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

Comparing and Using Very Large and Very Small Numbers

B1.1 A1About 40 minutes

Learning goals

I can compare and order numbers written in scientific notation.
I can explain how many times as large one power of ten is than another.
I can connect metric prefixes like kilo, mega, giga, milli and micro to powers of ten.
I can describe how very large and very small numbers are used in everyday life.

Words to know

order of magnitude
Roughly how big a number is, shown by its power of 10. In scientific notation it is the exponent: 4 × 10⁶ and 7 × 10⁶ have the same order of magnitude; 4 × 10⁹ is three orders of magnitude bigger.

metric prefix
A word part that tells the power of ten: kilo = 10³, mega = 10⁶, giga = 10⁹, milli = 10⁻³, micro = 10⁻⁶, nano = 10⁻⁹.

byte
A unit of computer storage. Phone and computer storage is often measured in gigabytes (GB).

Which is bigger?

Which is greater: 9.8 × 10⁵ or 1.2 × 10⁶? The coefficient 9.8 is bigger than 1.2, but that is not the whole story.

9.8 × 10⁵ = 980 000 and 1.2 × 10⁶ = 1 200 000. So 1.2 × 10⁶ is greater. The power of 10 matters most!

Drawn to scale: 9.8 × 10⁵ has the bigger coefficient, but 1.2 × 10⁶ has the bigger exponent, so its bar is longer. The exponent matters most.

How to compare numbers in scientific notation

For positive numbers written correctly in scientific notation:

Compare the exponents first. The number with the greater exponent is greater. (Remember: −3 is greater than −5.)
If the exponents are the same, compare the coefficients.

Why does this work? Each coefficient is at least 1 and less than 10. So a number with exponent 6 is somewhere from 1 000 000 up to (but not including) 10 000 000, and every number with exponent 5 is less than 1 000 000.

How many times as large? Each step up in the exponent multiplies by 10. So 10⁸ is 10 × 10 × 10 = 1 000 times as large as 10⁵, because it has three more factors of 10.

Worked example

Example 1: Order large numbers

Order from least to greatest: 4.5 × 10⁶, 9.8 × 10⁵, 4.05 × 10⁶, 1.2 × 10⁷

Sort by exponent first: 10⁵ is smallest, then 10⁶, then 10⁷.
Two numbers have exponent 6: 4.05 and 4.5. Compare coefficients: 4.05 < 4.50.
Check in standard form: 980 000 < 4 050 000 < 4 500 000 < 12 000 000. ✓

9.8 × 10⁵, 4.05 × 10⁶, 4.5 × 10⁶, 1.2 × 10⁷

Worked example

Example 2: Compare small numbers

Which is greater: 2.6 × 10⁻³ or 8.1 × 10⁻⁴?

Compare exponents: −3 is greater than −4.
So 2.6 × 10⁻³ is greater, even though 2.6 < 8.1.
Check: 2.6 × 10⁻³ = 0.002 6 and 8.1 × 10⁻⁴ = 0.000 81. 0.002 6 > 0.000 81. ✓

2.6 × 10⁻³ is greater.

Worked example

Example 3: How many times as large?

A city has about 6 × 10⁵ people. A town has about 2 × 10³ people. How many times as many people does the city have?

Divide the coefficients: 6 ÷ 2 = 3.
Compare the powers: 10⁵ is 10² = 100 times 10³ (two more factors of 10).
Multiply: 3 × 100 = 300. Check: 600 000 ÷ 2 000 = 300. ✓

The city has about 300 times as many people.

Metric prefixes on a power-of-ten scale. Each step to the right is 1 000 times as large. Equal spaces on this scale mean multiplying by 1 000, not adding.

Where you see it	Example	Scientific notation
Computer storage	A 64 GB phone (1 GB = 10⁹ bytes)	6.4 × 10¹⁰ bytes
Distances in space	Earth to the Sun, about 150 million km	1.5 × 10⁸ km
Population	The world, about 8 billion people	8 × 10⁹ people
Medicine labels	A 500 mg tablet	5 × 10⁻¹ g
Science	A red blood cell, about 7.5 µm across	7.5 × 10⁻⁶ m

Very large and very small numbers are all around us. Many of the numbers are rounded estimates.

