Chapter 2 — Measurement, Units, and Scientific Notation

2.1 Why Measurement Matters in Physics

Physics is a quantitative science. The difference between describing motion as “fast” and describing it as “28 meters per second” is not merely a matter of precision; it is the difference between an impression and a testable claim. A statement that can be measured can be checked, refined, and built upon. A statement that cannot be measured remains an opinion.

Measurement connects ideas to reality. When Newton proposed that force equals mass times acceleration, that claim was not meaningful until physicists could measure force in newtons, mass in kilograms, and acceleration in meters per second squared — and verify that the equation held to within the uncertainty of their instruments. The entire structure of physics rests on this foundation: every law, every theory, every model must ultimately make predictions that can be tested by measurement.

Measurement also allows communication. A physicist in Brazil and a physicist in Japan can compare results and build on each other’s work only because they are using the same units, the same standards, and the same conventions for reporting precision. Without that shared language, science could not be cumulative.

Historical Voice: Galileo Galilei is often associated with the principle that measurable description strengthens scientific knowledge.

This chapter teaches you that shared language. By the end, you will be able to read any measurement in physics, express it in proper scientific form, convert between units, and report it with the precision the situation demands.

2.2 Physical Quantities and Units

A physical quantity is any property of matter or energy that can be measured and expressed as a number. Length, mass, time, temperature, speed, force, and electric charge are all physical quantities. Each physical quantity has two components: a numerical value and a unit.

For example, saying that a table is “1.8” tells us nothing. Saying that it is “1.8 meters” tells us its length precisely. The unit — the meter — is the standard against which the measurement is made. Changing the unit changes the number: the same table is 180 centimeters long, or 5.9 feet long. The physical reality does not change; only the way we express it in numbers does.

Common Mistake: Never write a measurement without its unit. In physics, “5.0” is not a length, a mass, or anything else. “5.0 m” is a length. The unit is not optional.

Physical quantities fall into two broad categories. Base quantities are the fundamental, independently defined quantities from which all others are derived. Derived quantities are combinations of base quantities. Speed, for instance, is a derived quantity: it is defined as distance divided by time, combining two base quantities.

2.3 The International System of Units (SI)

Modern science uses one unified system of units: the Système International d’Unités, universally abbreviated SI. Adopted internationally in 1960 and continuously refined by the International Bureau of Weights and Measures, the SI is the agreed-upon standard for scientific measurement worldwide.

Before SI, different countries — and different fields within science — used incompatible systems. Engineers in the United States measured length in feet and force in pounds; scientists in Europe measured in centimeters and dynes; navigators used nautical miles. The resulting confusion cost real resources and, in several famous cases, caused engineering disasters. The Mars Climate Orbiter was lost in 1999 because one engineering team used metric units and another used imperial units — a $328 million lesson in the importance of unit consistency.

Historical Note: The meter was originally defined as one ten-millionth of the distance from the North Pole to the equator. Today it is defined precisely in terms of the speed of light: one meter is the distance light travels in a vacuum in exactly 1/299,792,458 of a second.

The SI has several important features. It is decimal: all prefixes are powers of ten, so conversions never require awkward factors like 12 or 5,280. It is coherent: derived units are defined as simple products or quotients of base units with no numerical factors. And it is universal: every country’s national standards laboratory is traceable back to the same international definitions.

Bridge Note

2.4 Base Units and Derived Units

The SI is built on seven base units. Every other unit in physics can be expressed as a combination of these seven. You do not need to memorize the precise definitions of all seven immediately, but you should know them by name, symbol, and what they measure.

Derived units are built from base units using multiplication and division. A few derived units are used so frequently that they have been given their own names and symbols to avoid writing out the full combination every time.

Why This Matters: Knowing that the newton equals kg·m/s² is not just trivia. When you check whether an equation makes sense by examining its units — a process called dimensional analysis — you will need to break named units like newtons and joules back into their base components.

