Use scientific notation, significant figures, and approximation

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Number and Algebra

Representing very large and very small numbers clearly and choosing sensible numerical accuracy

Measurements and calculations often produce numbers that are inconvenient to write in full. A distance may be many millions of metres, while a tiny mass may require several zeros after the decimal point. Scientific notation makes these values easier to read and compare. Significant figures communicate how precisely a value is stated, while approximation helps us present a suitable answer and judge whether a calculation is reasonable. This lesson uses familiar place value and powers of ten; no special calculator or graphing method is needed.

What you will learn

Write numbers in scientific notation and convert them back to ordinary notation.
Round values to a stated number of significant figures or decimal places.
Use approximation to estimate and check calculations.
Interpret calculator displays and communicate appropriately rounded answers with units.

1. Place value and scientific notation

A digit's place depends on powers of ten. Moving a decimal point one place to the left divides a number by ten; moving it one place to the right multiplies it by ten. For example,

4{,}800=4.8\times10^3

because the decimal point has moved three places to the left to make

4.8

.

Scientific notation writes a non-zero number as

a\times10^n

, where

n

is an integer and

1\leq |a|<10

. The coefficient

a

contains the significant digits, and the exponent

n

records the decimal-point shift. For a positive number greater than or equal to ten, the exponent is positive. For a positive number less than one, it is negative.

To convert back, use the exponent to move the decimal point: a positive exponent moves it right, and a negative exponent moves it left. Keep track of the number of places, not just the direction. For example,

6.2\times10^{-4}=0.00062

.

x=a\times10^n,\quad 1\leq |a|<10

Scientific notation has one non-zero digit before the decimal point.
A larger positive exponent represents a larger scale; a more negative exponent represents a smaller positive scale.
The notation is useful for comparing quantities with many zeros.

2. Significant figures and rounding

Significant figures are the digits that show the stated precision of a value. Non-zero digits are significant. Zeros between non-zero digits are significant, but zeros at the start of a decimal number only locate the decimal point and are not significant. For example,

0.00470

has three significant figures:

4

,

7

, and the final zero.

A zero at the end of a whole number can be ambiguous when no decimal point or other precision information is shown. Scientific notation removes that ambiguity:

5.20\times10^3

has three significant figures, whereas

5.2\times10^3

has two.

To round to a required number of significant figures, begin counting at the first non-zero digit. Keep the required number of digits, then inspect the next digit. If it is

5

or greater, increase the last kept digit by one; if it is less than

5

, leave it unchanged. Remove later digits, adjusting place value as needed. Decimal places are counted from the decimal point, so significant figures and decimal places are not interchangeable.

A rounded value is an approximation rather than the exact original number. The symbol

\approx

means “approximately equal to.” State the requested accuracy, such as “to three significant figures,” and include units when the quantity has them.

Leading zeros are not significant; zeros between significant digits and final decimal zeros are significant.
Count significant figures from the first non-zero digit.
Use $\approx$ for a rounded value, not $=$ .

3. Approximation in calculations and technology

An estimate gives a quick sense of the size of an answer. Round input values to convenient numbers, calculate mentally or with simple arithmetic, and use the result to check whether a more precise answer is plausible. An estimate is not a replacement for a requested calculation; it is a way to detect misplaced decimal points or incorrect scale.

Scientific notation also supports efficient calculations. For multiplication, multiply the coefficients and add the exponents. For division, divide the coefficients and subtract the exponents. Then rewrite the result so the coefficient satisfies the scientific-notation range. For addition or subtraction, first express both values with the same power of ten, then combine the coefficients.

A calculator may show scientific notation using an E display. For example,

3.6\mathrm{E}{-5}

means

3.6\times10^{-5}

. Enter powers of ten using the calculator's scientific-notation key where available, and check the display's exponent sign. Retain enough digits during intermediate calculations; round the final result to the requested accuracy. A calculator supplies numerical output, but the notation, rounding decision, and reasonableness check remain your responsibility.

In context, precision should match the information given and the question asked. A measured length stated to the nearest centimetre should not be reported as though it were measured to a tiny fraction of a centimetre. When a problem explicitly requests a particular number of significant figures, follow that request and retain the unit.

(a\times10^m)(b\times10^n)=(ab)\times10^{m+n}

For multiplication, add powers; for division, subtract powers.
For addition and subtraction, align powers of ten before combining.
Check calculator exponents and round only after the calculation is complete.

4. A reliable exam-style routine

Read the requested form and accuracy before calculating. Convert values carefully, keep units attached to contextual quantities, and avoid rounding intermediate results unless the question asks for an estimate. After obtaining a result, put it in the requested form and verify its scale with a quick estimate.

A numerical answer can be checked in two representations: ordinary decimal form makes the size familiar, while scientific notation makes the scale and significant digits clear. These are two ways of writing the same value, not different answers.

Identify whether the task asks for scientific notation, significant figures, or decimal places.
Keep adequate intermediate precision and round at the end.
Check the answer's magnitude and include units where appropriate.

Rounding guide

Task	What to count	Example
Three significant figures	Start at the first non-zero digit	$0.006284\to0.00628$
Two decimal places	Count two digits after the decimal point	$7.236\to7.24$
Scientific notation	One non-zero digit before the decimal point	$62{,}000=6.2\times10^4$

Worked example

Convert a small measurement

Write

0.0000836

in scientific notation, then round it to two significant figures.

