66:

−13+56=−26+56=−2+56=36=12.-\frac{1}{3} + \frac{5}{6} = -\frac{2}{6} + \frac{5}{6} = \frac{-2 + 5}{6} = \frac{3}{6} = \frac{1}{2}.
Side note
Where it goes wrong

Putting the sign on the denominator and adding it: −13+56=−1+5−3+6=43-\tfrac13 + \tfrac56 = \tfrac{-1 + 5}{-3 + 6} = \tfrac43. Denominators are never added; rewrite over 66 first: −2+56=12\tfrac{-2 + 5}{6} = \tfrac12.

Example
Subtracting a Negative Fraction

Compute 14−(−38)\dfrac{1}{4} - \left(-\dfrac{3}{8}\right). Subtracting a negative adds:

14+38=28+38=58.\frac{1}{4} + \frac{3}{8} = \frac{2}{8} + \frac{3}{8} = \frac{5}{8}.
Tip

Tip: The denominator only sets the “size” of the pieces — all the sign work happens up in the numerators. Combine numerators using the same rules you learned for integers.

Try it
Try it: signed fractions

Compute −34−56-\dfrac{3}{4} - \dfrac{5}{6} and 25−(−710)\dfrac{2}{5} - \left(-\dfrac{7}{10}\right).

Show the answer

Over 1212: −912−1012=−1912-\tfrac{9}{12} - \tfrac{10}{12} = -\tfrac{19}{12}. Subtracting a negative adds: 410+710=1110\tfrac{4}{10} + \tfrac{7}{10} = \tfrac{11}{10}.

Adding and Subtracting Decimals

Concept
Line Up the Points, Then Use Sign Rules

To add or subtract decimals, line up the decimal points and work as usual. The signs follow the exact same integer rules: same signs add, different signs subtract and keep the larger sign.

Signed decimals combine exactly like signed integers, only with smaller steps. −4.5+1.8-4.5 + 1.8 is a move of 4.54.5 to the left followed by 1.81.8 back to the right; the net move is 2.72.7 to the left, because the left move was longer.

Example
Different Signs
−5−4−3−2−101−4.5+1.8
The longer move is to the left, so the result, −2.7-2.7, is negative.

Compute −4.5+1.8-4.5 + 1.8. Different signs, so subtract the absolute values: 4.5−1.8=2.74.5 - 1.8 = 2.7. Since 4.5>1.84.5 > 1.8, keep the negative sign:  −2.7\ -2.7.

Side note
Where it goes wrong

Subtracting and keeping the wrong sign: −4.5+1.8=2.7-4.5 + 1.8 = 2.7. The number with the larger absolute value is −4.5-4.5, so the answer takes its sign: −2.7-2.7.

Example
Subtracting a Negative Decimal

Compute 2.3−(−0.9)2.3 - (-0.9). Subtracting a negative adds:

2.3−(−0.9)=2.3+0.9=3.2.2.3 - (-0.9) = 2.3 + 0.9 = 3.2.
Tip

Tip: Decimals and fractions obey the same sign rules as integers. The only extra care with decimals is keeping the decimal points lined up neatly.

Try it
Try it: signed decimals

Compute −6.25+9.4-6.25 + 9.4, −1.08−3.5-1.08 - 3.5 and 0.7−(−2.35)0.7 - (-2.35).

Show the answer

−6.25+9.4-6.25 + 9.4: different signs, 9.40−6.25=3.159.40 - 6.25 = 3.15, positive: 3.153.15. −1.08−3.5=−1.08+(−3.50)=−4.58-1.08 - 3.5 = -1.08 + (-3.50) = -4.58. 0.7−(−2.35)=0.70+2.35=3.050.7 - (-2.35) = 0.70 + 2.35 = 3.05.

Multiplying and Dividing Rational Numbers

Concept
Sign Rule Is the Same; Method Depends on Form

For both multiplication and division: same signs give a positive, different signs give a negative. Fractions: multiply straight across; to divide, multiply by the reciprocal (“flip the second and multiply”). Decimals: multiply as whole numbers then place the point; to divide, you may shift both points to make the divisor whole.

Every rational number is a signed count, so a product has a sign part and a size part that can be handled separately. The sign follows the integer rule (it came from the distributive property), and the size follows the fraction or decimal method. Deciding the sign first keeps the arithmetic in the second step free of negatives.

