Math Survival Kit

ASTR 201 • Tools of the Trade Companion

A quick-reference + practice set for scientific notation, exponent rules, unit conversions, and order-of-magnitude reasoning used throughout ASTR 201.

1 Why this exists

Astronomy is full of numbers that are too big, too small, or too many units to handle by “gut feel” alone. In ASTR 201, you’re expected to be able to:

• read and manipulate scientific notation
• use exponent rules fluently
• do unit conversions using factor-label (units-as-algebra)
• make order-of-magnitude estimates and sanity checks

WarningExpectations (friendly but firm)

You are responsible for these math skills in this course.

That does not mean you already have them. It means you will build them, and you will practice them until they’re reliable.

2 How to use this handout

1. Take the Core diagnostic below (~5 min). If you miss 2+, do the practice set.
2. Try the Extended diagnostic if you want to check roots/scaling fluency.
3. Keep this open while you do homework. It’s a reference, not a punishment.

Section guide:

• Core (used constantly): scientific notation, exponents, PEMDAS, units, OOM
• Extended (used often): roots/fractional exponents, logarithms, proportionality

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3 Core diagnostic (~5 minutes)

Instructions: Work without a calculator if you can. Circle anything you’re unsure about.

Scientific notation & exponents

1. Rewrite in standard decimal form: \(3\times 10^4\) = ____________________
2. Rewrite in scientific notation: \(0.00072\) = ____________________
3. Compare: which is larger? \(4\times 10^6\) or \(9\times 10^5\) Answer: ____________________
4. \(10^3\times 10^5 = 10^{\_\_}\)
5. \(10^{12}/10^7 = 10^{\_\_}\)
6. \((10^4)^2 = 10^{\_\_}\)
7. \(2\times 10^3 \times 5\times 10^{-2} =\) ____________________

Unit conversions

8. Convert: \(3\,\mathrm{km} =\) ____________________ \(\mathrm{cm}\)
9. Convert: \(1\,\mathrm{m}^2 =\) ____________________ \(\mathrm{cm}^2\)

OOM

10. Order of magnitude: \(7\times 10^{10}\) rounds to \(10^{\_\_}\)

TipCore diagnostic answer key

1. 30,000
2. \(7.2\times 10^{-4}\)
3. \(4\times 10^6\) (bigger exponent wins)
4. 8
5. 5
6. 8
7. \((2\cdot 5)\times 10^{3-2} = 10\times 10^1 = 10^2\)
8. \(3\times 10^5\,\mathrm{cm}\)
9. \(10^4\,\mathrm{cm}^2\)
10. 11 (rule of 3)

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4 Extended diagnostic (~5 minutes)

Order of operations & roots

11. Evaluate: \(-2^4 =\) ____________________
12. Evaluate: \((-2)^4 =\) ____________________
13. \(\sqrt{10^{8}} = 10^{?}\) → ____________________
14. \((10^{6})^{1/2} = 10^{?}\) → ____________________

Proportionality

15. If \(A \propto B^2\) and \(B\) triples, by what factor does \(A\) change? ____________________

TipExtended diagnostic answer key

11. \(-16\) (exponent binds first, then negative)
12. \(16\) (parentheses make the base negative)
13. 4 (square root halves the exponent)
14. 3
15. \(3^2 = 9\) (ninefold increase)

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5 Scientific notation (the language of astronomy)

5.1 What \(a\times 10^n\) means

• \(a\) is the coefficient (usually \(1\le a < 10\))
• \(10^n\) tells you how many places the decimal moves
  • \(n>0\): big number (moves right)
  • \(n<0\): small number (moves left)

Example: \(3.2\times 10^{5} = 320{,}000\)

5.2 Normalizing (getting \(1\le a < 10\))

• \(32\times 10^5 = 3.2\times 10^6\)
• \(0.32\times 10^5 = 3.2\times 10^4\)

WarningCommon mistake

Forgetting to change the exponent when you move the decimal.

5.3 Comparing numbers quickly

When comparing \(a\times 10^n\) values: - Exponent first (bigger \(n\) wins) - If exponents match, compare coefficients

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6 Exponent rules you must know

These are not “math trivia.” They’re the rules that make astronomy manageable.