Locate the first significant digit
The first non-zero digit is $8$ . Moving the decimal point four places right changes the coefficient to $8.36$ , so the power of ten must be negative four to preserve the original value.
$0.0000836=8.36\times10^{-5}$
Round to two significant figures
Keep the first two significant digits, $8$ and $3$ . The next digit is $6$ , so increase the $3$ to $4$ .
$8.36\times10^{-5}\approx8.4\times10^{-5}$

Answer: The scientific notation is

8.36\times10^{-5}

; to two significant figures it is

8.4\times10^{-5}

.

Check: The rounded value is

0.000084

, which is close to the original

0.0000836

and has the same scale.

Worked example

Multiply and round

Calculate

(3.2\times10^5)(4.6\times10^{-3})

. Give the answer to two significant figures.

Multiply coefficients and combine exponents
Multiply the coefficients and add the exponents. This gives a coefficient of $14.72$ and a power of ten of $2$ .
$(3.2\times4.6)\times10^{5+(-3)}=14.72\times10^2$
Normalize and round
The coefficient must be at least $1$ and less than $10$ . Move the decimal point one place left and increase the exponent by one. For two significant figures, the next digit after $1.5$ is $7$ , so round up.
$14.72\times10^2=1.472\times10^3\approx1.5\times10^3$

Answer: To two significant figures, the product is

1.5\times10^3

.

Check: The coefficients multiply to a value near

15

, and the combined power is near

10^2

, so a result near

1{,}500

is reasonable.

Worked example

Estimate a contextual quantity

A delivery service transports

48

boxes, each with a mass of

19.6

kg. Estimate the total mass by rounding each value to one significant figure, then calculate the total and give it to three significant figures.

Estimate using rounded inputs
To one significant figure, $48$ is about $50$ , and $19.6$ kg is about $20$ kg. Their product gives a quick scale check.
$50\times20\text{ kg}=1{,}000\text{ kg}$
Calculate with the original values
Multiply the original number of boxes by the mass of each box. The unit is kilograms because a count of boxes multiplied by kilograms per box gives kilograms.
$48\times19.6\text{ kg}=940.8\text{ kg}$
Round the calculated result
For three significant figures, keep $9$ , $4$ , and $0$ . The next digit is $8$ , so round upward. Writing the answer in scientific notation makes the three significant figures clear.
$940.8\text{ kg}\approx9.41\times10^2\text{ kg}$

Answer: The estimate is

1{,}000

kg, and the calculated total to three significant figures is

9.41\times10^2

kg.

Check: The calculated total is close to the estimate of

1{,}000

kg, so its magnitude is plausible.

Common mistakes and how to avoid them

Using a positive exponent for a small decimal such as

0.0007

.

Correction: The decimal point moves right to make a coefficient between

1

and

10

, so the exponent is negative:

7\times10^{-4}

.

Counting leading zeros as significant figures.

Correction: Leading zeros only locate the decimal point. In

0.00308

, the significant digits are

3

,

0

, and

8

.

Rounding each intermediate step before finishing a calculation.

Correction: Keep adequate digits during the calculation and round the final answer to the requested accuracy.

Adding scientific-notation coefficients while their powers of ten differ.

Correction: Rewrite both numbers with the same power of ten before adding or subtracting.

Treating a rounded approximation as exactly equal to the original.

Correction: Use

\approx

for a rounded value and reserve

=

for exact equality.

Lesson summary

Scientific notation is $a\times10^n$ , with $1\leq|a|<10$ .
A positive exponent shifts the decimal point right; a negative exponent shifts it left.
Significant figures begin at the first non-zero digit; apply the next-digit rule when rounding.
Approximation helps communicate suitable accuracy and check a result's scale.
Use calculator output carefully, retain intermediate precision, and report units when relevant.

Check your understanding

Question 1

Which is the scientific notation for

0.000562

?

$5.62\times10^{-4}$
$5.62\times10^4$
$56.2\times10^{-5}$
$0.562\times10^{-3}$

Show answer and explanation

5.62\times10^{-4}

Moving the decimal point four places right gives

5.62

, so the exponent is

-4

. The coefficient is in the required range.

Question 2

How many significant figures are in

0.02040

?

Two
Three
Four
Five

Show answer and explanation

Four

The leading zeros are not significant. The digits

2

,

0

,

4

, and the final decimal zero are significant, giving four.

Question 3

What is

7.846

rounded to three significant figures?

$7.84$
$7.85$
$7.8$
$7.846$

Show answer and explanation

7.85

Keep

7

,

8

, and

4

. The next digit is

6

, so the final kept digit increases, giving

7.85

.

Key terms

Scientific notation: A way to write a non-zero number as a coefficient multiplied by a power of ten, with the coefficient's absolute value at least one and less than ten.
Significant figures: The digits in a stated value that communicate its precision, counted from the first non-zero digit.
Approximation: A value close to an exact or more detailed value, often used after rounding.
Coefficient: The number multiplied by a power of ten in scientific notation.
Exponent: The small integer that states the power to which ten is raised.

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