Grade by Grade
B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

Scientific Notation for Very Large Numbers

B1.1About 40 minutes
Learning goals
  • I can explain what scientific notation is and why scientists use it.
  • I can write a very large number in scientific notation.
  • I can change a number in scientific notation back to standard form.
  • I can tell whether a number is written correctly in scientific notation.

Words to know

standard form
A number written out with all its digits, like 150 000 000.
scientific notation
A way to write a number as a number from 1 up to (but not including) 10, multiplied by a power of 10. Example: 1.5 × 10⁸.
coefficient
The first part of a number in scientific notation. In 1.5 × 10⁸ the coefficient is 1.5. It must be at least 1 and less than 10.
power of ten
10 with an exponent, like 10⁸ = 100 000 000.

Remember from Grade 7

In Grade 7 you read and wrote numbers up to one billion and wrote them in expanded form with powers of ten, like 3 × 10⁸ + 4 × 10⁶. You also learned that 10⁶ means 10 × 10 × 10 × 10 × 10 × 10 = 1 000 000: the exponent tells how many zeros follow the 1. Today you will use powers of ten in a new way.

Too many zeros!

The Sun is about 150 000 000 km from Earth. Light travels about 300 000 km every second. One light-year (the distance light travels in a year) is about 9 460 000 000 000 km.

Writing all those zeros is slow, and it is easy to miss one. Missing a single zero makes a number 10 times too small. Scientists, engineers and computers use a shorter way: scientific notation.

150 000 000 = 1.5 × 100 000 000 = 1.5 × 10⁸

The Milky Way stretches across a starry night sky above a calm lake with pine trees on the shore.
Every one of those points of light is a star, and even the nearest one is about 40 000 000 000 000 km away. Distances like that are much easier to write in scientific notation. AI-generated image

What scientific notation looks like

A number in scientific notation has two parts:

coefficient × power of 10

  • The coefficient must be at least 1 and less than 10. So it has exactly one non-zero digit before the decimal point: 1.5, 3, 9.46, 7.08 are all fine.
  • The power of 10 tells how many places to move the decimal point.

Writing a large number in scientific notation:

  1. Put the decimal point right after the first non-zero digit. That gives the coefficient.
  2. Count how many places the decimal point moved to the left. That count is the exponent.
  3. Drop the zeros at the end of the coefficient (1.50 is just 1.5).
One light-year ≈ 9 460 000 000 000 km9.46×1012coefficientat least 1, less than 10basealways 10exponentthe point moved 12 places
A number in scientific notation has three parts. The coefficient is at least 1 and less than 10, the base is always 10, and the exponent counts how many places the decimal point moved.
150000000The decimal point hops 8 places to the left150 000 000 = 1.5 × 10⁸
The decimal point starts at the end of 150 000 000 (open circle). It hops 8 places to the left to sit after the first digit, 1 (filled circle). 8 hops means the exponent is 8.
Standard formThinkScientific notation
4 0004 × 1 0004 × 10³
52 0005.2 × 10 0005.2 × 10⁴
730 0007.3 × 100 0007.3 × 10⁵
6 080 0006.08 × 1 000 0006.08 × 10⁶
150 000 0001.5 × 100 000 0001.5 × 10⁸
The exponent counts the places the decimal point moved.
Worked example

Example 1: Large number → scientific notation

Write 4 700 000 in scientific notation.

  1. The first non-zero digit is 4. Put the decimal point after it: 4.7.
  2. Count the places the decimal point moved: from the end of 4 700 000 back to after the 4 is 6 places.
  3. So the exponent is 6. Check: 4.7 × 1 000 000 = 4 700 000. ✓

4 700 000 = 4.7 × 10⁶

Worked example

Example 2: Scientific notation → standard form

Write 8.03 × 10⁵ in standard form.

  1. The exponent 5 means multiply by 10⁵ = 100 000. That moves the decimal point 5 places to the right.
  2. 8.03 → 80.3 → 803 → 8 030 → 80 300 → 803 000. Fill empty places with zeros.
  3. Check: 8.03 × 100 000 = 803 000. ✓

8.03 × 10⁵ = 803 000

Worked example

Example 3: Fix a number that is not in scientific notation

Is 47 × 10⁵ in scientific notation? If not, fix it.