2.5 Metric Prefixes and Powers of Ten

One of the great advantages of the SI is its use of decimal prefixes. Rather than inventing a new unit for every scale — as imperial units do with inches, feet, yards, and miles — the metric system applies standard prefixes to a base unit to indicate multiples or fractions of that unit. A prefix represents multiplication by a power of ten.

To convert between prefixes, you shift the decimal point or equivalently multiply or divide by the appropriate power of ten. The key skill is to always convert to base units first when doing any calculation.

Converting with Prefixes

The method is always the same: replace the prefix with its power of ten, then simplify.

Example: 45 km = 45 × 10³ m = 45,000 m = 4.5 × 10⁴ m.

Example: 320 μm = 320 × 10⁻⁶ m = 3.20 × 10⁻⁴ m.

Example: 2.5 GHz = 2.5 × 10⁹ Hz = 2,500,000,000 Hz.

Common Mistake: Prefix symbols are case-sensitive. “m” is milli, or 10⁻³; “M” is mega, or 10⁶. Writing “Mm” means megameters, not millimeters.

2.6 Scientific Notation

Physics deals with quantities that span an enormous range of magnitudes. The diameter of a proton is about 0.000000000000001 meters. The distance to the nearest star beyond our Sun is about 40,000,000,000,000,000 meters. Writing these numbers in ordinary decimal form is cumbersome and error-prone. Scientific notation solves this problem.

The coefficient a must satisfy 1 ≤ a < 10. This constraint means there is exactly one non-zero digit to the left of the decimal point. The exponent n tells you how many places the decimal point has been moved, and in which direction.

The Rules of Scientific Notation

• The coefficient must be at least 1 and less than 10.
• The exponent is positive if the original number is 10 or greater.
• The exponent is negative if the original number is between 0 and 1.
• The exponent counts the number of places the decimal point moves.

Example: 6,400,000 = 6.4 × 10⁶.

Example: 0.00000320 = 3.20 × 10⁻⁶.

Example: 1 = 1.0 × 10⁰.

Arithmetic with Scientific Notation

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents. When dividing, divide the coefficients and subtract the exponents. When adding or subtracting, first convert both numbers to the same power of ten.

Example: (3.0 × 10⁴) × (2.0 × 10³) = 6.0 × 10⁷.

Example: (9.6 × 10⁸) ÷ (3.2 × 10³) = 3.0 × 10⁵.

Common Mistake: After multiplying or dividing, check that your coefficient is still between 1 and 10. If you get 24 × 10⁵, rewrite it as 2.4 × 10⁶.

2.7 Converting Between Standard and Scientific Notation

Converting between ordinary decimal notation and scientific notation is a mechanical skill that should become automatic. The key is to count decimal places carefully and track the sign of the exponent.

From Standard to Scientific Notation

Move the decimal point until only one non-zero digit remains to its left. Count the number of places moved. If you moved left, the exponent is positive. If you moved right, the exponent is negative.

From Scientific to Standard Notation

Move the decimal point in the coefficient by the number of places given by the exponent. Positive exponent: move right. Negative exponent: move left. Fill in zeros as needed.

Example: 4.7 × 10⁵ = 470,000.

Example: 8.3 × 10⁻⁴ = 0.00083.

2.8 Significant Figures

Not all digits in a number carry equal meaning. The significant figures of a measurement are the digits that carry meaningful information about the precision of the measurement. Reporting too many significant figures implies more precision than the instrument can provide; reporting too few discards information.

Rules for Counting Significant Figures

1. All non-zero digits are significant.
2. Zeros between non-zero digits are significant.
3. Leading zeros before the first non-zero digit are never significant.
4. Trailing zeros after a decimal point are significant.
5. Trailing zeros in a whole number without a decimal point are ambiguous.

Significant Figures in Calculations

When you multiply or divide measurements, round the result to the same number of significant figures as the measurement with the fewest significant figures.

Example: 4.52 m × 3.1 m = 14 m². The result is rounded to 2 significant figures because 3.1 has 2 significant figures.

