Day 2 — Fractions & rationals

Day 2 — Fractions & rationals

Stage I · concept day
Goal: Define \(\mathbb{Q}\) as a field of fractions; prove equality via \(ad=bc\); reduce by gcd; master mixed numbers, complex fractions, rates/proportions; sketch density of \(\mathbb{Q}\); see one Egyptian fraction decomposition.

Why this matters

Rationals are the language of slopes, probabilities (ratios of counts), normalized quantities, unit conversions, and fixed-point thinking. If fraction arithmetic is shaky, every later manipulation of \(\frac{p(x)}{q(x)}\) will feel harder than it is. Division algorithms, sampling rates, and “\(k\) of \(n\) succeed” all live in \(\mathbb{Q}\).

Theory

Rational numbers as a field

A rational number is a quotient of integers with nonzero denominator:

\[\mathbb{Q} = \Bigl\{\frac{p}{q} : p\in\mathbb{Z},\; q\in\mathbb{Z}\setminus\{0\}\Bigr\}.\]

The same rational has many names: \(\frac{1}{2}=\frac{2}{4}=\frac{-3}{-6}\).

\(\mathbb{Q}\) with \(+\) and \(\times\) is a field: closed under the four arithmetic operations (except divide-by-zero), associative and commutative as appropriate, distributive, with identities \(0,1\) and additive inverses, and every nonzero element has a multiplicative inverse.

Multiplicative inverse: if \(\frac{a}{b}\neq 0\) (so \(a\neq 0\)), then \(\bigl(\frac{a}{b}\bigr)^{-1}=\frac{b}{a}\).

Equality (cross-multiplication test)

For \(b,d\neq 0\),

\[\frac{a}{b}=\frac{c}{d}\quad\Longleftrightarrow\quad ad=bc.\]

Proof. \[ \frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}. \] This difference is \(0\) if and only if the numerator is \(0\) (since \(bd\neq 0\)). So \(\frac{a}{b}=\frac{c}{d}\) iff \(ad-bc=0\) iff \(ad=bc\). \(\square\)

Corollary. \(\frac{a}{b}=\frac{ka}{kb}\) for any integer \(k\neq 0\) (cancel common factors / scale).

Lowest terms and gcd

A fraction \(\frac{p}{q}\) with \(q>0\) is in lowest terms (canonical form) when \(\gcd(|p|,q)=1\).
Any nonzero rational has a unique such representation with positive denominator (unique \(p\in\mathbb{Z}\), \(q\in\mathbb{Z}_{>0}\)).

To reduce: compute \(d=\gcd(|p|,|q|)\), divide numerator and denominator by \(d\), then fix the sign so the denominator is positive.

Example of uniqueness idea: if \(\frac{p}{q}=\frac{r}{s}\) with \(\gcd(|p|,q)=\gcd(|r|,s)=1\) and \(q,s>0\), then \(ps=qr\). Since \(p\) and \(q\) share no common prime factors, \(q\) divides \(s\), and similarly \(s\) divides \(q\), so \(q=s\) and \(p=r\).

Field operations (formulas)

For \(b,d\neq 0\) (and \(c\neq 0\) when dividing by \(\frac{c}{d}\)):

\[ \begin{align*} \frac{a}{b}+\frac{c}{d}&=\frac{ad+bc}{bd},\\ \frac{a}{b}-\frac{c}{d}&=\frac{ad-bc}{bd},\\ \frac{a}{b}\cdot\frac{c}{d}&=\frac{ac}{bd},\\ \frac{a}{b}\div\frac{c}{d}&=\frac{a}{b}\cdot\frac{d}{c}=\frac{ad}{bc}. \end{align*} \]

Strategy: reduce intermediate fractions when possible; for addition, use least common multiple of denominators for smaller intermediates: \[\frac{a}{b}+\frac{c}{d}=\frac{a(d/\gcd)+c(b/\gcd)}{\operatorname{lcm}(b,d)}\] when \(b,d>0\) (using \(\operatorname{lcm}(b,d)=bd/\gcd(b,d)\) — Day 4).

Mixed numbers

A mixed number \(n\frac{a}{b}\) (with \(n\in\mathbb{Z}_{\ge 0}\), \(0\le a<b\)) means \(n+\frac{a}{b}=\frac{nb+a}{b}\).

Convert mixed \(\to\) improper: \(3\frac{2}{5}=\frac{17}{5}\).
Convert improper \(\to\) mixed: \(\frac{17}{5}=3+\frac{2}{5}=3\frac{2}{5}\).