  1. The coefficient 47 is not less than 10, so this is not scientific notation.
  2. Write 47 as 4.7 × 10. Then 47 × 10⁵ = 4.7 × 10 × 10⁵.
  3. 10 × 10⁵ is one more factor of 10, so it is 10⁶.
  4. Check: 47 × 100 000 = 4 700 000, and 4.7 × 10⁶ = 4 700 000. ✓

47 × 10⁵ = 4.7 × 10⁶

Calculators show it too

Try 99 999 × 99 999 on a calculator. The answer is 9 999 800 001. Many calculators show very large answers in a short form like 9.9998E9 or 9.9998 × 10⁹. The E9 means "× 10⁹". (The calculator rounded the coefficient.) Knowing scientific notation helps you read these screens.

Watch out!

  • The coefficient must be at least 1 and less than 10. 0.5 × 10⁸ and 15 × 10³ are not in scientific notation.
  • The exponent is not always the number of zeros in the original number. 52 000 has 3 zeros, but it is 5.2 × 10⁴. Count the places the decimal point moved.
Just for fun: a kid-sized word for a giant number

The word googol means 1 × 10¹⁰⁰: a 1 followed by 100 zeros. It was made up by a 9-year-old, the nephew of the mathematician Edward Kasner. Try writing that one in standard form during a test. Actually, please don't.

Hands-on activity

How many grains of rice?

You need: a bag of rice (any size, the label shows its mass), a teaspoon, a small plate, a calculator.

  1. Scoop one level teaspoon of rice onto a plate. Count the grains. (Make groups of 10 to keep track.)
  2. Count how many level teaspoons fill a cup (or ask a grown-up which measure to use).
  3. Find how many cups of rice are in the bag. Ask a grown-up to help you estimate this from the bag size if you don't want to measure it all.
  4. Multiply to estimate the number of grains in the whole bag.
  5. Write your estimate in standard form and in scientific notation, rounded to one or two digits in the coefficient.

What to look for: answers will vary. For a full bag, your estimate will likely be in the tens of thousands or hundreds of thousands, so somewhere between 10⁴ and 10⁶. For example, 85 000 grains = 8.5 × 10⁴.

Math talk: "Why is it fine to round the coefficient for an estimate like this?"

Your turn

1

Write 6 000 000 in scientific notation.

2

Write 35 000 in scientific notation.

3

Write 920 000 000 in scientific notation.

4

Write 4.1 × 10⁵ in standard form.

5

Write 7.25 × 10⁷ in standard form.

6

Is 12.5 × 10³ in scientific notation? If not, fix it.

7

Canada's total area (land and fresh water) is about 9 980 000 km². Write this in scientific notation.

More practice: warm-up

1

Write 300 000 in scientific notation.

2

Write 50 000 000 in scientific notation.

3

Write 2 × 10³ in standard form.

4

Write 8.5 × 10⁴ in standard form.

5

Write 1 000 000 in scientific notation.

More practice: on your own

1

Write 6 470 000 in scientific notation.

2

Write 3.008 × 10⁹ in standard form.

3

Write 52 × 10⁶ in scientific notation.

4

Fix 0.9 × 10⁵ so it is in scientific notation.

5

A light-year is about 9.46 × 10¹² km. Write this in standard form.

Challenge

1

One online video has 2.5 million views. Another has 4 billion views. Write both in scientific notation. How many times as many views does the second video have?

2

The Sun is about 1.5 × 10⁸ km from Earth. Light travels about 3 × 10⁵ km per second. About how many seconds does sunlight take to reach Earth? About how many minutes?

3

Without writing them in standard form, decide which is greater: 9.9 × 10⁶ or 1.1 × 10⁷. Explain.

Check your understanding

Quiz

1.

What is 7 300 000 in scientific notation?

2.

What is 4.06 × 10⁴ in standard form?

3.

Which number is written in scientific notation?

4.

800 000 000 = 8 × 10ⁿ. What is n?

5.

A calculator shows 3.2E9. What number is this?

6.

In scientific notation a × 10ⁿ, what must be true about a?

For parents and teachers

Big idea: The Ontario curriculum (B1.1) asks Grade 8 students to represent and compare very large and very small numbers, including through the use of scientific notation, and to describe how these numbers are used in everyday life. This lesson handles very large numbers; the next lesson handles very small ones.

Common difficulties:

  • Counting zeros instead of decimal places. 52 000 has three zeros but is 5.2 × 10⁴. Have your child mark the decimal point at the end of the number and draw the hops.
  • Coefficients that are too big or too small, like 47 × 10⁵. Ask: "Is the first part at least 1 and less than 10?"
  • Losing zeros when converting back. Encourage a check by multiplying: 8.03 × 100 000.

How to help: Look for big numbers together in news stories, science books or sports statistics and rewrite them in scientific notation. If you have a calculator or phone calculator, multiply large numbers until it shows "E" notation and read the screen together.