When you add or subtract measurements, round the result to the same number of decimal places as the measurement with the fewest decimal places.

Example: 12.3 m + 0.456 m = 12.8 m. The result is rounded to 1 decimal place.

Common Mistake: Never round in the middle of a calculation. Carry one or two extra digits through intermediate steps, then round only the final answer.

Bridge Note

2.9 Precision vs. Accuracy

Precision and accuracy are often used interchangeably in everyday speech, but in physics they have distinct meanings that are essential to understanding what a measurement tells you.

Accuracy refers to how close a measurement is to the true value of the quantity being measured. Precision refers to how reproducible a measurement is — how close repeated measurements are to each other, regardless of whether they are close to the true value.

A measurement can be precise without being accurate. A miscalibrated balance might give the same reading every time while being systematically off by 5 grams. Conversely, a measuring technique might on average give the right answer while individual measurements scatter widely.

Key Distinction: Precision is limited by random error. Accuracy is limited by systematic error.

In experimental physics, you want both. The goal is to reduce random error by repeating measurements and averaging, and eliminate systematic error by calibrating instruments and checking assumptions.

2.10 Measurement Error and Instrument Limits

Every measuring instrument has a limit of resolution — the smallest difference it can detect. A ruler marked in millimeters cannot reliably measure lengths smaller than about half a millimeter. A stopwatch that displays hundredths of a second cannot be read to thousandths. The instrument’s resolution sets a lower bound on the uncertainty of a direct measurement.

Reading Analog Instruments

For analog instruments, the convention is to estimate to one decimal place beyond the smallest marked division. If a ruler has millimeter markings, you read to the nearest 0.1 mm. This estimated digit is uncertain, but it still carries information.

Example: A ruler marked in millimeters shows an object between 4.3 cm and 4.4 cm, appearing closer to 4.4 cm. A good reading is 4.37 cm ± 0.05 cm.

Reading Digital Instruments

Digital instruments display a fixed number of digits. The last displayed digit is typically uncertain by ±1. A digital balance reading 12.34 g implies an uncertainty of about ±0.01 g.

Expressing Uncertainty

For example: L = 3.52 m ± 0.01 m. This tells the reader that the true value is believed to lie between 3.51 m and 3.53 m.

Percent Error and Percent Uncertainty

Example: If a mass is measured as 48.5 g ± 0.5 g, the percent uncertainty is (0.5 ÷ 48.5) × 100% ≈ 1.0%.

2.11 Estimation and Order of Magnitude

Not every physical problem requires a precise measurement. Sometimes the most useful question is: roughly how big is this? Estimation — the art of making reasonable guesses without exact data — is one of the most important skills in physics.

An order of magnitude is the nearest power of ten that represents the size of a quantity. The mass of a proton is about 10⁻²⁷ kg. The mass of the Earth is about 10²⁵ kg. These are order-of-magnitude estimates; the exact values differ by factors of two or three, but the powers of ten capture the essential scale.

The Fermi Method

The physicist Enrico Fermi was famous for making accurate estimates of seemingly impossible quantities from limited information. His approach — now called Fermi estimation — breaks a problem into smaller sub-questions that can each be estimated from common knowledge, then combines them.

Classic Fermi Problem: How many piano tuners are there in Chicago? Estimate the population, the fraction of households with a piano, the number of tunings per year per piano, and the number of tunings a tuner can perform per year. The value is not the exact answer but the process.

Order-of-Magnitude Estimates in Practice

• Always carry units through your estimate.
• Round aggressively.
• Check your answer against physical intuition.
• An estimate off by a factor of two or three is usually acceptable; an estimate off by a factor of 1,000 is not.

2.12 Recording and Reporting Measurements Properly

The conventions for recording and reporting measurements exist for one reason: to communicate the information content of a measurement as clearly and honestly as possible. A reader of your lab report should be able to reproduce your measurement and understand exactly what precision it has.

