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Numbers, Units and Estimation

Why this chapter

Why this chapter exists

Every answer in physics is a number with a unit. The numbers run from the size of a nucleus to the size of the Earth and beyond. This chapter teaches the language in which physics writes them. You will write any number in one short line, keep only the digits a measurement can support, and change any unit into another. You will also check an equation by its dimensions, and estimate a quantity that no table lists. Every later chapter uses these tools on almost every page.

You will learn

By the end of this chapter you will

Write any number in scientific notation, calculate with powers of ten, and state the order of magnitude of a quantity.
Count the significant figures of a measured value, round a number, and give a result with the figures its data support.
Convert units with factors equal to one, read the SI prefixes, and turn km/h into m/s and kilowatt-hours into joules.
Find the dimensions of a quantity, and check an equation by the principle of homogeneity.
Derive the form of a formula by dimensional analysis, and make a Fermi estimate of a quantity that no table lists.

Before you begin

No physics is assumed. You should be able to multiply and divide decimals, work with simple fractions, and know that \(10 \times 10 \times 10 = 1000\). Lesson 5 solves three small equations for three unknown powers, one at a time.

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Lessons

Scientific Notation and Orders of Magnitude

Physics measures a nucleus and the Earth with the same metre. Writing any number as \(a \times 10^b\), multiplying and dividing powers of ten, and reading the order of magnitude of a quantity.

Scientific notation and orders of magnitudeMultiplying and dividing powers of tenThe order of magnitude of a quantity

14 min0%

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Significant Figures and Rounding

A calculator gives eight digits, but a measurement supports only a few. Which digits of a measured number are significant, how to round a number, and how many figures a sum, a product or a quotient may keep.

Significant figures and roundingRounding off a numberTrailing zeros and exact numbers

14 min0%

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Converting Units and Prefixes

A physical quantity is a number times a unit. The seven SI base units, the prefixes from femto to tera, conversion by factors equal to one, kilometres per hour and metres per second, and the "unit" on the electricity bill.

Converting units and prefixesThe seven si base unitsKilometres per hour and metres per secondThe kilowatt hour on the bill

15 min0%

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Dimensions and Homogeneity

The dimensions of a quantity say which base quantities build it and with what powers. Dimensional formulas, dimensionless quantities, the principle of homogeneity, and checking an equation of the book before trusting it.

Dimensions and homogeneityThe principle of homogeneityChecking an equation by its dimensions

15 min0%

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Dimensional Analysis Derives a Formula

If you know what a quantity depends on, its dimensions often fix the formula except for a pure number. The period of a pendulum and Stokes' law derived by dimensions, a unit carried between the SI and CGS systems, and what the method cannot do.

Dimensional analysis derives a formulaConverting a unit between systemsThe limits of dimensional analysis

15 min0%

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Estimating by Orders of Magnitude

A Fermi estimate breaks a question you cannot measure into factors you can guess. Rounding each factor, why the errors partly cancel, the fuel of a reactor in four numbers, and how an estimate catches a wrong answer.

Estimating by orders of magnitudeChecking an answer by its size

13 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

base quantity: One of the seven quantities from which the SI builds all others: length, mass, time, electric current, temperature, amount of substance and luminous intensity.
derived unit: A unit built from the base units by a defining equation, such as the newton, \(1\,\text{N} = 1\,\text{kg m s}^{-2}\).
dimensional analysis: A method that finds the form of a formula. It writes a quantity as a product of powers and matches the dimensions of the two sides.
dimensional formula: The expression \(\text{M}^a\text{L}^b\text{T}^c\) that gives the dimensions of a quantity, such as \(\text{ML}^2\text{T}^{-2}\) for energy.
dimensionless: Having no dimensions, \(\text{M}^0\text{L}^0\text{T}^0\). A ratio of two like quantities, such as a strain or an angle in radians, is dimensionless.
dimensions: The powers of mass M, length L and time T that build a quantity. Force has dimensions \(\text{MLT}^{-2}\).
Fermi estimate: An estimate of the order of magnitude of a quantity. It writes the quantity as a product of factors that can be guessed, and multiplies them.
kilowatt-hour: The energy used by a 1 kW appliance in one hour, \(1\,\text{kWh} = 3.6 \times 10^6\,\text{J}\). It is the "unit" of an electricity bill.
order of magnitude: The power of ten nearest to a quantity. For \(a \times 10^b\) it is \(b\) when \(a \le 5\) and \(b + 1\) when \(a > 5\).
prefix: A syllable in front of a unit that multiplies it by a power of ten: kilo \(10^3\), milli \(10^{-3}\), micro \(10^{-6}\), nano \(10^{-9}\).
principle of homogeneity: The rule that every term of a correct physical equation has the same dimensions.
scientific notation: A number written as \(a \times 10^b\), with \(1 \le a < 10\) and \(b\) a whole number. The Earth's radius is \(6.4 \times 10^6\,\text{m}\).
SI: The International System of Units, used by scientists everywhere. It builds every unit from seven base units, such as the metre, the kilogram and the second.
significant figures: The digits of a measured number that are known reliably, together with the first uncertain digit. Leading zeros are not significant.
unit: The chosen standard with which a quantity is compared. A quantity is a number times a unit, such as \(5\,\text{m}\).