In algebra and CS, prefer improper fractions for calculation; mixed form is for human display.

Complex fractions

A complex fraction has a fraction in numerator, denominator, or both:

\[\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\div\frac{c}{d}=\frac{ad}{bc},\qquad \frac{a+\frac{b}{c}}{d}=\frac{\frac{ac+b}{c}}{d}=\frac{ac+b}{cd}.\]

Clear by multiplying top and bottom by a common denominator of all “inner” denominators.

Rates and proportions

A ratio \(a:b\) is the rational \(\frac{a}{b}\) (when \(b\neq 0\)).
A proportion is an equality of ratios: \(\frac{a}{b}=\frac{c}{d}\).

Cross-multiply to solve: if \(\frac{x}{12}=\frac{5}{8}\), then \(8x=60\), \(x=\frac{15}{2}\).

Rate: quantity per unit (speed \(= \frac{\text{distance}}{\text{time}}\)). Unit analysis is fraction cancellation of units.

Density of \(\mathbb{Q}\) (sketch)

Theorem (density of rationals in reals — sketch). Between any two reals \(x<y\) there exists a rational \(q\) with \(x<q<y\).

Idea. Choose integer \(n\) large enough that \(\frac{1}{n}<y-x\). Among the multiples of \(\frac{1}{n}\), some \(\frac{m}{n}\) lands in \((x,y)\). (Full real-analysis construction uses Archimedean property.)

Consequence for CS intuition: finite decimals and floats are a discrete grid; true \(\mathbb{Q}\) is dense — there is always a rational between two values, so “next rational” is meaningless without a fixed denominator.

Egyptian fractions (lite)

An Egyptian fraction representation writes a positive rational as a sum of distinct unit fractions \(\frac{1}{n_i}\).

Example. \(\frac{3}{4}=\frac{1}{2}+\frac{1}{4}\).
Greedy idea (Fibonacci–Sylvester): for \(\frac{a}{b}\) in \((0,1)\), take \(\frac{1}{\lceil b/a\rceil}\), subtract, repeat.

One step: \(\frac{3}{7}\): \(\lceil 7/3\rceil=3\), so \(\frac{1}{3}\); remainder \(\frac{3}{7}-\frac{1}{3}=\frac{2}{21}\); then \(\lceil 21/2\rceil=11\), etc. You need only one full worked example today, not a complete algorithm proof.

Ordering rationals

\(\frac{a}{b}<\frac{c}{d}\) (with \(b,d>0\)) iff \(ad<bc\). Signs and negative denominators require care: always reduce to positive denominators first.

Worked examples

Example 1 — Equality test

Is \(\frac{14}{21}=\frac{10}{15}\)? Cross: \(14\cdot 15=210\), \(21\cdot 10=210\). Yes. Both reduce to \(\frac{2}{3}\).

Example 2 — Reduce

Reduce \(\frac{84}{126}\). \(\gcd(84,126)=42\), so \(\frac{84/42}{126/42}=\frac{2}{3}\).

Example 3 — Add with LCD

\(\frac{2}{15}+\frac{4}{9}\). \(\operatorname{lcm}(15,9)=45\).
\(\frac{2\cdot 3}{45}+\frac{4\cdot 5}{45}=\frac{6+20}{45}=\frac{26}{45}\).

Example 4 — Subtract negatives

\(\frac{-3}{4}-\frac{5}{6}=\frac{-9-10}{12}=\frac{-19}{12}\).

Example 5 — Multiply and divide

\(\frac{6}{35}\cdot\frac{14}{9}=\frac{6\cdot 14}{35\cdot 9}\). Cancel \(7\): \(\frac{6\cdot 2}{5\cdot 9}=\frac{12}{45}=\frac{4}{15}\).
\(\frac{6}{35}\div\frac{9}{14}=\frac{6}{35}\cdot\frac{14}{9}=\frac{6\cdot 14}{35\cdot 9}=\frac{4}{15}\) after canceling.

Example 6 — Mixed numbers

\(2\frac{1}{3}+1\frac{3}{4}=\frac{7}{3}+\frac{7}{4}=\frac{28+21}{12}=\frac{49}{12}=4\frac{1}{12}\).

Example 7 — Complex fraction

Simplify \(\dfrac{1+\frac{1}{2}}{1-\frac{1}{3}}\).
Numerator \(\frac{3}{2}\), denominator \(\frac{2}{3}\), quotient \(\frac{3}{2}\cdot\frac{3}{2}=\frac{9}{4}\).
Alternatively multiply top and bottom by \(6\): \(\frac{6+3}{6-2}=\frac{9}{4}\).

