Day 3 — Decimals, percent & scientific notation

Day 3 — Decimals, percent & scientific notation

Stage I · concept day
Goal: Convert between fractions and decimals; classify terminating vs repeating; master percent and compound percent; operate in scientific notation; use order of magnitude and significant-figure discipline.

Why this matters

Humans read decimals and percents; machines store binary fractions and scaled exponents. Misreading \(0.1\), “\(20\%\) off then \(20\%\) off,” or \(3.2\times 10^{-3}\) causes off-by-factor bugs in metrics, money display, and logs. This day builds reliable conversion and estimation.

Theory

Place value (base 10)

A finite decimal \(d_k\ldots d_0.d_{-1}d_{-2}\ldots d_{-m}\) means

\[\sum_{i=-m}^{k} d_i\,10^{i},\qquad d_i\in\{0,1,\ldots,9\}.\]

Example: \(32.07=3\cdot 10+2\cdot 1+0\cdot 10^{-1}+7\cdot 10^{-2}\).

Terminating vs repeating decimals

A terminating decimal has finitely many nonzero digits after the point: \(0.25=\frac{1}{4}\).
A repeating (recurring) decimal eventually cycles: \(0.\overline{3}=0.333\ldots=\frac{1}{3}\), \(0.1\overline{6}=0.1666\ldots=\frac{1}{6}\).

Theorem (characterization, sketch). A reduced fraction \(\frac{p}{q}\) with \(q>0\) and \(\gcd(p,q)=1\) has a terminating decimal expansion iff the prime factors of \(q\) lie in \(\{2,5\}\) only (i.e. \(q=2^a 5^b\) for some \(a,b\ge 0\)).

Why. Terminating means \(\frac{p}{q}=\frac{n}{10^m}=\frac{n}{2^m 5^m}\) for some \(n,m\), so \(q\) divides \(2^m 5^m\) after canceling factors with \(p\)—hence only primes \(2\) and \(5\) in \(q\).

Otherwise the expansion is infinite and eventually periodic (long division of \(p\) by \(q\) has finitely many remainders \(0,\ldots,q-1\); once a remainder repeats, the digit cycle repeats). Remainder \(0\) \(\Leftrightarrow\) terminates.

Fraction \(\to\) decimal

Long division: divide \(p\) by \(q\), record remainders.
Examples: \(\frac{3}{8}=0.375\) (terminates). \(\frac{2}{11}=0.\overline{18}\).

Decimal \(\to\) fraction

Terminating: \(0.d_1\ldots d_m=\frac{d_1\ldots d_m}{10^m}\), then reduce.
Pure repeat \(0.\overline{d_1\ldots d_k}=\frac{N}{10^k-1}\) where \(N\) is the \(k\)-digit block as integer.
Example: \(0.\overline{27}=\frac{27}{99}=\frac{3}{11}\).

Mixed repeat \(0.ab\overline{cde}\): let \(x\) be the value; multiply by powers of \(10\) to align periods, subtract (standard algebra).

Percent

Percent means “per hundred”:

\[r\% = \frac{r}{100}.\]

So \(15\%=0.15=\frac{3}{20}\).
“\(p\%\) of \(A\)” means \(\frac{p}{100}\cdot A\).

Percent change

If a quantity goes from \(A\) to \(B\):

\[\text{relative change}=\frac{B-A}{A},\qquad \text{percent change}=100\cdot\frac{B-A}{A}\%.\]

Increase if \(B>A\); decrease if \(B<A\). Base matters: \(100\to 80\) is \(-20\%\); \(80\to 100\) is \(+25\%\).

Compound percent

Applying successive factors multiplies:

\[A\mapsto A\bigl(1+\tfrac{r_1}{100}\bigr)\bigl(1+\tfrac{r_2}{100}\bigr)\cdots.\]

Not additive: \(10\%\) then \(10\%\) is factor \(1.1\times 1.1=1.21\), i.e. \(21\%\) total, not \(20\%\).

