Grade by Grade
B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Scientific Notation: Calculating with Very Large and Very Small Numbers

B2.1 A1About 70 minutes
Learning goals
  • I can write numbers in scientific notation and in standard form, using the sign and size of the exponent.
  • I can compare and order numbers written in scientific notation.
  • I can multiply and divide numbers in scientific notation, with and without a calculator.
  • I can read calculator E notation and solve real problems with very large and very small numbers.

Words to know

scientific notation
A number written as a × 10ⁿ, where 1 ≤ a < 10 (ignoring the sign) and n is an integer. Example: 3.2 × 10⁵.
coefficient
The number a in a × 10ⁿ. In 3.2 × 10⁵ the coefficient is 3.2.
standard form
A number written out with all its digits, like 320 000.
E notation
How calculators and spreadsheets show scientific notation: 3.2E5 means 3.2 × 10⁵.
order of magnitude
The power of 10 closest to a number's size. 1.27 × 10⁷ has order of magnitude 10⁷.

Remember from Grade 8

In Grade 8 you wrote large numbers like 150 000 000 km as 1.5 × 10⁸ and small numbers like 0.000 007 5 m as 7.5 × 10⁻⁶. You compared them and used metric prefixes. In Grade 9 you will connect this to what you just learned about powers, and you will calculate with numbers in scientific notation.

The sign of the exponent tells you big or small

  • Positive exponent (10⁵): the number is 10 or more (ignoring sign). 10⁵ = 100 000.
  • Exponent 0 (10⁰ = 1): the number is between 1 and 10.
  • Negative exponent (10⁻⁵): the number is between 0 and 1 (ignoring sign). 10⁻⁵ = 1/100 000 = 0.000 01.

The size of the exponent tells you how many places the decimal point moves.

00000045startnew point1234560.000 004 5 = 4.5 × 10⁻⁶The point moved 6 places right, so the exponent is −6.
To write 0.000 004 5 in scientific notation, move the decimal point until exactly one non-zero digit is in front of it. It moves 6 places right, so the number is 4.5 × 10⁻⁶. A small number gets a negative exponent.

Writing a number in scientific notation

  1. Move the decimal point so the coefficient is at least 1 and less than 10.
  2. Count the places moved: that is the size of the exponent.
  3. Big number (10 or more)? The exponent is positive. Small number (between 0 and 1)? The exponent is negative.
  4. A negative number keeps its sign on the coefficient: −45 000 = −4.5 × 10⁴.
Worked example

Example 1: Convert both ways

a) Write 384 400 km (the average Earth–Moon distance) and 0.000 72 m in scientific notation. b) Write 6.05 × 10⁻³ and 2.1 × 10⁶ in standard form.

  1. a) 384 400: move the point 5 places left to get 3.844. Big number, so 3.844 × 10⁵ km.
  2. 0.000 72: move the point 4 places right to get 7.2. Small number, so 7.2 × 10⁻⁴ m.
  3. b) 6.05 × 10⁻³: move the point 3 places left: 0.006 05.
  4. 2.1 × 10⁶: move the point 6 places right: 2 100 000.

3.844 × 10⁵ km; 7.2 × 10⁻⁴ m; 0.006 05; 2 100 000

Worked example

Example 2: Compare

Which is larger: 3.9 × 10⁻⁴ or 4.1 × 10⁻⁵?

  1. Compare the exponents first. −4 > −5, so 10⁻⁴ is 10 times as large as 10⁻⁵.
  2. In standard form: 3.9 × 10⁻⁴ = 0.000 39 and 4.1 × 10⁻⁵ = 0.000 041.

3.9 × 10⁻⁴ is larger, even though its coefficient is smaller.

10⁻⁶10⁻³10⁰10³10⁶10⁹10¹²metres (each labelled tick is 1000 times the one before)red blood cell7.5 × 10⁻⁶ mperson1.7 mCN Tower5.53 × 10² mEarth's diameter1.27 × 10⁷ mEarth to Sun1.5 × 10¹¹ m
Sizes from tiny to huge on one line (a powers-of-ten scale, so equal spaces mean multiplying by the same amount). The Earth–Sun distance is about 2 × 10¹⁶ times the width of a red blood cell: (1.5 × 10¹¹) ÷ (7.5 × 10⁻⁶) = 0.2 × 10¹⁷.

Multiplying and dividing in scientific notation

To multiply, multiply the coefficients and add the exponents of 10. To divide, divide the coefficients and subtract the exponents. (You will see why in the next lesson, on exponent laws: 10³ × 10⁴ = 10⁷ because that is 3 + 4 factors of 10.)

If the coefficient ends up outside 1 to 10, adjust it: 12 × 10³ = 1.2 × 10⁴ and 0.5 × 10³ = 5 × 10².

Worked example

Example 3: Multiply

Calculate (3 × 10⁵)(4 × 10⁻²).

  1. Multiply the coefficients: 3 × 4 = 12.
  2. Add the exponents: 10⁵ × 10⁻² = 10⁵⁺⁽⁻²⁾ = 10³.
  3. 12 × 10³ is not in scientific notation (12 ≥ 10). 12 = 1.2 × 10¹, so 12 × 10³ = 1.2 × 10⁴.

1.2 × 10⁴ (= 12 000)

Worked example

Example 4: How long does sunlight take to reach Earth?