Example
Multiplying and Dividing Fractions

(−23)(94)\left(-\dfrac{2}{3}\right)\left(\dfrac{9}{4}\right): different signs →\rightarrow negative. Multiply across and simplify:

−2×93×4=−1812=−32.-\frac{2 \times 9}{3 \times 4} = -\frac{18}{12} = -\frac{3}{2}.

−35÷(−67)-\dfrac{3}{5} \div \left(-\dfrac{6}{7}\right): same signs →\rightarrow positive. Flip and multiply:

35×76=2130=710.\frac{3}{5} \times \frac{7}{6} = \frac{21}{30} = \frac{7}{10}.
Side note
Where it goes wrong

Flipping the sign when taking a reciprocal: the reciprocal of −67-\tfrac67 written as 76\tfrac76. A reciprocal flips the fraction, not the sign: it is −76-\tfrac76, since −67×−76=1-\tfrac67 \times -\tfrac76 = 1.

Example
Multiplying Decimals

(−1.2)(0.4)(-1.2)(0.4): different signs →\rightarrow negative. Multiply 12×4=4812 \times 4 = 48, then place two decimal places: 0.480.48. Answer:  −0.48\ -0.48.

Tip

Tip: One motto covers integers, fractions, and decimals: same signs →\rightarrow positive, different signs →\rightarrow negative. Decide the sign first, then do the arithmetic.

Try it
Try it: sign first

Compute (−49)÷23\left(-\dfrac{4}{9}\right) \div \dfrac{2}{3} and (−0.25)(−1.6)(−2)(-0.25)(-1.6)(-2).

Show the answer

Different signs, so negative: 49×32=1218=23\tfrac49 \times \tfrac32 = \tfrac{12}{18} = \tfrac23, giving −23-\tfrac23. Three negative factors: negative. 0.25×1.6=0.40.25 \times 1.6 = 0.4 and 0.4×2=0.80.4 \times 2 = 0.8, giving −0.8-0.8.

Terminating vs. Repeating Decimals

Concept
Every Fraction Becomes One or the Other

Convert a fraction to a decimal by dividing the numerator by the denominator. The result is always either:

  • Terminating — the division ends, like 38=0.375\dfrac{3}{8} = 0.375; or
  • Repeating — a block of digits repeats forever, like 23=0.6‾\dfrac{2}{3} = 0.\overline{6}.

We write the repeating block with a bar over it: 0.16‾0.1\overline{6} means 0.1666…0.1666\ldots

Long division by 66 can only leave the remainders 0,1,2,3,4,50, 1, 2, 3, 4, 5. If a remainder of 00 appears, the division stops (terminates). If not, a remainder must eventually come back, and from that moment the same digits are produced again: the decimal repeats. So every fraction has to do one or the other; there is no third possibility.

Example
A Terminating Decimal

Convert 720\dfrac{7}{20}. Divide 7÷20=0.357 \div 20 = 0.35. It terminates. Shortcut: if the denominator's only prime factors are 22 and 55, the decimal terminates. Here 20=2×2×520 = 2 \times 2 \times 5.

Side note
Where it goes wrong

Placing the bar over every digit: 56=0.83‾\tfrac56 = \overline{0.83}. Only the part that repeats gets the bar: 0.8333…=0.83‾0.8333\ldots = 0.8\overline{3}.

Example
A Repeating Decimal

Convert 56\dfrac{5}{6}. Divide 5÷6=0.8333…=0.83‾5 \div 6 = 0.8333\ldots = 0.8\overline{3}. The 33 repeats forever because 6=2×36 = 2 \times 3 contains a prime other than 22 or 55.

Tip

Tip: Reduce the fraction first. If the reduced denominator has only 22s and 55s as factors, the decimal terminates; otherwise it repeats.

Try it
Try it: predict, then divide

Predict whether 1140\dfrac{11}{40} and 29\dfrac{2}{9} terminate, then write their decimals.

Show the answer

40=23×540 = 2^3 \times 5 has only the primes 22 and 55, so 1140\tfrac{11}{40} terminates: 0.2750.275. 9=329 = 3^2 contains 33, so 29\tfrac29 repeats: 0.222…=0.2‾0.222\ldots = 0.\overline{2}.

Order of Operations with Rational Numbers

Concept
PEMDAS, Now with Fractions and Signs

Use the same order every time: Parentheses, Exponents, Multiply/Divide (left to right), Add/Subtract (left to right). The new challenge is tracking signs and fractions at each step.