Discount then tax order matters in general if rates apply to different bases; track the actual quantity each rate multiplies.

Scientific notation

A positive number in scientific notation is \(m\times 10^{e}\) with \(1\le m<10\) (normalized) and \(e\in\mathbb{Z}\).
(Engineering notation sometimes uses exponents multiple of \(3\).)

Operations:

\[ \begin{align*} (a\times 10^{m})(b\times 10^{n})&=(ab)\times 10^{m+n},\\ \frac{a\times 10^{m}}{b\times 10^{n}}&=\frac{a}{b}\times 10^{m-n},\\ (a\times 10^{m})+(b\times 10^{n})&=\text{align exponents first}. \end{align*} \]

Renormalize so the mantissa is in \([1,10)\).

Order of magnitude

The order of magnitude is roughly the power of ten: \(\lfloor\log_{10} x\rfloor\) for \(x>0\).
\(3\times 10^{4}\) and \(7\times 10^{4}\) are the same order of magnitude; \(3\times 10^{5}\) is one higher. Used for sanity checks (“is this megabytes or gigabytes?”).

Significant figures (discipline)

Significant figures count digits that carry meaning in a measurement or rounded value.

Rules of thumb:

• Nonzero digits are significant.
  
• Zeros between nonzero digits are significant.
  
• Leading zeros after the decimal are not significant (\(0.0032\) has \(2\) sig figs).
  
• Trailing zeros in a decimal are significant (\(1.200\) has \(4\)).
  
• Trailing zeros in integers may be ambiguous (\(1200\)); scientific notation \(1.20\times 10^{3}\) clarifies \(3\) sig figs.

Arithmetic discipline (intro): when multiplying/dividing, result sig figs \(\approx\) min of inputs; when adding, align decimal places (least precise place). Today: awareness and honest reporting, not legalistic lab protocol.

Worked examples

Example 1 — Terminating conversion

\(\frac{7}{20}=0.35\) because \(\frac{7}{20}=\frac{35}{100}\). Denominator \(20=2^2\cdot 5\), only primes \(2,5\).

Example 2 — Repeating conversion

\(\frac{1}{6}=0.1\overline{6}\). Long division: \(1.000\div 6\) → \(0.1666\ldots\).

Example 3 — Decimal to fraction

\(0.125=\frac{125}{1000}=\frac{1}{8}\).
\(0.\overline{1}=\frac{1}{9}\).
\(0.\overline{09}=\frac{9}{99}=\frac{1}{11}\).

Example 4 — Mixed repeat

Let \(x=0.1\overline{6}=0.1666\ldots\). Then \(10x=1.666\ldots\), \(100x=16.666\ldots\), so \(100x-10x=15\Rightarrow 90x=15\Rightarrow x=\frac{15}{90}=\frac{1}{6}\).

Example 5 — Percent of

\(18\%\) of \(250\): \(0.18\times 250=45\).

Example 6 — Percent change

Price \(80\to 100\): \(\frac{20}{80}=0.25=25\%\) increase.
Price \(100\to 80\): \(\frac{-20}{100}=-20\%\) decrease.

Example 7 — Compound percent

Start \(200\). Up \(10\%\), then down \(10\%\): \(200\times 1.1\times 0.9=200\times 0.99=198\neq 200\). Order both ways here gives same product of factors.

Example 8 — Two discounts

\(20\%\) off then extra \(10\%\) off list \(L\): pay \(L\times 0.8\times 0.9=0.72L\), i.e. \(28\%\) total off, not \(30\%\).

Example 9 — Scientific multiply

\((3.0\times 10^{4})(2.5\times 10^{-2})=7.5\times 10^{2}=750\).

Example 10 — Scientific divide and add

\(\frac{6.0\times 10^{-3}}{2.0\times 10^{2}}=3.0\times 10^{-5}\).
\(3.2\times 10^{3}+4.5\times 10^{2}=3.2\times 10^{3}+0.45\times 10^{3}=3.65\times 10^{3}\).