Example 8 — Nested complex

\(\dfrac{\frac{2}{3}-\frac{1}{4}}{\frac{5}{6}}=\frac{\frac{8-3}{12}}{\frac{5}{6}}=\frac{\frac{5}{12}}{\frac{5}{6}}=\frac{5}{12}\cdot\frac{6}{5}=\frac{1}{2}\).

Example 9 — Proportion

If \(3\) of \(8\) packets fail and the failure rate stays proportional, about how many of \(120\) fail? \(\frac{x}{120}=\frac{3}{8}\Rightarrow x=45\).

Example 10 — Rate

A link transfers \(150\) MB in \(40\) seconds. Rate \(= \frac{150}{40}=\frac{15}{4}\) MB/s \(=3.75\) MB/s. Time for \(90\) MB: \(90\div\frac{15}{4}=90\cdot\frac{4}{15}=24\) s.

Example 11 — Density sketch

Between \(0.1\) and \(0.11\), the rational \(\frac{1}{10}+\frac{1}{200}=\frac{21}{200}=0.105\) works. Or \(\frac{105}{1000}=\frac{21}{200}\).

Example 12 — Egyptian fraction

\(\frac{3}{4}=\frac{1}{2}+\frac{1}{4}\). Check: \(\frac{2+1}{4}=\frac{3}{4}\).
Greedy for \(\frac{4}{13}\): \(\lceil 13/4\rceil=4\), take \(\frac{1}{4}\); remainder \(\frac{4}{13}-\frac{1}{4}=\frac{3}{52}\); continue if desired.

Example 13 — Order

Compare \(\frac{5}{7}\) and \(\frac{7}{10}\): \(5\cdot 10=50\), \(7\cdot 7=49\), so \(\frac{5}{7}>\frac{7}{10}\).

Example 14 — Inverse and field

Multiplicative inverse of \(-\frac{3}{8}\) is \(-\frac{8}{3}\). Product: \(\bigl(-\frac{3}{8}\bigr)\bigl(-\frac{8}{3}\bigr)=1\).

Exercises

Easy

1. Test equality: \(\frac{15}{25}\) vs \(\frac{9}{15}\); reduce both.
2. Compute \(\frac{2}{3}+\frac{5}{6}\) and reduce.
3. Compute \(\frac{7}{8}\cdot\frac{4}{21}\).
4. Convert \(5\frac{2}{7}\) to improper; convert \(\frac{47}{6}\) to mixed.
5. Compute \(\frac{5}{9}\div\frac{10}{3}\).

Medium

6. Prove carefully that \(\frac{a}{b}=\frac{c}{d}\) iff \(ad=bc\) for \(b,d\neq 0\).
7. Simplify \(\dfrac{\frac{3}{4}+\frac{1}{6}}{\frac{5}{12}-\frac{1}{8}}\).
8. Evaluate \(\frac{-2}{15}-\frac{4}{9}+\frac{1}{5}\); reduce.
9. Solve the proportion \(\frac{x-1}{6}=\frac{2x+1}{15}\).
10. A job takes \(12\) person-hours. What fraction of the job do \(5\) people do in \(2\) hours (assuming equal constant rate)?
11. Show \(\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\) by using a common denominator.
12. Reduce \(\frac{360}{504}\) by prime factors or Euclidean gcd steps.
13. Write \(\frac{5}{6}\) as an Egyptian sum of distinct unit fractions (not necessarily greedy).

Hard

14. Prove: if \(\frac{p}{q}\) is in lowest terms and \(q>0\), and \(\frac{p}{q}=\frac{r}{s}\) with \(s>0\) and \(\gcd(|r|,s)=1\), then \(p=r\) and \(q=s\).
15. Between \(\frac{1}{3}\) and \(\frac{1}{2}\), find three distinct rationals. Explain a general method using midpoints repeatedly.
16. Prove that if \(b,d>0\), then \(\frac{a}{b}<\frac{c}{d}\) iff \(ad<bc\).
17. Simplify \(\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{2}}}\) to a single fraction.
18. Rate: \(A\) finishes a task in \(6\) hours, \(B\) in \(4\) hours. Working together at constant rates, what fraction per hour, and how long for one task?
19. Show that the sum of two rationals is rational; the product of two rationals is rational (closure proofs with formulas).
20. Disprove: “between two rationals there are only finitely many rationals.”