6.1 The Big Three

\[ 10^a\times 10^b = 10^{a+b} \] \[ \frac{10^a}{10^b} = 10^{a-b} \] \[ (10^a)^b = 10^{ab} \]

6.2 Negative exponents

\[ 10^{-3} = \frac{1}{10^3} = 0.001 \]

6.3 Mixing coefficients and powers of ten

Treat it like: (numbers) × (powers of ten).

Example: \[ (2\times 10^3)(5\times 10^{-2}) = (2\cdot 5)\times 10^{3-2} = 10\times 10^1 = 10^2 \]

TipSanity trick

If you get \(a\ge 10\), rewrite (normalize) so the coefficient is between 1 and 10.

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7 Order of operations (PEMDAS)

When you see \(F = GMm/r^2\), how do you parse it?

7.1 The hierarchy

Parentheses → Exponents → Multiplication/Division → Addition/Subtraction

Operations at the same level go left to right.

7.2 The traps that catch students

Trap 1: Negatives and exponents \[ -3^2 = -(3^2) = -9 \qquad \text{but} \qquad (-3)^2 = 9 \] The exponent binds tighter than the negative sign unless you use parentheses.

Trap 2: Division chains

The expression \(a/bc\) is ambiguous. It could mean either: \[ \frac{a}{bc} \qquad \text{or} \qquad \left(\frac{a}{b}\right)c = \frac{ac}{b} \] Never write it that way. Use parentheses or a stacked fraction.

Trap 3: Fractions of fractions \[ \frac{a/b}{c} = \frac{a}{bc} \qquad \text{and} \qquad \frac{a}{b/c} = \frac{ac}{b} \] Dividing by a fraction = multiplying by its reciprocal.

7.3 Astronomy example

Parse \(F = GMm/r^2\):

1. Exponent first: \(r^2\)
2. Then multiplication/division left to right: \(G \cdot M \cdot m / r^2\)
3. Result: \(\displaystyle F = \frac{GMm}{r^2}\)

WarningThe fix

When in doubt, add parentheses. They cost nothing and prevent errors.

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8 Fractional exponents and roots

Roots are just exponents in disguise.

8.1 The connection

\[ a^{1/2} = \sqrt{a} \qquad a^{1/3} = \sqrt[3]{a} \qquad a^{1/n} = \sqrt[n]{a} \]

8.2 The power rule still works

\[ (10^6)^{1/2} = 10^{6 \times 1/2} = 10^3 \]

Translation: Taking a square root halves the exponent.

8.3 Kepler’s Law uses this constantly

\[ P \propto r^{3/2} = r^{1.5} = r \cdot \sqrt{r} \]

If \(r\) doubles: \[ P_{\text{new}} = P_{\text{old}} \times 2^{3/2} = P_{\text{old}} \times 2\sqrt{2} \approx 2.8 \, P_{\text{old}} \]

8.4 Breaking apart roots

\[ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \]

Example: Simplify \(\sqrt{\dfrac{r^3}{GM}}\) \[ \sqrt{\frac{r^3}{GM}} = \frac{\sqrt{r^3}}{\sqrt{GM}} = \frac{r^{3/2}}{\sqrt{G}\sqrt{M}} \]

TipThe pattern

To take a square root of \(10^n\): divide the exponent by 2.

\(\sqrt{10^{6}} = 10^3\), \(\sqrt{10^{33}} = 10^{16.5} \approx 3\times 10^{16}\)

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9 Unit conversions: factor-label method (units as algebra)

9.1 The core idea

A conversion factor is just multiplying by 1.

If \(1\,\mathrm{m}=100\,\mathrm{cm}\), then both are true: \[ \frac{100\,\mathrm{cm}}{1\,\mathrm{m}} = 1 \qquad \text{and} \qquad \frac{1\,\mathrm{m}}{100\,\mathrm{cm}} = 1 \]

9.2 The template

1. Write the quantity with units.
2. Multiply by conversion factors so the unwanted unit cancels.
3. Cancel units explicitly.
4. Combine powers of ten.

Example: \(30\,\mathrm{km/s}\rightarrow \mathrm{cm/s}\) \[ 30\,\frac{\mathrm{km}}{\mathrm{s}} \times \frac{10^3\,\mathrm{m}}{1\,\mathrm{km}} \times \frac{10^2\,\mathrm{cm}}{1\,\mathrm{m}} = 30\times 10^5\,\frac{\mathrm{cm}}{\mathrm{s}} = 3\times 10^6\,\mathrm{cm/s} \]

WarningThe #1 unit mistake

Dropping units mid-calculation.