The Sun is about 1.5 × 10⁸ km from Earth. Light travels about 3.0 × 10⁵ km/s. How many seconds does sunlight take to reach us?

  1. time = distance ÷ speed = (1.5 × 10⁸) ÷ (3.0 × 10⁵).
  2. Divide coefficients: 1.5 ÷ 3.0 = 0.5. Subtract exponents: 10⁸⁻⁵ = 10³.
  3. 0.5 × 10³ = 5 × 10² = 500 s.
  4. 500 s ÷ 60 ≈ 8.3 min.

About 5 × 10² s, a little over 8 minutes.

Calculator E notation

Most calculators have a key labelled EE, EXP or ×10ˣ for scientific notation. To enter 1.5 × 10⁸, press 1.5 EE 8. Answers that are very big or very small appear like 1.2E4 (meaning 1.2 × 10⁴) or 3.3E−5 (meaning 3.3 × 10⁻⁵). Do not type "× 10 ^" and EE together, or you will multiply by 10 twice.

Hands-on activity

Kitchen scientific notation

You need: a food package with a Nutrition Facts table, a ruler, a calculator.

  1. Find the mass of one serving in grams (for example 30 g). Write it in kilograms in scientific notation: 30 g = 0.030 kg = 3.0 × 10⁻² kg.
  2. Find the sodium in one serving in milligrams (for example 160 mg). Write it in grams and in scientific notation: 160 mg = 0.160 g = 1.6 × 10⁻¹ g.
  3. Measure the thickness of a stack of 100 sheets of printer paper (for example about 1 cm). Find the thickness of one sheet in metres in scientific notation: 1 cm ÷ 100 = 0.01 cm = 0.0001 m = 1 × 10⁻⁴ m.
  4. How many sheets would make a stack as tall as the CN Tower (553 m)? 553 ÷ (1 × 10⁻⁴) = 5.53 × 10⁶ sheets (if each sheet is 1 × 10⁻⁴ m thick).

Your own numbers will be different; check that the coefficient is between 1 and 10.

Just for fun

A googol is 10¹⁰⁰, a 1 followed by 100 zeros. The name of a certain search engine company comes from a playful misspelling of "googol". There are fewer atoms in the observable universe than a googol, so you will never count to one.

Your turn

1

Write 52 000 000 in scientific notation.

2

Write 0.000 091 in scientific notation.

3

Write 7.03 × 10⁻⁴ in standard form.

4

Write −6.4 × 10³ in standard form.

5

Which is greater, 8 × 10⁻⁷ or 2 × 10⁻⁶?

6

Calculate (2.5 × 10⁴)(6 × 10³).

7

Calculate (8.4 × 10⁻³) ÷ (2.1 × 10⁻⁷).

8

A calculator shows 2.7E−6. What does it mean in standard form?

More practice: warm-up

1

Write 4000 in scientific notation.

2

Write 0.06 in scientific notation.

3

Write 3 × 10⁵ in standard form.

4

Is 1.9 × 10⁻² greater or less than 1?

5

Fix 25 × 10⁴ so it is in scientific notation.

More practice: on your own

1

Order from least to greatest: 3.1 × 10⁴, 9.9 × 10³, 1.0 × 10⁵, 5.5 × 10⁻¹.

2

Canada's population passed 40 000 000 in 2023, and its area is about 9.98 × 10⁶ km². About how many people per square kilometre is that?

3

Calculate (5 × 10⁻³)(4 × 10⁻⁵).

4

A red blood cell is about 7.5 × 10⁻⁶ m wide. How many would fit side by side across 1 cm (1 × 10⁻² m)?

5

Light travels about 3.0 × 10⁵ km/s. How far does it travel in one minute?

Challenge

1

A water molecule has a mass of about 3.0 × 10⁻²⁶ kg. About how many molecules are in 1 kg of water (about 1 L)?

2

Explain why 0.5 × 10⁻³ is not in scientific notation, and fix it.

3

Light takes about 500 s to reach Earth from the Sun. Neptune is about 4.5 × 10⁹ km from the Sun. About how long does sunlight take to reach Neptune, in hours? (Light: 3.0 × 10⁵ km/s.)

Check your understanding

Quiz

1.

What is 0.000 038 in scientific notation?

2.

What is 4.06 × 10³ in standard form?

3.

Which number is largest?

4.

What is (6 × 10⁴)(5 × 10²) in scientific notation?

5.

A calculator shows 5.2E−3. What does it mean?

6.

In a × 10ⁿ, what does a negative n tell you about a positive number?

For parents and teachers

Big idea: Expectation B2.1 connects the sign and size of an exponent to the value of a power, and uses that to express numbers in scientific notation. Grade 8 introduced scientific notation; Grade 9 adds calculation and calculator skills that Grade 9 and 10 science courses use constantly.

Common difficulties:

  • Moving the decimal the wrong way or getting the sign of the exponent backward. Ask: "Is the number big or small?"
  • Leaving a coefficient outside 1 to 10 (like 15 × 10⁷).
  • Comparing coefficients before exponents. The exponent matters most.
  • Calculator entry: pressing × 10 then EE, which multiplies by 10 twice.

How to help: Find very large or very small numbers in the news or on food labels and ask your teen to write them in scientific notation. Check calculator entry with a known example: 3 EE 2 should show 300.