Some Basic Concepts of Chemistry · Lesson 4 of 11

Scientific Notation and Reliable Calculations

“Handle very large and small quantities while preserving the meaning of units and measurement precision.”

Learning Objectives

• Write and calculate with numbers in scientific notation. • Determine significant figures, including zeros and exact counts. • Distinguish precision from accuracy and explain measurement uncertainty. • Round calculated results appropriately for the operation performed. • Use unit factors to perform and check conversions.

Making large and small numbers manageable

A tiny sample can contain an enormous number of particles, while one atom has an extremely small mass. Writing all the zeros is inconvenient and makes mistakes easy. Scientific notation separates a manageable numerical coefficient from a power of ten.

Scientific notationLaTeX

a \times 1 0^{k} 1 \leq ∣ a ∣ < 10

For a non-zero number,

a

is the coefficient and

k

is an integer exponent. Positive exponents move the scale upward; negative exponents describe small magnitudes.

The number $232.508$ becomes $2.32508 \times 1 0^{2}$ : multiplying by $1 0^{2}$ restores the two places moved in the decimal. The number $0.00016$ becomes $1.6 \times 1 0^{- 4}$ . The exponent records scale, not the number of significant figures.

Example — Multiplication and division of powers

Problem
Evaluate $(5.6 \times 1 0^{5}) (6.9 \times 1 0^{8})$ before rounding to measurement precision.

1.Multiply the coefficients: $5.6 \times 6.9 = 38.64$ . Add exponents because $1 0^{5} \times 1 0^{8} = 1 0^{13}$ .
2.The unrounded product is $38.64 \times 1 0^{13} = 3.864 \times 1 0^{14}$ . The extra factor of ten appears when the coefficient is brought below 10.
3.For division, divide coefficients and subtract exponents. For example, $(2.7 \times 1 0^{- 3}) / (5.5 \times 1 0^{4}) = 4.90909 \dots \times 1 0^{- 8}$ before rounding.

Addition and subtraction require a common exponent, just as amounts must be in common units before being combined. Thus $(6.65 \times 1 0^{4}) + (8.95 \times 1 0^{3}) = (6.65 + 0.895) \times 1 0^{4} = 7.545 \times 1 0^{4}$ before rounding. Similarly, $(2.5 \times 1 0^{- 2}) - (4.8 \times 1 0^{- 3}) = (2.5 - 0.48) \times 1 0^{- 2} = 2.02 \times 1 0^{- 2}$ before rounding.

What a measured number can claim

A measuring instrument does not provide unlimited detail. A platform balance might report 9.4 g, while an analytical balance reports 9.4213 g for the same object. Extra digits are justified only if the instrument and procedure support them. The final reported digit is an estimate at the resolution of the measurement.

Definition

Significant figures

The meaningful digits of a measured value: the digits regarded as certain together with the final estimated digit.

Measurement uncertainty can arise from instrument resolution, calibration, reading technique, and variation in the procedure. Significant figures communicate reporting precision, but a fully evaluated uncertainty may need a separate statement. They do not automatically prove that a measurement is close to the true value.

Rule	Illustration	Significant figures
Non-zero digits count	$285$	3
Leading zeros locate the decimal	$0.0052$	2
Zeros between non-zero digits count	$2.005$	4
Trailing zeros after a decimal can count	$0.200$	3
Scientific notation makes trailing zeros explicit	$1.00 \times 1 0^{2}$	3
A counted number is exact	Exactly 20 objects	Does not limit precision

A whole-number report such as 100 can be ambiguous about trailing zeros. Use $1 \times 1 0^{2}$ , $1.0 \times 1 0^{2}$ , or $1.00 \times 1 0^{2}$ to state one, two, or three significant figures. Exact unit definitions and exact counts do not restrict the number of significant figures in a calculation; measured quantities do.

Precision and accuracy answer different questions

Repeating a measurement provides information about consistency. Comparing the result with a reliable reference provides information about correctness. A balance with a constant calibration error can give tightly clustered readings that are all too low, so repeatability alone is insufficient.

Definition

Precision

The closeness of repeated measurements to one another under specified conditions.

Definition

Accuracy

The closeness of a measured result to the reference or true value.

Repeatability and closeness to a reference— A tight cluster can miss the centre. A good average can also hide widely scattered readings.

Student	Two measurements in grams	Average in grams	Interpretation for a 2.00 g reference
A	1.95, 1.93	1.94	Close together but both below the reference
B	1.94, 2.05	1.995	Widely separated; a close average does not show good repeatability
C	2.01, 1.99	2.00	Close together and close to the reference

Always specify whether you are judging individual observations or an average. Student B’s average is close to 2.00 g, but that does not erase the scatter in the observations. This is why reporting several readings can reveal more than reporting only their mean.