The Standard Format

Every measurement you record in physics should include the numerical value, the unit, the uncertainty, and the number of significant figures consistent with the uncertainty.

Example: m = 125.3 g ± 0.1 g

Consistency Between Value and Uncertainty

The uncertainty must be expressed to the same number of decimal places as the measurement. You would never write 12.3456 m ± 0.1 m. You would write 12.3 m ± 0.1 m, or 12.35 m ± 0.05 m if your instrument supports that precision.

Common Mistake: Never report a measurement as “12.3456 cm” from a ruler marked in centimeters. The instrument cannot support that precision.

Laboratory Notebook Standards

• Record measurements in ink, not pencil.
• Record measurements at the time you make them, not from memory.
• Always note the instrument you used and its stated precision or resolution.
• Include units immediately after every numerical entry.
• If a measurement seems wrong, note it as suspect and repeat. Never erase or destroy original data.

Bridge Note

Chapter Summary

This chapter established the mathematical language physicists use to quantify the physical world. Every measurement in physics must state a numerical value, a unit, and an uncertainty. The SI provides a universal, coherent, decimal-based system of units built on seven base quantities.

• Every physical quantity has a numerical value and a unit.
• The SI is the universal system of measurement in science.
• All other units in physics are derived units.
• Metric prefixes are powers of ten applied to base units.
• Scientific notation expresses any number as a coefficient between 1 and 10 multiplied by a power of ten.
• Significant figures communicate the precision of a measurement.
• Accuracy measures closeness to the true value; precision measures reproducibility.
• Every measurement has uncertainty.
• Order-of-magnitude estimation is a core physics skill.

Key Terms

physical quantity

Any property of matter or energy that can be measured and expressed as a numerical value with a unit.

unit

A standard reference amount against which a physical quantity is measured.

SI

The internationally agreed-upon system of units built on seven base units, used universally in science.

base unit

One of the seven fundamental SI units from which all other units are derived.

derived unit

A unit formed by combining base units through multiplication or division.

metric prefix

A prefix applied to an SI base or derived unit to indicate multiplication by a power of ten.

scientific notation

A way of expressing any number as a coefficient between 1 and 10 multiplied by a power of ten.

significant figures

The digits in a measurement that carry meaningful information about its precision.

accuracy

How close a measurement is to the true or accepted value.

precision

How reproducible a measurement is.

random error

Unpredictable variation in measurements caused by inherent limits of measurement.

systematic error

Consistent deviation from the true value caused by miscalibration, faulty technique, or flawed assumptions.

uncertainty

The range of values within which the true value of a measurement is believed to lie.

percent error

A measure of accuracy: |measured − accepted| ÷ accepted × 100%.

percent uncertainty

Uncertainty expressed as a percentage of the measured value.

order of magnitude

The nearest power of ten representing the size of a quantity.

resolution

The smallest difference an instrument can reliably detect or display.

Fermi estimation

A problem-solving method in which a complex quantity is estimated by breaking it into reasonably guessed sub-quantities.

Questions and Practice Problems

Concept Review Questions

10. Why is a number without a unit meaningless in physics? Give two examples.
11. What are the seven SI base units? List each one, its symbol, and what physical quantity it measures.
12. Explain the difference between a base unit and a derived unit.
13. Why does the metric system make unit conversion easier than the imperial system?
14. What does the exponent in scientific notation tell you, and what does its sign indicate?
15. Why is the rule that the coefficient in scientific notation must be between 1 and 10 useful?
16. State the rules for counting significant figures. Apply each rule to one example of your own.
17. Explain the difference between accuracy and precision.
18. What is a systematic error? Give one realistic laboratory example.
19. What does it mean to say a measurement has an uncertainty of ±0.05 m?
20. Why is order-of-magnitude estimation useful even when precise calculations are possible?
21. A student records a length as “42 centimeters” in her lab notebook. What is missing?