Challenge / CS-flavored

21. A progress bar shows \(\frac{7}{32}\) complete. What percent (exact fraction then decimal)? How many of \(256\) equal parts?
22. Fixed-point: values stored as integer thousandths. What rational is the integer \(137\) representing? Add “\(0.25\)” in this system.
23. Why is \(\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1\) useful for splitting a whole into unit fractions? Connect to Egyptian style.
24. Prove \(\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}\).
25. If a hash table load factor is \(\alpha=\frac{n}{m}=\frac{5}{8}\), and \(m=1024\), find \(n\). If \(n\) grows by \(10\%\), what is new \(\alpha\) as a fraction in lowest terms?

Additional identities and techniques

Negatives. \(\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\). Prefer a single minus in the numerator with \(b>0\).

Multiplicative cancellation. If \(k\neq 0\), then \(\frac{ka}{kb}=\frac{a}{b}\). Never cancel addends: \(\frac{a+c}{b+c}\neq\frac{a}{b}\) in general.

Mediant (optional curiosity). The mediant of \(\frac{a}{b}\) and \(\frac{c}{d}\) (positive denominators) is \(\frac{a+c}{b+d}\), which lies between them. Used in Farey sequences and continued-fraction intuition—not a field operation.

Continued fraction lite. \(1+\frac{1}{2+\frac{1}{2}}=\frac{7}{5}\). Nested unit structure connects to Egyptian/greedy ideas and Diophantine approximation (advanced).

More worked reasoning (patterns)

Pattern A — clear complex fraction by LCD.
\(\dfrac{\frac{a}{b}+\frac{c}{d}}{\frac{e}{f}}\): multiply top and bottom by \(bdf\) in one stroke when all letters nonzero.

Pattern B — proportion cross.
\(\frac{a}{b}=\frac{c}{d}\Rightarrow ad=bc\) even when solving for a letter inside \(a\) or \(c\).

Pattern C — rate composition.
If \(A\) does a job in \(p\) hours and \(B\) in \(q\) hours, together rate \(\frac{1}{p}+\frac{1}{q}=\frac{p+q}{pq}\) jobs per hour; time \(\frac{pq}{p+q}\).

Pattern D — density construction.
Given \(x<y\) rationals, midpoint \(\frac{x+y}{2}\) is rational and strictly between them; iterate for infinitely many.

Pattern E — Egyptian non-uniqueness.
\(\frac{3}{4}=\frac{1}{2}+\frac{1}{4}=\frac{1}{3}+\frac{1}{4}+\frac{1}{6}\) (check). Representations need not be unique.

Mini-proofs to internalize

1. Closure under \(+\). \(\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\) with \(bd\neq 0\) is a ratio of integers.
   
2. Closure under \(\times\). \(\frac{ac}{bd}\) likewise.
   
3. Inverse. Nonzero \(\frac{a}{b}\) has inverse \(\frac{b}{a}\) because product is \(1\).
   
4. Cross-equality. Already proved via difference formula.

Extra exercises (for volume practice)

26. Prove \(\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}\) from the addition formula.
    
27. Simplify \(\dfrac{1-\frac{1}{n}}{1+\frac{1}{n}}\) for integer \(n>1\).
    
28. A tank fills at \(\frac{1}{6}\) per hour and leaks at \(\frac{1}{8}\) per hour; net rate and hours to fill from empty.
    
29. Show that \(\frac{p}{q}\) in lowest terms is an integer iff \(q=1\).
    
30. Between \(\frac{22}{7}\) and \(\frac{333}{106}\), find one rational (midpoint OK).

CS connection

Fractions appear as load factors, sampling rates, aspect ratios, time quanta, and rational slopes in graphics. Exact rational arithmetic (big integers for numerator/denominator) avoids floating-point drift when it matters (geometry predicates, some financial ledgers). Proportions solve “scale this recipe” problems in resource allocation. Density of \(\mathbb{Q}\) reminds you that “the next number after \(x\)” only makes sense on a discrete type (integers, fixed-precision floats), not on true rationals.