Example 11 — Order of magnitude

\(8.1\times 10^{6}\) bytes \(\approx\) order \(10^{6}\) (megabyte scale if \(10^{6}\) definition). \(2^{20}=1048576\approx 1.05\times 10^{6}\).

Example 12 — Fraction \(\leftrightarrow\) percent

\(\frac{3}{8}=0.375=37.5\%\).
\(12.5\%=\frac{12.5}{100}=\frac{1}{8}\).

Example 13 — Characterization check

\(\frac{1}{7}\) has denominator \(7\notin\{2,5\}\) primes → nonterminating (period \(6\): \(0.\overline{142857}\)).
\(\frac{3}{40}=\frac{3}{2^3\cdot 5}\) terminates: \(0.075\).

Example 14 — Sig figs

\(1.20\times 10^{3}\) has three significant figures. Product \(1.2\times 3.45\) with \(2\) and \(3\) sig figs → report \(\approx 4.1\) (two sig figs) as a rough rule.

Exercises

Easy

1. Write \(0.45\) and \(2.05\) as reduced fractions.
2. Convert \(\frac{5}{16}\) to a decimal.
3. Convert \(36\%\) and \(2.5\%\) to fractions in lowest terms.
4. What is \(15\%\) of \(80\)?
5. Write \(45000\) and \(0.00032\) in normalized scientific notation.

Medium

6. Convert \(0.\overline{36}\) to a reduced fraction.
7. Convert \(0.2\overline{7}\) to a fraction (algebra method).
8. Does \(\frac{9}{24}\) terminate? Reduce first, then apply the \(2\)–\(5\) criterion.
9. A value rises from \(40\) to \(50\), then falls back to \(40\). Percent changes each step? Net percent from original?
10. Apply \(+5\%\) then \(+5\%\) to \(1000\); compare to a single \(+10\%\).
11. Compute \((1.2\times 10^{-3})+(3.4\times 10^{-4})\) in scientific notation.
12. \(\frac{6.0\times 10^{5}}{1.5\times 10^{-2}}\) in scientific notation.
13. Express \(\frac{7}{11}\) as a repeating decimal (long division).

Hard

14. Prove: if \(\gcd(p,q)=1\) and \(q=2^a 5^b\), then \(\frac{p}{q}\) terminates (produce an explicit denominator \(10^m\)).
15. Explain why \(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\) in terminating decimal display can look like \(0.999\ldots\) vs \(1\) — connect to repeating nines.
16. A store marks up cost \(C\) by \(25\%\) to get list price, then discounts list by \(25\%\). Express final price in terms of \(C\).
17. Order of magnitude: estimate \(2^{30}\) using \(2^{10}\approx 10^{3}\).
18. From \(A\) to \(B\) is \(+r\%\). What percent change goes from \(B\) back to \(A\)? (Formula in \(r\).)
19. Classify \(\frac{21}{28}\) after reducing: terminating or not?
20. Significant figures: which of \(0.00450\), \(450\), \(4.50\times 10^{2}\) has unambiguous three sig figs?

Challenge / CS-flavored

21. Why can \(0.1+0.2\) in binary floating point fail to equal \(0.3\) exactly? (Connect to “denominator primes other than \(2\).”)
22. A metric reports \(99.5\%\) success on \(n=200\) trials. How many successes? Can you recover an exact fraction?
23. Latency \(3.2\times 10^{-3}\) s in ms? in μs?
24. Compound: software license grows \(8\%\) per year for \(3\) years on base \(B\). Factor and expanded form \(B(1.08)^3\).
25. Percent points vs percent: explain why “grew from \(10\%\) to \(12\%\)” is \(+2\) percentage points but \(+20\%\) relative change.