Rule: If units disappear before the last line, something went wrong.

9.3 Squared and cubed units (the sneaky trap)

If \(1\,\mathrm{m}=10^2\,\mathrm{cm}\), then: \[ 1\,\mathrm{m}^2 = (10^2\,\mathrm{cm})^2 = 10^4\,\mathrm{cm}^2 \] \[ 1\,\mathrm{m}^3 = (10^2\,\mathrm{cm})^3 = 10^6\,\mathrm{cm}^3 \]

Translation: You must raise the conversion factor to the same power.

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10 Order-of-magnitude (OOM) thinking

Astronomy often rewards being roughly right.

10.1 Rule of 3 (fast rounding)

When estimating: - coefficient \(<3\) → round down to 1 - coefficient \(>3\) → round up to 10

Examples: - \(2\times 10^{33} \approx 10^{33}\) - \(7\times 10^{10} \approx 10^{11}\)

10.2 What OOM is (and isn’t)

• OOM means you care about the exponent most.
• Being off by a factor of 2–3 is often fine.
• Being off by a factor of 10^8 means your model or units are broken.

TipThe sanity-check question

“What real-world scale should this live near?”

If your result puts a star at \(10^{11}\,\mathrm{cm}\), that’s Sun-sized, not star-distance.

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11 Logarithms (just enough)

Logarithms extract exponents. That’s the whole idea.

11.1 The definition

\[ \log_{10}(10^n) = n \]

If \(x = 10^n\), then \(\log_{10}(x) = n\).

11.2 Why astronomers care

1. “Order of magnitude” literally means “the log.” When we say two quantities differ by 3 orders of magnitude, we mean \(\log_{10}(A/B) = 3\).

2. Magnitudes use logs. The brightness scale: \[ m_1 - m_2 = -2.5\log_{10}\left(\frac{F_1}{F_2}\right) \]

3. Log scales compress huge ranges. A plot from \(10^{-13}\) to \(10^{28}\) cm is impossible on a linear axis but simple on a log axis.

11.3 The rules you need

Rule	Formula	Example
Log of a product	\(\log(ab) = \log a + \log b\)	\(\log(10^3 \times 10^5) = 3+5 = 8\)
Log of a quotient	\(\log(a/b) = \log a - \log b\)	\(\log(10^{12}/10^7) = 12-7 = 5\)
Log of a power	\(\log(a^n) = n\log a\)	\(\log(10^{3\times 2}) = 6\)

NoteYou won’t need to compute logs by hand

Calculators handle the numbers. You need to understand what logs mean and why they appear in formulas.

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12 Proportionality notation

The symbol \(\propto\) means “is proportional to.” It’s everywhere in physics.

12.1 What it means

\[ A \propto B \quad \Longleftrightarrow \quad A = k \cdot B \text{ for some constant } k \]

“A is proportional to B” means if you double B, A doubles too.

12.2 What it does NOT mean

\(\propto\) does not tell you the value of the constant \(k\). It only tells you the relationship.

12.3 The power of ratios

If \(A \propto B^n\), then for two systems: \[ \frac{A_2}{A_1} = \left(\frac{B_2}{B_1}\right)^n \]

The constant \(k\) cancels. You don’t need to know it.

12.4 Examples from ASTR 201

Relationship	Meaning	If input doubles…
\(P \propto r^{3/2}\)	Period scales with radius	\(P\) increases by \(2^{1.5} \approx 2.8\times\)
\(F \propto 1/r^2\)	Inverse-square law	\(F\) decreases by \(4\times\)
\(L \propto R^2 T^4\)	Stefan-Boltzmann	Double \(T\) → \(L\) increases \(16\times\)
\(R_s \propto M\)	Schwarzschild radius	\(R_s\) doubles

TipThe workflow

1. Write down the proportionality.
2. Form a ratio between two cases.
3. Cancel constants and compute.

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13 Calculator survival (optional, but useful)

You do not need a fancy calculator, but you do need to avoid common traps.

• Use the EE/EXP key for scientific notation input.
  • \(3.8\times 10^{26}\) should be typed like 3.8 EE 26.
• Use parentheses aggressively.
• After a calculation, ask: does the exponent make sense?