Common pitfalls

Pitfall	What to do instead
Adding denominators: \(\frac{1}{2}+\frac{1}{3}=\frac{2}{5}\)	Common denominator: \(\frac{3+2}{6}=\frac{5}{6}\)
Canceling terms across \(+\)	Only cancel factors of whole numerator/denominator
Forgetting signs when reducing	Put the minus in the numerator; keep \(q>0\)
Mixed number \(2\frac{1}{3}\) as \(2\cdot\frac{1}{3}\)	It means \(2+\frac{1}{3}\)
Complex fraction paralysis	Multiply top and bottom by LCD of inner denominators
Cross-multiplying inequalities with negatives	Flip inequality if multiplying by negative quantity
Assuming “unique next rational”	Density: infinitely many between any two

End-of-day synthesis problems

S1. Prove \(\frac{a}{b}=\frac{c}{d}\Leftrightarrow ad=bc\) carefully; apply to \(\frac{21}{28}\) vs \(\frac{15}{20}\).

S2. Compute \(\frac{5}{6}-\frac{7}{15}+\frac{1}{10}\) reduced.

S3. Complex fraction \(\dfrac{\frac{3}{4}-\frac{1}{6}}{1+\frac{1}{2}}\) simplified.

S4. Work rates: \(A\) alone \(8\) h, \(B\) alone \(12\) h; hours together.

S5. Midpoint method: three rationals between \(\frac{1}{5}\) and \(\frac{1}{4}\).

S6. Egyptian: write \(\frac{5}{6}\) as sum of distinct unit fractions two different ways if possible.

S7. Invert \(-\frac{14}{9}\); verify product \(1\).

S8. Load factor \(\frac{n}{m}=\frac{3}{5}\) with \(m=512\); find \(n\).

Checkpoint

• Define \(\mathbb{Q}\) and state what “field” means at a practical level
  
• Prove equality \(\Leftrightarrow ad=bc\)
  
• Add, multiply, divide fractions and reduce by gcd
  
• Convert mixed \(\leftrightarrow\) improper
  
• Simplify a complex fraction
  
• Solve a proportion / rate problem
  
• Sketch why rationals are dense
  
• Write one Egyptian decomposition
  
• S1–S3 exam-ready

Write two takeaways in your own words.

Deep dive — more worked reasoning

D1 — Clear a nested complex fraction in one LCD.
Simplify \(\dfrac{\dfrac{1}{2}+\dfrac{1}{3}}{\dfrac{1}{4}-\dfrac{1}{6}}\).
Inner numerator \(\frac{3+2}{6}=\frac{5}{6}\); denominator \(\frac{3-2}{12}=\frac{1}{12}\); quotient \(\frac{5}{6}\cdot 12=\frac{5}{6}\cdot\frac{12}{1}=10\).
One stroke: multiply top and bottom by \(12\): \(\dfrac{6+4}{3-2}=\dfrac{10}{1}=10\).

D2 — Proportion with a linear expression.
\(\frac{2x+1}{5}=\frac{x-3}{2}\). Cross-multiply: \(2(2x+1)=5(x-3)\), \(4x+2=5x-15\), \(17=x\). Check: \(\frac{35}{5}=7\), \(\frac{14}{2}=7\).

D3 — Work rates three agents (optional stretch).
\(A\) alone \(6\) h, \(B\) alone \(8\) h, \(C\) alone \(24\) h. Together rate \(\frac{1}{6}+\frac{1}{8}+\frac{1}{24}=\frac{4+3+1}{24}=\frac{8}{24}=\frac{1}{3}\) job/h → \(3\) hours for one job.

D4 — Ordering with negatives.
Compare \(-\frac{5}{6}\) and \(-\frac{4}{5}\) with positive denominators: \(-\frac{5}{6}<-\frac{4}{5}\) because \(\frac{5}{6}>\frac{4}{5}\) (cross: \(25>24\)). Taking negatives reverses order.

Extra practice set (post-checkpoint)

31. Simplify \(\dfrac{\frac{2}{5}-\frac{1}{10}}{\frac{3}{4}}\) and reduce.
    
32. Solve \(\frac{3x-1}{4}=\frac{x+5}{6}\).
    
33. Prove closure of \(\mathbb{Q}\) under division by a nonzero rational.
    
34. Between \(\frac{2}{5}\) and \(\frac{3}{7}\), produce two distinct rationals (midpoint and mediant both OK).
    
35. A pipeline delivers \(\frac{2}{3}\) of capacity for \(3\) hours and \(\frac{1}{2}\) for \(2\) hours; total “capacity-hours” as a single fraction.

Tomorrow

Day 3 — Decimals, percent, scientific notation. Terminating vs repeating decimals, fraction\(\leftrightarrow\)decimal, percent change, compound percent, scientific notation operations, order of magnitude, significant figures.