Nothing about the order changes with fractions and decimals; the order of operations is about operations, not about what kind of numbers they act on. What changes is bookkeeping, so write each step on its own line and resolve each sign as it is created.

Example
Mixing Fractions and Signs

Evaluate −12+34×(−2)-\dfrac{1}{2} + \dfrac{3}{4} \times (-2).

−12+34×(−2)=−12+(−64)multiply first=−12−32simplify 64=32=−42=−2then add\begin{aligned} -\frac{1}{2} + \frac{3}{4} \times (-2) &= -\frac{1}{2} + \left(-\frac{6}{4}\right) &&\text{multiply first}\\ &= -\frac{1}{2} - \frac{3}{2} &&\text{simplify } \tfrac{6}{4}=\tfrac{3}{2}\\ &= -\frac{4}{2} = -2 &&\text{then add} \end{aligned}
Side note
Where it goes wrong

Adding before multiplying: −12+34×(−2)-\tfrac12 + \tfrac34 \times (-2) done as 14×(−2)=−12\tfrac14 \times (-2) = -\tfrac12. The multiplication comes first: 34×(−2)=−32\tfrac34 \times (-2) = -\tfrac32, then −12−32=−2-\tfrac12 - \tfrac32 = -2.

Example
Decimals and Parentheses

Evaluate 2.5−0.5(3−7)2.5 - 0.5(3 - 7).

2.5−0.5(3−7)=2.5−0.5(−4)parentheses first=2.5+2multiply, add the opposite=4.5\begin{aligned} 2.5 - 0.5(3 - 7) &= 2.5 - 0.5(-4) &&\text{parentheses first}\\ &= 2.5 + 2 &&\text{multiply, add the opposite}\\ &= 4.5 \end{aligned}
Tip

Tip: Do one step at a time and write each line. Decide each sign before you compute, and keep fractions in a common denominator when you add or subtract.

Try it
Try it: evaluate

Evaluate (12−56)2÷13\left(\dfrac{1}{2} - \dfrac{5}{6}\right)^2 \div \dfrac{1}{3}.

Show the answer

Parentheses: 36−56=−26=−13\tfrac36 - \tfrac56 = -\tfrac26 = -\tfrac13. Exponent: (−13)2=19\left(-\tfrac13\right)^2 = \tfrac19. Divide: 19÷13=19×3=13\tfrac19 \div \tfrac13 = \tfrac19 \times 3 = \tfrac13.

Word Problems with Rational Numbers

Concept
Translate, Then Compute

Real situations mix signs and fractions: temperatures, money owed, elevation, and recipes. Gaining, rising, and depositing are positive; losing, falling, and spending are negative. Decide the sign of each amount, then add, subtract, multiply, or divide.

Signed rational numbers let one calculation cover every direction of change: a rise and a fall are both "add the change," with the fall written as a negative. That is why a single expression like −2.5+6.8-2.5 + 6.8 can describe the whole morning's temperature, and why "how much did it change?" is always new−old\text{new} - \text{old}.

Example
Temperature with Decimals
−3−2−1012345rose 6.8°6 a.m.noon
From −2.5∘-2.5^\circ up 6.8∘6.8^\circ to 4.3∘4.3^\circ.

At 66 a.m. it was −2.5∘-2.5^\circC. By noon it rose 6.86.8 degrees. Rising means adding:

−2.5+6.8=4.3.-2.5 + 6.8 = 4.3.

The noon temperature was 4.3∘4.3^\circC.

Side note
Where it goes wrong

Giving the change from −2.5∘-2.5^\circ to 4.3∘4.3^\circ as 4.3−2.5=1.84.3 - 2.5 = 1.8. Keep the sign of the starting value: 4.3−(−2.5)=6.84.3 - (-2.5) = 6.8 degrees.

Example
Sharing with Fractions

A recipe needs 34\dfrac{3}{4} cup of sugar, but you want to make only half a batch. Multiply:

12×34=38.\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}.

You need 38\dfrac{3}{8} cup of sugar.

Tip

Tip: Read carefully to spot the sign of each quantity. “Below,” “owes,” “loses,” and “drops” all signal negative numbers.

Try it
Try it: a bank balance

An account holds −$42.50-\$42.50 (overdrawn). A deposit of $120\$120 is made, then 13\tfrac13 of the new balance is spent. What is the final balance?