WarningCommon calculator failure mode

Typing \(3.8\times 10^{26}\) as \(3.8\times 10\) then \(+26\).

That is not how exponents work. (The calculator will happily let you be wrong.)

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14 Quick reference tables

14.1 SI prefixes (the ones we actually use)

Prefix	Symbol	Power
tera	T	\(10^{12}\)
giga	G	\(10^{9}\)
mega	M	\(10^{6}\)
kilo	k	\(10^{3}\)
centi	c	\(10^{-2}\)
milli	m	\(10^{-3}\)
micro	\(\mu\)	\(10^{-6}\)
nano	n	\(10^{-9}\)

14.2 High-use conversions

Conversion	Value
\(1\,\mathrm{km}\)	\(10^5\,\mathrm{cm}\)
\(1\,\mathrm{m}\)	\(10^2\,\mathrm{cm}\)
\(1\,\mathrm{kg}\)	\(10^3\,\mathrm{g}\)
\(1\,\mathrm{J}\)	\(10^7\,\mathrm{erg}\)
\(1\,\mathrm{W}\)	\(10^7\,\mathrm{erg/s}\)

14.3 Course anchor values (CGS where appropriate)

Quantity	Approx value
Speed of light \(c\)	\(3\times 10^{10}\,\mathrm{cm/s}\)
Gravitational constant \(G\)	\(6.67\times 10^{-8}\,\mathrm{cm^3\,g^{-1}\,s^{-2}}\)
1 AU	\(1.50\times 10^{13}\,\mathrm{cm}\)
1 pc	\(3.09\times 10^{18}\,\mathrm{cm}\)
Solar mass \(M_\odot\)	\(2.0\times 10^{33}\,\mathrm{g}\)
Solar radius \(R_\odot\)	\(7.0\times 10^{10}\,\mathrm{cm}\)
Solar luminosity \(L_\odot\)	\(3.8\times 10^{33}\,\mathrm{erg/s}\)

NoteReality check

These values are the “mental yardsticks” you’ll use all semester. You do not need to memorize everything at once.

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15 Practice set

15.1 Level 0 — warm-up (scientific notation + exponents)

1. Normalize: \(45\times 10^6\) → ____________________
2. Normalize: \(0.008\times 10^5\) → ____________________
3. Compute: \(10^{-3}\times 10^{8} = 10^{\;\_\_}\)
4. Compute: \(10^{12}/10^{-4} = 10^{\;\_\_}\)
5. Compute: \((10^2)^5 = 10^{\;\_\_}\)

15.2 Level 1 — order of operations and roots

6. Evaluate: \(-5^2\) = ____________________
7. Evaluate: \((-5)^2\) = ____________________
8. Simplify: \(\dfrac{12/4}{3}\) = ____________________
9. Simplify: \(\sqrt{10^{12}} = 10^{?}\) → ____________________
10. Compute: \((10^9)^{1/3} = 10^{?}\) → ____________________
11. If \(P \propto r^{3/2}\) and \(r = 4\), what is \(r^{3/2}\)? ____________________

15.3 Level 2 — unit conversions (factor-label)

12. Convert \(12\,\mathrm{km}\) to cm.
13. Convert \(0.35\,\mathrm{m}\) to cm.
14. Convert \(250\,\mathrm{m/s}\) to cm/s.
15. Convert \(2.0\,\mathrm{m^2}\) to \(\mathrm{cm^2}\).
16. Convert \(5\,\mathrm{J}\) to erg.

15.4 Level 3 — astronomy-flavored problems

17. The Sun’s luminosity is \(3.8\times 10^{26}\,\mathrm{W}\). Convert to \(\mathrm{erg/s}\).
18. OOM round: \(2.7\times 10^{33}\,\mathrm{g}\) → ____________________
19. OOM round: \(9.1\times 10^{-28}\,\mathrm{g}\) → ____________________
20. If \(R_s \propto M\) and \(R_s(1\,M_\odot)\approx 3\,\mathrm{km}\), estimate \(R_s(10\,M_\odot)\).
21. A quantity comes out to \(4\times 10^{11}\,\mathrm{cm}\). Is that closer to an Earth radius (\(\sim 10^9\,\mathrm{cm}\)) or a solar radius (\(\sim 10^{11}\,\mathrm{cm}\))? Explain in one sentence.