More conversion techniques

Long division remainder map. Remainders when dividing by \(q\) live in \(\{0,1,\ldots,q-1\}\). Period length of \(\frac{1}{q}\) (reduced, \(q\) coprime to \(10\)) divides \(\varphi(q)\) (Euler)—optional fact for culture, not required proof.

Scientific comparison. To compare \(a\times 10^{m}\) and \(b\times 10^{n}\) with \(1\le a,b<10\): if \(m>n\) then first is larger (same for normalized positives); if \(m=n\) compare mantissas.

Percent composition identities.

• Successive factors: overall multiplier \(\prod(1+r_i)\).
  
• “\(p\%\) of \(q\%\) of \(A\)” means \(\frac{p}{100}\cdot\frac{q}{100}\cdot A\), not \(\frac{pq}{100}A\) unless one percent is \(1\).
  
• Reverse percent: if after \(+r\%\) you have \(B\), original is \(B/(1+r/100)\) for \(r\neq -100\).

Order-of-magnitude arithmetic. Rough product: add exponents and rough-mantissa multiply. Sanity check algorithms and data sizes (\(2^{10}\approx 10^3\), \(2^{20}\approx 10^6\), \(2^{30}\approx 10^9\)).

Sig-fig worked discipline. \(12.3\times 4.56\) has three and three sig figs → product \(56.088\) reported as \(56.1\) under the min-sig-fig product rule of thumb. Addition: \(12.3+0.045=12.345\) → \(12.3\) if tenths place limits.

Extra numeric drills

Drill 1. Convert \(\frac{5}{6}\), \(\frac{5}{7}\), \(\frac{5}{8}\) to decimals; classify terminating/repeating.
Drill 2. \(0.\overline{142857}=\frac{1}{7}\) check by \(1\div 7\).
Drill 3. Price \(P\), tax \(8\%\), then \(10\%\) tip on post-tax vs pre-tax—write both formulas.
Drill 4. \((6.02\times 10^{23})\times(3.0\times 10^{-2})\) order of magnitude.
Drill 5. Relative error when approximating \(\frac{1}{3}\) by \(0.333\).

Extra exercises

26. Prove terminating criterion for reduced \(\frac{p}{q}\): if only primes \(2,5\) in \(q\), construct \(m\) with \(q\mid 10^m\).
    
27. Convert \(0.1\overline{6}\) two ways (algebra alignment vs known \(\frac{1}{6}\)).
    
28. A metric drops from \(99\%\) to \(97\%\) availability: percentage points vs relative percent change of downtime.
    
29. Expand \((1+r)^2\) and interpret compound two-period growth.
    
30. Write \(2^{-10}\) in scientific notation approximately using \(2^{10}\approx 10^3\).

CS connection

Floating-point numbers are binary scientific notation with fixed mantissa width (Day 9). Decimal \(0.1\) is repeating in binary—hence classic sum errors. Logs, metrics dashboards, and SLOs speak percent; capacity planning uses orders of magnitude. Scientific notation is how you read 1e-6 in code and SI prefixes (milli, micro, mega). Sig-fig discipline is honesty about measurement and rounded intermediates—not the same as exact integer arithmetic.

Common pitfalls

Pitfall	What to do instead
Adding percents of different bases	Convert each step to a multiply factor
\(20\%\) then \(20\%\) = \(40\%\)	\(1.2\times 1.2=1.44\)
Moving decimal wrong in scientific notation	Count places; renormalize mantissa to \([1,10)\)
Assuming all fractions terminate	Only \(2\)–\(5\) denominators (reduced)
Confusing percentage points with relative \(\%\)	State base and formula
Treating \(0.999\ldots\) and \(1\) as “different reals”	They are equal as real numbers
Spurious precision (many decimal places)	Match significant figures to data quality

End-of-day synthesis problems (show full work)

S1. A laptop costs \(\$900\). Markup \(20\%\) to list, then holiday discount \(15\%\) off list. Final price? Overall percent change from cost?

S2. Convert \(0.\overline{27}\) and \(0.2\overline{7}\) to fractions; compare.