Show the answer

After the deposit: −42.50+120=77.50-42.50 + 120 = 77.50. One third of that is 77.50÷3≈25.8377.50 \div 3 \approx 25.83. Final balance: 77.50−25.83=$51.6777.50 - 25.83 = \$51.67.

Going Deeper: Advanced Rational-Number Ideas

Concept
The Rationals Are Dense

Between any two different rational numbers there is always another rational number — in fact, infinitely many. This property is called density. The integers are not dense (there is no integer strictly between 33 and 44), but the rationals are. The quickest way to find one is the average (midpoint): the number halfway between aa and bb is a+b2\dfrac{a+b}{2}, and it is always rational when aa and bb are.

Example
Squeezing a Rational Between Two Others
2/59/201/2average
The average of 25\tfrac25 and 12\tfrac12 sits exactly halfway: 920\tfrac{9}{20}.

Find a rational number between 25\dfrac{2}{5} and 12\dfrac{1}{2}. Take the average:

12(25+12)=12(410+510)=12⋅910=920.\frac{1}{2}\left(\frac{2}{5} + \frac{1}{2}\right) = \frac{1}{2}\left(\frac{4}{10} + \frac{5}{10}\right) = \frac{1}{2}\cdot\frac{9}{10} = \frac{9}{20}.

Check: 25=820\dfrac{2}{5} = \dfrac{8}{20} and 12=1020\dfrac{1}{2} = \dfrac{10}{20}, so indeed 820<920<1020\dfrac{8}{20} < \dfrac{9}{20} < \dfrac{10}{20}. We could repeat this forever, so there is no “next” rational number.

Concept
The Mediant: Another In-Between Number

There is a second trick for landing between two positive fractions ab\dfrac{a}{b} and cd\dfrac{c}{d}: add the tops and add the bottoms to form the mediant a+cb+d\dfrac{a+c}{b+d}. Whenever ab<cd\dfrac{a}{b} < \dfrac{c}{d}, the mediant sits strictly between them:

ab<a+cb+d<cd.\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}.

Warning: this is not how you add fractions! It is only a way to build an in-between value.

Example
Using the Mediant

Between 13\dfrac{1}{3} and 12\dfrac{1}{2}, the mediant is 1+13+2=25\dfrac{1+1}{3+2} = \dfrac{2}{5}. Check with a common denominator of 3030: 13=1030\dfrac{1}{3} = \dfrac{10}{30},  25=1230\ \dfrac{2}{5} = \dfrac{12}{30},  12=1530\ \dfrac{1}{2} = \dfrac{15}{30}. So 1030<1230<1530\dfrac{10}{30} < \dfrac{12}{30} < \dfrac{15}{30}, exactly as promised.

Concept
Not Every Number Is Rational: 2\sqrt{2}
012√2√211
Swing the diagonal of a unit square down: it lands at 2\sqrt2, a point no fraction names.

Some numbers can never be written as a fraction of integers; we call them irrational. The famous first example is 2\sqrt{2}. Its decimal 1.41421356…1.41421356\ldots never terminates and never repeats. Here is a gentle proof that no fraction equals it.

Example
Why 2\sqrt{2} Cannot Be a Fraction

Suppose, for contradiction, that 2=ab\sqrt{2} = \dfrac{a}{b} where ab\dfrac{a}{b} is fully reduced (no common factors). Squaring both sides:

2=a2b2⟹a2=2b2.2 = \frac{a^2}{b^2} \quad\Longrightarrow\quad a^2 = 2b^2.

Then a2a^2 is even, which forces aa itself to be even, so write a=2ka = 2k. Substituting:

(2k)2=2b2⟹4k2=2b2⟹b2=2k2.(2k)^2 = 2b^2 \quad\Longrightarrow\quad 4k^2 = 2b^2 \quad\Longrightarrow\quad b^2 = 2k^2.

Now b2b^2 is even, so bb is even too. But then aa and bb are both even — they share the factor 22 — contradicting “fully reduced.” No such fraction can exist, so 2\sqrt{2} is irrational.

Concept
Closure: Staying Inside the Family

A set is closed under an operation if combining two members always gives another member. The rationals are closed under addition, subtraction, and multiplication — add, subtract, or multiply any two fractions and you get a fraction. They are closed under division too, except by 00. The integers, by contrast, are not closed under division: 3÷4=343 \div 4 = \dfrac{3}{4} escapes the integers. That gap is exactly why we needed to build the rationals.