15.5 Level 4 — proportionality and scaling

22. If \(L \propto R^2\) and a star’s radius doubles, by what factor does its luminosity change?
23. If \(F \propto 1/r^2\) and distance triples, by what factor does flux change?
24. Using \(P \propto r^{3/2}\): Earth orbits at 1 AU with \(P = 1\) yr. Mars orbits at 1.52 AU. Estimate Mars’s period.
25. The Schwarzschild radius scales as \(R_s \propto M\). Sgr A* has mass \(4\times 10^6\,M_\odot\). If \(R_s(1\,M_\odot) \approx 3\,\mathrm{km}\), estimate \(R_s\) for Sgr A* in km.

TipPractice answer key

Level 0

1. \(4.5\times 10^7\)
2. \(8\times 10^2\)
3. 5
4. 16
5. 10

Level 1

6. \(-25\) (exponent binds before negative)
7. \(25\)
8. \(1\) (it’s \((12/4)/3 = 3/3 = 1\))
9. 6 (square root halves the exponent: \(\sqrt{10^{12}} = 10^6\))
10. 3 (cube root divides by 3: \((10^9)^{1/3} = 10^3\))
11. \(4^{3/2} = 4 \cdot \sqrt{4} = 4 \cdot 2 = 8\)

Level 2

12. \(12\times 10^5 = 1.2\times 10^6\,\mathrm{cm}\)
13. \(35\,\mathrm{cm}\)
14. \(250\times 10^2 = 2.5\times 10^4\,\mathrm{cm/s}\)
15. \(2.0\times 10^4\,\mathrm{cm^2}\)
16. \(5\times 10^7\,\mathrm{erg}\)

Level 3

17. \(3.8\times 10^{33}\,\mathrm{erg/s}\)
18. \(10^{33}\,\mathrm{g}\) (rule of 3)
19. \(10^{-27}\,\mathrm{g}\) (rule of 3)
20. \(\approx 30\,\mathrm{km}\)
21. Solar-radius scale (\(10^{11}\,\mathrm{cm}\)) — exponent matches Sun scale.

Level 4

22. \(2^2 = 4\) (luminosity quadruples)
23. \(1/3^2 = 1/9\) (flux decreases by factor of 9)
24. \(P^2 = 1.52^3 \approx 3.5\), so \(P \approx \sqrt{3.5} \approx 1.9\) years
25. \(R_s = 3\,\mathrm{km} \times 4\times 10^6 = 1.2\times 10^7\,\mathrm{km}\)

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16 What “show your work” means in ASTR 201

When math is part of the physics, your work needs to be readable—and checkable.

16.1 Why this matters

Small mistakes are easy to make: a dropped exponent, a flipped fraction, a unit that didn’t cancel. The only reliable way to catch them is to write every step explicitly.

If you skip steps, you can’t find your error. Neither can we.

16.2 The template

1. Write the equation first — before plugging in numbers. This separates “what physics am I using?” from “what numbers go where?”
2. Carry units through every step — not just at the start and end. Units that don’t cancel are a red flag.
3. Show exponent arithmetic explicitly — write \(10^{26}\times 10^7 = 10^{33}\), not just the answer.
4. Simplify before you calculate — cancel terms, combine exponents, reduce fractions on paper first. The calculator is the last step, not the first.
5. Box the final answer with units — make it easy to find.
6. Sanity check — one sentence. Does the magnitude make sense?

WarningCalculator errors are sneaky

Typing mistakes are easy to make and hard to spot. A misplaced parenthesis or a wrong exponent key can silently wreck your answer.

Always check your calculator work. Re-enter the calculation a second way, or estimate the answer by hand first so you know what ballpark to expect. If the calculator says \(10^{47}\) but your estimate says \(10^{33}\), trust the estimate—you probably made a typo.

WarningThe hard truth

Most errors aren’t conceptual—they’re mechanical. A sign flip. A forgotten square. A unit that should have been squared but wasn’t.

Showing your work isn’t about proving you know the material. It’s about building a trail you can follow when something goes wrong. And something will go wrong.

NoteThe real skill

You’re not being graded on “being fast.” You’re being graded on being trustworthy—and trustworthy work can be checked.