Think like a mathematician

When you compare two numbers in different forms, put them in the same form first. Either change both to scientific notation or both to standard form. Then estimate: "Is my answer the right size?" If one number is a billion and the other is a million, the answer should be about 1 000 times.

Just for fun: count to a billion?

If you counted out loud, one number every second without stopping, you would reach one million (10⁶) in about 11.6 days. Reaching one billion (10⁹) would take about 31.7 years. Better pack snacks.

Hands-on activity

Big and small number hunt

You need: packages and labels around the home (cereal boxes, a phone or tablet's storage settings, the internet plan on a bill, a measuring tape), paper and a pencil.

Safety: Only read medicine and cleaning-product labels with a grown-up. Do not open any containers.

Find five numbers with metric prefixes, like 128 GB, 2 kg, 750 mL, 200 mg or 5 km.
Change each to the base unit (bytes, grams, litres, metres) using the prefix scale.
Write each in scientific notation.
Order your five numbers from smallest to largest within the same base unit.

Model answers: 128 GB = 128 000 000 000 bytes = 1.28 × 10¹¹ bytes. 750 mL = 0.75 L = 7.5 × 10⁻¹ L. 200 mg = 0.2 g = 2 × 10⁻¹ g. 2 kg = 2 000 g = 2 × 10³ g. (Some computer settings use 1 GB = 1 024 MB instead of 1 000 MB. For this activity, use powers of ten.)

Math talk: "Which prefix did you find most often? Why do you think products use prefixes instead of scientific notation?"

Your turn

Which is greater: 3 × 10⁸ or 9 × 10⁷?

Which is greater: 5.2 × 10⁻³ or 5.2 × 10⁻²?

Order from least to greatest: 7.1 × 10⁴, 2.3 × 10⁵, 7.01 × 10⁴, 9.9 × 10³

Order from least to greatest: 4 × 10⁻⁵, 3 × 10⁻⁴, 6 × 10⁻⁵, 1 × 10⁻³

Write 4.6 billion in scientific notation.

Write 25 micrometres in metres, in scientific notation.

How many times as large is 10⁷ as 10⁴?

More practice: warm-up

Which is greater: 10⁶ or 10⁹?

Which is greater: 10⁻² or 10⁻⁵?

1 km = 10ⁿ m. What is n?

1 mg = 10ⁿ g. What is n?

Write 2.5 million in scientific notation.

More practice: on your own

Which is smaller: 8.8 × 10⁻⁶ or 1.2 × 10⁻⁵?

Order from greatest to least: 6.5 × 10⁹, 6.5 × 10⁸, 7 × 10⁸, 1.1 × 10¹⁰

A tablet stores 64 GB. Using 1 GB = 10⁹ bytes, write its storage in bytes in scientific notation.

A sheet of paper is 1 × 10⁻⁴ m thick. How thick is a stack of 500 sheets? Give your answer in scientific notation and in centimetres.

How many times as large is 8 × 10⁶ as 2 × 10³?

Challenge

The world has about 8 × 10⁹ people. Canada has a little more than 4 × 10⁷ people. Using these round numbers, about how many times as many people live in the world as in Canada? About what percent of the world's people live in Canada?

Mia says 9 × 10⁻³ is greater than 2 × 10⁻² because 9 > 2. Explain her mistake.

Suppose a computer can do 3 × 10⁹ simple steps each second. How many steps could it do in one minute? Write your answer in scientific notation.

Check your understanding

Quiz

Which number is greatest?

Which number is smallest?

What does the prefix kilo mean?

Which is equal to 3.5 million?

How many times as large is 10⁹ as 10⁶?

Which list is in order from least to greatest?

For parents and teachers

Big idea: B1.1 asks students to compare very large and very small numbers and describe how they are used in everyday life. The key strategy: compare exponents first, then coefficients. Students also learn that each step in the exponent is a factor of 10, which explains metric prefixes.

Common difficulties:

Comparing coefficients before exponents (thinking 9 × 10⁻³ > 2 × 10⁻²).
Ordering negative exponents: −3 is greater than −5. A quick number-line sketch helps.
Mixing units when comparing (kilometres with metres). Change to the same unit first.

How to help: Do the number hunt together and talk about where big and small numbers show up: phone storage, data plans, distances on road signs, sizes in science articles. This lesson builds A1 skills too: making connections between math and everyday life and checking whether an answer is reasonable.