Tip

Tip: To see closure at work, note that ab+cd=ad+bcbd\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}. Since integers are closed under ×\times and ++, the top and bottom are integers and bd≠0bd \neq 0, so the sum is again rational.

Concept
Absolute Value as Distance Between Two Numbers

You already know ∣x∣|x| measures distance from 00. More powerfully, ∣a−b∣|a - b| measures the distance between aa and bb on the number line — and the order does not matter, since ∣a−b∣=∣b−a∣|a - b| = |b - a|.

∣a−b∣=∣b−a∣.|a - b| = |b - a|.

This is why distance is always positive: it does not care which point you start from.

Example
Distance Between Two Rationals

How far apart are −34-\dfrac{3}{4} and 12\dfrac{1}{2}?

∣−34−12∣=∣−34−24∣=∣−54∣=54.\left|-\frac{3}{4} - \frac{1}{2}\right| = \left|-\frac{3}{4} - \frac{2}{4}\right| = \left|-\frac{5}{4}\right| = \frac{5}{4}.

Reversing the order gives the same distance: ∣12−(−34)∣=∣24+34∣=54\left|\dfrac{1}{2} - \left(-\dfrac{3}{4}\right)\right| = \left|\dfrac{2}{4} + \dfrac{3}{4}\right| = \dfrac{5}{4}.

Example
A Multi-Step Signed Expression

Evaluate −23−12(34−52)2-\dfrac{2}{3} - \dfrac{1}{2}\left(\dfrac{3}{4} - \dfrac{5}{2}\right)^2.

=−23−12(34−52)2=−23−12(34−104)2inside parentheses first=−23−12(−74)2combine to −74=−23−12⋅4916square: a negative squared is positive=−23−4932multiply=−6496−14796=−21196common denominator 96\begin{aligned} &\phantom{={}} -\frac{2}{3} - \frac{1}{2}\left(\frac{3}{4} - \frac{5}{2}\right)^2 \\ &= -\frac{2}{3} - \frac{1}{2}\left(\frac{3}{4} - \frac{10}{4}\right)^2 &&\text{inside parentheses first}\\ &= -\frac{2}{3} - \frac{1}{2}\left(-\frac{7}{4}\right)^2 &&\text{combine to } -\tfrac{7}{4}\\ &= -\frac{2}{3} - \frac{1}{2}\cdot\frac{49}{16} &&\text{square: a negative squared is positive}\\ &= -\frac{2}{3} - \frac{49}{32} &&\text{multiply}\\ &= -\frac{64}{96} - \frac{147}{96} = -\frac{211}{96} &&\text{common denominator } 96 \end{aligned}

The result −21196-\dfrac{211}{96} is still a rational number — closure at work.

Tip

Tip: With exponents and signs together, resolve the parentheses before squaring, and remember that squaring a negative gives a positive. Track the leading minus sign in front of a whole term all the way to the last line.

Why It Works

A repeating decimal is rational, and here is the trick that proves it: let x=0.27‾x = 0.\overline{27}. Then 100x=27.27‾100x = 27.\overline{27}, and subtracting, 99x=2799x = 27, so x=2799=311x = \tfrac{27}{99} = \tfrac{3}{11}. Multiplying by a power of 1010 that matches the period lines the repeating tails up so they cancel — every repeating decimal yields to the same two lines.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.

Formulas, Proofs & Tips

Tip
Fraction arithmetic
ab±cd=ad±bcbd,ab⋅cd=acbd,ab÷cd=ab⋅dc\frac{a}{b}\pm\frac{c}{d}=\frac{ad\pm bc}{bd},\qquad \frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd},\qquad \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}

What it means. Add with a common denominator, multiply straight across, divide by flipping the second fraction.

Example. 23+16=46+16=56\tfrac23+\tfrac16=\tfrac46+\tfrac16=\tfrac56.

Why it works. Rewriting over the common denominator bdbd makes the pieces the same size so they can be counted together. Division asks "how many cd\tfrac{c}{d} fit?", and multiplying by the reciprocal answers it because cd⋅dc=1\tfrac{c}{d}\cdot\tfrac{d}{c}=1.

Tip. Simplify before multiplying — cancelling early keeps the numbers small.