S3. Compute \(\dfrac{(4.0\times 10^{-3})(3.0\times 10^{5})}{6.0\times 10^{-2}}\) in scientific notation.

S4. Explain with prime factors why \(\frac{11}{28}\) does not terminate but \(\frac{11}{40}\) does.

S5. Population \(2.5\times 10^{4}\) grows \(4\%\) then \(4\%\) again. Exact factor and new population.

S6. Absolute and relative error when \(2^{10}\) is approximated by \(10^{3}\).

S7. Write a \(3\)-significant-figure scientific notation for \(0.00045678\).

S8. Chain: \(50\) increased by \(10\%\), decreased by \(10\%\), increased by \(10\%\). Final value?

Checkpoint

• Convert fraction \(\leftrightarrow\) terminating/repeating decimal
  
• State the \(2\)–\(5\) criterion for terminating decimals
  
• Compute percent of and percent change
  
• Apply compound percent as product of factors
  
• Multiply/divide/add in scientific notation
  
• Estimate order of magnitude
  
• Report a result with sensible significant figures
  
• Complete synthesis problems S1–S4 without notes

Write two takeaways in your own words.

Deep dive — conversions, compound percent, sci-notation

D1 — Mixed repeating to fraction (full algebra).
Let \(x=0.23\overline{45}=0.23454545\ldots\). Then \(100x=23.454545\ldots\), \(10000x=2345.4545\ldots\).
Subtract: \(10000x-100x=2345.45\ldots-23.45\ldots=2322\), so \(9900x=2322\), \(x=\frac{2322}{9900}=\frac{387}{1650}=\frac{129}{550}\) after dividing by \(6\). (Reduce fully for your write-up.)

D2 — Terminating criterion with reduction first.
\(\frac{14}{35}\) reduces by \(7\) to \(\frac{2}{5}\); denominator \(5=5^1\) → terminates: \(0.4\). Without reducing you might wrongly stare at the \(7\) in \(35\).

D3 — Compound percent chain with three factors.
Start \(500\). Up \(20\%\), down \(10\%\), up \(5\%\):
\(500\times 1.2\times 0.9\times 1.05=500\times 1.134=567\).
Overall factor \(1.134\) means \(+13.4\%\) net, not \(20-10+5=15\%\).

D4 — Scientific addition with different exponents.
\((7.2\times 10^{-3})+(4.5\times 10^{-4})=7.2\times 10^{-3}+0.45\times 10^{-3}=7.65\times 10^{-3}\).
Renormalize only if mantissa leaves \([1,10)\).

D5 — Order of magnitude for \(2^{40}\).
\(2^{10}\approx 10^3\), so \(2^{40}=(2^{10})^4\approx (10^3)^4=10^{12}\). More tightly \(2^{10}=1024\approx 1.024\times 10^3\), so \(2^{40}\approx 1.1\times 10^{12}\).

D6 — Percentage points vs relative change (SLO story).
Availability \(99.0\%\to 99.5\%\) is \(+0.5\) percentage points. Relative change of availability is small; relative change of unavailability \(1.0\%\to 0.5\%\) is a \(50\%\) reduction in downtime fraction—always name the base quantity.

Extra practice set

31. Convert \(0.\overline{142857}\) recognition: show \(1/7\) by long division remainders cycling.
    
32. Markup \(30\%\) then discount \(30\%\) on cost \(C\): final price as fraction of \(C\).
    
33. \((2.5\times 10^{4})\div(5.0\times 10^{-3})\) in scientific notation.
    
34. Does \(\frac{15}{48}\) terminate? Reduce, then apply \(2\)–\(5\) criterion.
    
35. Round \(0.009876\) to \(3\) significant figures in scientific notation.

Tomorrow

Day 4 — Factors, primes & FTA. Primes and composites, sieve idea, fundamental theorem of arithmetic, Euclid’s infinite primes, gcd/lcm identities, coprimality.