Skills Beta

Math you need: the one-page sheet

Units and Dimensional Analysis

Every measurement is a number with a unit. Metric prefixes scale units by factors of ten, and conversion factors (ratios equal to 1) change the unit without changing the quantity. Chain factors so that units cancel; density converts between mass and volume, and K = °C + 273.15.

Every measured number needs its unit; "12.6" means nothing until it is 12.6 g or 12.6 mL.
Metric prefixes scale a unit: kilo (k) = 1000, centi (c) = 1/100, milli (m) = 1/1000, micro (µ) = 1/1,000,000, nano (n) = 1/1,000,000,000.
Dimensional analysis: multiply by conversion factors, each with the unit you have on the bottom, until only the unit you want is left.
Density = mass ÷ volume (g/mL or g/cm³), and works as a conversion factor between mass and volume. Kelvin = °C + 273.15.

changing the unit must change the number so the quantity stays the same multiplying by it is multiplying by 1, so the quantity does not change choosing which way up each factor goes decides which units are left the setup is right, and the arithmetic gives the answer in that unit

unit of measurement: The agreed-on amount that a measurement counts in, such as the gram, the liter or the second; a number without its unit is incomplete.
metric prefix: A prefix that scales a unit by a factor of ten, such as kilo (1000), centi (1/100), milli (1/1000), micro (one millionth) and nano (one billionth).
dimensional analysis: Solving a problem by multiplying by conversion factors, ratios equal to 1, so that unwanted units cancel and only the wanted unit is left.
kelvin: The SI unit of temperature. Its zero is absolute zero, the coldest possible temperature; K = °C + 273.15.
density: Mass divided by volume, usually in g/mL or g/cm³; it can be used to convert between the mass and the volume of a substance.

Exponents

An exponent counts repeated multiplication. Multiplying powers of the same base adds exponents, dividing subtracts them, and raising a power to a power multiplies them. A negative exponent means a reciprocal, and any nonzero base to the zero power is 1. These rules do not apply to adding.

An exponent says how many times to multiply the base by itself: 10⁴ = 10,000.
Multiply: add exponents (10² × 10³ = 10⁵). Divide: subtract (10² ÷ 10⁵ = 10⁻³). Power of a power: multiply (10²)³ = 10⁶.
Negative exponent = reciprocal: 10⁻³ = 1/1000 = 0.001. Zero exponent: 10⁰ = 1.
Roots are fractional powers: √(10⁸) = 10⁴. Adding numbers does not combine exponents: 10⁴ + 10⁴ = 2 × 10⁴.

10³ means 10 × 10 × 10 = 1000 their exponents add the exponents subtract, and a larger bottom exponent leaves a negative exponent 10⁻ⁿ is 1/10ⁿ, a small positive number

exponent: The small raised number that says how many times a base is multiplied by itself; 10³ = 10 × 10 × 10 = 1000.
exponent rules: Multiplying powers of the same base adds the exponents, dividing subtracts them, a power of a power multiplies them, 10⁰ = 1, 10⁻ⁿ = 1/10ⁿ, and a square root halves the exponent.

Scientific Notation

Scientific notation writes a number as a coefficient from 1 to just under 10 times a power of ten. Large numbers have positive exponents and small numbers negative ones. To multiply, multiply coefficients and add exponents; to divide, divide and subtract; to add, match exponents first. Each factor of ten is one order of magnitude.

Scientific notation: coefficient × 10ⁿ, with 1 ≤ coefficient < 10. 0.00560 = 5.60 × 10⁻³; 149,600,000 = 1.496 × 10⁸.
Numbers above 10 get positive exponents; numbers below 1 get negative exponents.
Multiply: multiply coefficients, add exponents. Divide: divide coefficients, subtract exponents. Then fix the coefficient if it left the 1-10 range.
Add or subtract only after giving both numbers the same exponent. One order of magnitude is a factor of 10.

they are slow to write and easy to miscount the zeros move into the exponent the exponent goes up by one for each place, so a large number gets a positive exponent comparing, multiplying and dividing become exponent arithmetic

scientific notation: A way to write a number as a coefficient of at least 1 and less than 10 multiplied by a power of ten, such as 6.50 × 10⁻⁷.
order of magnitude: A factor of ten. Two quantities that differ by a factor of about 1000 differ by three orders of magnitude.

Significant Figures

A measurement records every certain digit plus one estimated digit; these are its significant figures. Leading zeros never count. Products and quotients keep the fewest significant figures, sums and differences the fewest decimal places, and exact numbers never limit. Precision is agreement between readings; accuracy is closeness to the true value.

Count significant figures: all nonzero digits; zeros between them; trailing zeros after a decimal point. Leading zeros never count. 0.004050 has four.
Multiply or divide: keep the fewest significant figures. Add or subtract: keep the fewest decimal places.
Exact numbers (counts, defined equalities like 1 kg = 1000 g) never limit an answer.
Precision is how closely repeated readings agree; accuracy is how close they are to the true value. Round once, at the end.

the last recorded digit of a measurement is an estimate they show how precise the measurement is it is rounded to match the least precise measurement products keep the fewest significant figures, and sums keep the fewest decimal places

significant figures: The digits in a measurement that carry meaning: every certain digit plus the one estimated digit. Leading zeros are not significant.
measurement uncertainty: The doubt in the last digit of every measurement, set by how finely the instrument can be read.
precision: How closely repeated measurements agree with each other. Accuracy, a different idea, is how close they are to the true value.
exact number: A number known with no uncertainty, such as a count of objects or a defined equality like 1 kg = 1000 g; it never limits significant figures.

Logarithms

A base-10 logarithm is the exponent of ten that gives a number; the natural logarithm uses base e. Logs turn multiplying into adding, so a factor of ten adds 1 to log x. ln x = 2.303 log x. An antilog undoes a log: 10ˣ for log, eˣ for ln. The decimal places of a log match the significant figures of the number.

log x (base 10) is the power of ten that gives x: log 10⁻⁵ = −5. Numbers between 0 and 1 have negative logs.
log (A × B) = log A + log B; log (A/B) = log A − log B; log (Aⁿ) = n log A.
ln x is the natural log, base e. ln x = 2.303 log x. Use the key the equation names.
Digits after the decimal point in a log are its significant figures: log (3.2 × 10⁻⁴) = −3.49. Undo log with 10ˣ and ln with eˣ (the antilog).

log 1000 = 3 and log 0.001 = −3 the log of a product is the sum of the logs, so a factor of 10 adds 1 ln x = 2.303 log x, and mixing them up gives answers 2.303 times off 10ˣ undoes log and eˣ undoes ln

logarithm: The exponent that 10 must be raised to in order to give a number: log 1000 = 3 because 10³ = 1000. Also called the base-10 or common logarithm.
natural logarithm: A logarithm with base e (about 2.718), written ln. ln x is about 2.303 times log x.
antilog: The inverse of a logarithm: 10ˣ undoes log x and eˣ undoes ln x.