Reporting arithmetic without inventing detail

A calculator can display more digits than the data justify. In addition and subtraction, the limiting issue is decimal place. In multiplication and division, the limiting issue is the number of significant figures. Keep extra working digits during intermediate steps and round the final result appropriately.

Example — Adding measured quantities

Problem
Add $12.11 g$ , $18.0 g$ , and $1.012 g$ .

1.The unrounded sum is $31.122 g$ .
2.The least detailed measurement, $18.0 g$ , extends only to tenths of a gram.
3.Report $31.1 g$ . Addition is controlled by decimal place, not by choosing the fewest significant figures.

Example — Multiplying measured quantities

Problem
Find the area of a rectangle with measured sides $2.5 cm$ and $1.25 cm$ .

1.The unrounded area is $2.5 \times 1.25 = 3.125 c m^{2}$ .
2.The first length has two significant figures, while the second has three.
3.Report $3.1 c m^{2}$ , with two significant figures. Extra displayed digits would suggest unsupported precision.

When rounding, increase the retained digit if the discarded part is greater than half of one unit in that place; leave it if less. For an exact halfway case, use the round-to-even convention here: keep an even last digit or raise an odd one to the next even digit. If a discarded 5 is followed by further non-zero digits, the number is above halfway.

To two significant figures, $6.35$ becomes $6.4$ , while $6.25$ becomes $6.2$ . But $6.251$ becomes $6.3$ because it is greater than the halfway value. To three significant figures, $1.386$ becomes $1.39$ and $4.334$ becomes $4.33$ .

Do not round every intermediate step

Rounding too early can alter the final answer. Carry guard digits, keep units attached, and state the final result at a precision justified by the measurements. Exact counts and defined conversions do not impose an artificial rounding limit.

Let units guide the conversion

Dimensional analysis, also called the factor-label or unit-factor method, treats units algebraically. An equality between two descriptions of the same amount produces a ratio equal to one. Choose the orientation of that ratio so the unwanted unit cancels.

Example — A length conversion

Problem
Convert a reported length of $3.00 in$ to centimetres, using the exact relation $1 in = 2.54 cm$ .

1.The factor $2.54 cm /1 in$ equals one and puts inches in the denominator.
2. $3.00 in \times (2.54 cm /1 in) = 7.62 cm$ .
3.The input has three significant figures. The exact inch definition adds no measurement uncertainty.

Example — Converting volume

Problem
Express $2.00 L$ of milk in cubic metres.

1.Use $1 L = 1000 c m^{3}$ and $1 m = 100 cm$ .
2. $2.00 L \times (1000 c m^{3} /1 L) \times (1 m /100 cm)^{3} = 2.00 \times 1 0^{- 3} m^{3}$ .
3.The cube applies to the number 100 and to the centimetre unit. All unwanted units cancel.

Example — Chaining factors

Problem
How many seconds are in exactly two days?

1.Use the defined relations $1 day = 24 h$ , $1 h = 60 min$ , and $1 min = 60 s$ .
2. $2 day \times (24 h / day) \times (60 min / h) \times (60 s / min) = 172800 s$ .
3.Here two days is specified as an exact count of defined intervals. A measured time reported to limited precision would need an appropriate precision in its answer.

Quick check

How is $0.00072$ written in scientific notation?

Quick check

How many significant figures are in $0.02050$ ?

Quick check

A balance gives nearly identical readings, all 0.20 g too low. What does this suggest?

Quick check

What should $2.50 \times 1.2$ be reported as when both are measured?

Quick check

Which factor converts $c m^{3}$ to $m^{3}$ ?

Quick check

Using exact-halfway round-to-even, what is $6.25$ to two significant figures?

Practice Problems

Write $8008$ and $0.00480$ in scientific notation without losing their stated significant figures.
Calculate $(9.8 \times 1 0^{- 2}) (2.5 \times 1 0^{- 6})$ and report two significant figures.
Report $18.0 + 1.012$ with appropriate decimal precision.
Round $34.216$ , $10.4107$ , and $0.04597$ to three significant figures.
Explain why a close mean does not guarantee precise individual readings.
Convert $28.7 pm$ to metres and $25.365 g$ to kilograms.
Light travels at $3.0 \times 1 0^{8} m s^{- 1}$ . Find the distance in $2.00 ns$ and justify the significant figures.

Key Takeaways

Scientific notation records scale using a power of ten. Significant figures describe justified digits; leading zeros do not add precision. Precision is repeatability; accuracy concerns closeness to a reference. Addition and subtraction use decimal-place limits; multiplication and division use significant-figure limits. Unit factors are ratios equal to one, and their orientation should cancel unwanted units.

Previous · Lesson 3

Properties, Measurement, and Units

Next · Lesson 5

Laws of Chemical Combination and Atomic Theory

Making large and small numbers manageable

What a measured number can claim

Precision and accuracy answer different questions

Reporting arithmetic without inventing detail

Let units guide the conversion

Quiz

Practice Problems

Key Takeaways