Measurement Practice Problems

22. Convert 5.50 km to meters. Express your answer in scientific notation.
23. Convert 250 mA to amperes.
24. Convert 3.2 × 10⁻⁴ m to micrometers.
25. Convert 1 light-year, approximately 9.461 × 10¹⁵ m, to kilometers.
26. A car travels 60.0 miles per hour. Convert this speed to meters per second. Use 1 mile = 1609 m.
27. The diameter of a human hair is about 70 μm. Express this in meters, millimeters, and nanometers.
28. A force of 35 N acts on an object with mass 5.0 kg. What is the acceleration?
29. A box has dimensions 0.40 m × 0.25 m × 0.15 m and a mass of 3.6 kg. Calculate its density.
30. An engine produces 4,500 J of work in 15 s. Calculate the power in watts.
31. A student measures a pendulum three times: 82.3 cm, 82.5 cm, 82.4 cm. Find the average, range, and uncertainty.
32. The accepted value for gravity is 9.81 m/s². A student measures 9.65 m/s². Calculate the percent error.
33. A mass is measured as 45.2 g ± 0.2 g. What is the percent uncertainty?

Scientific Notation Exercises

34. 0.000 000 167
35. 5,972,000,000,000,000,000,000,000
36. 0.00350
37. 186,000
38. 0.1
39. 1,000,000,000
40. 6.022 × 10²³
41. 1.67 × 10⁻²⁷
42. 3.00 × 10⁸
43. 9.11 × 10⁻³¹
44. (2.5 × 10³) × (4.0 × 10⁵)
45. (8.4 × 10¹²) ÷ (2.1 × 10⁴)
46. (6.0 × 10⁻³) × (5.0 × 10⁻⁴)
47. (3.0 × 10⁸) ÷ (6.0 × 10³)
48. (4.0 × 10³) + (2.5 × 10²)
49. (7.2 × 10⁻⁴) − (3.0 × 10⁻⁵)

Significant Figures Exercises

50. 0.00340 m
51. 1,050 kg
52. 2.000 × 10³ s
53. 10,000 g with no decimal point shown
54. 0.10800 mL
55. 3.14159
56. Round 7.8546 to 3 significant figures.
57. Round 0.004397 to 2 significant figures.
58. Round 12,850 to 3 significant figures using scientific notation.
59. Round 99.97 to 3 significant figures.
60. (4.25 m) × (1.7 m)
61. (100.5 g) − (2.35 g)
62. (6.3 m/s) × (4.00 s)
63. (9.6 × 10³ J) ÷ (3.00 × 10² s)
64. (5.280 km) + (342 m converted to km)

Lab / Measurement Activity

Measurement Reliability Lab — Comparing Instruments

Purpose: To practice taking measurements with instruments of different precision, to observe the effect of instrument resolution on significant figures, to calculate average values and uncertainties, and to distinguish random from systematic error.

Materials

• A small cylindrical object
• A ruler with millimeter markings
• A digital caliper, if available
• A triple-beam balance or digital scale
• A stopwatch

Procedure

65. Using the millimeter ruler, measure the diameter five times.
66. If a caliper is available, repeat with the caliper.
67. Measure the mass of the object five times.
68. Time how long it takes you to tap the desk ten times, five separate trials.

Data Analysis

69. For each set of five measurements, calculate the mean, range, uncertainty as half the range, and percent uncertainty.
70. Report each quantity in the format: value ± uncertainty (unit).
71. Compare the ruler and caliper results, if both were used.
72. Identify one possible source of random error and one possible source of systematic error.

Reflection Questions

73. If you repeated the diameter measurement 50 times instead of 5, would your precision improve? Your accuracy?
74. Suppose the caliper’s jaws were slightly bent, causing it to consistently read 0.3 mm too large. How would this affect accuracy and precision?
75. Express your average diameter in scientific notation. How many significant figures are justified by your uncertainty?

Chapter 2 Checkpoint

Before moving on to Chapter 3, you should be able to do the following without looking at your notes. If you cannot, review the relevant section.