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Number Sense: Big Numbers, Tiny Numbers and Square Roots

Number sense means you can feel how big a number is, compare numbers and estimate quickly. Each place is 10 times the one on its right, so very large and very small numbers are written with powers of 10 (scientific notation: a Γ— 10ⁿ with 1 ≀ a < 10). All real numbers β€” whole numbers, negatives, fractions, decimals and irrationals like √2 β€” sit in order on one number line. A square root can be estimated by finding the two perfect squares around the number. Fractions, decimals and percents are three ways to write the same amount.

🎬 Step-by-step story

  1. Each step up the stairs is 10 times bigger: 10, 100, 1000… A million is 10⁢.
  2. Each step down is 10 times smaller: 0.1, 0.01, 0.001. These are 10⁻¹, 10⁻², 10⁻³.
  3. Scientific notation: a number from 1 to 10, times a power of 10. 3 800 000 = 3.8 Γ— 10⁢.
  4. All real numbers sit in order on one line: negatives, fractions, √2 and Ο€ too.
  5. To estimate √50, find the squares around it: 49 and 64. So √50 is a bit more than 7.
  6. Free play: slide n and see between which two squares it sits.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

πŸ€” Common doubts, cleared

Why is 10⁻³ not negative?

The minus in the power means divide, not subtract. Going down the stairs, each step is a tenth of the last, but still above zero.

Why must the first part be between 1 and 10?

So every number has only one way to be written. 38 Γ— 10⁡ and 3.8 Γ— 10⁢ are equal, but only the second is standard.

How can βˆ’2 be smaller than βˆ’1/2?

On the line, βˆ’2 is further left. Owing β‚Ή2 is worse than owing 50 paise.

Where does √2 go on the number line?

√2 β‰ˆ 1.414, so between 1.4 and 1.5. It has a fixed place even though its decimals never end.

Why is √50 close to 7 and not 7.5?

50 is just 1 more than 49 but 14 less than 64, so the root sits near 7.

Very large and very small numbers

Our number system is based on 10. Each place is 10 times the place on its right.

A negative power does not make a number negative. It means "divide by 10 that many times". So 10⁻² = 0.01, a small positive number.

Scientific notation

Write the number as a Γ— 10ⁿ, where a is at least 1 and less than 10. Move the decimal point until one non-zero digit is left of it, and count the jumps.

Scientists, engineers and phone screens (for example 1.2e9) all use this short form.

Ordering numbers in the real number system

The real numbers are every number that has a place on the number line:

To order them, change each one to a decimal and compare place by place. On the line, smaller is always to the left. With negatives, the number farther from zero is smaller: βˆ’5 < βˆ’2.

Estimating and calculating square roots

The square root of n is the number that times itself gives n. √49 = 7 because 7 Γ— 7 = 49. Numbers like 1, 4, 9, 16, 25… are perfect squares.

For other numbers, estimate:

  1. Find the perfect squares just below and just above: 49 < 50 < 64.
  2. So 7 < √50 < 8.
  3. 50 is much nearer 49, so √50 is just above 7. Try 7.1Β² = 50.41. Good enough: √50 β‰ˆ 7.1.

A calculator gives √50 = 7.071… In real life, a square garden of area 50 mΒ² has a side of about 7.1 m.

Fractions, decimals and percents

One amount, three ways to write it:

To change a fraction to a decimal, divide the top by the bottom. To change a decimal to a percent, multiply by 100. To compare β…— and 0.62, change both to decimals: 0.60 < 0.62.

Try it

Take a 1-litre bottle. Fill it a quarter. Write the amount as a fraction, a decimal (in litres) and a percent. Then use the slider in the 3D to find √ of your age.

Key formulas and definitions

Worked examples

1. Write 4 600 000 in scientific notation.

Put the point after the first digit: 4.6. The point moved 6 places left, so 4 600 000 = 4.6 Γ— 10⁢.

2. Write 0.000 035 in scientific notation.

Move the point right until it sits after 3: 3.5. It moved 5 places right, so the power is βˆ’5: 3.5 Γ— 10⁻⁡.

3. Write 2.07 Γ— 10⁴ as an ordinary number.

Move the point 4 places right: 2.07 β†’ 20 700.

4. Order from least to greatest: 1.5, √3, 7/4, βˆ’2, βˆ’2.5.

Decimals: 1.5, √3 β‰ˆ 1.732, 7/4 = 1.75, βˆ’2, βˆ’2.5. Order: βˆ’2.5, βˆ’2, 1.5, √3, 7/4.

5. Estimate √90 to one decimal place.

81 < 90 < 100, so 9 < √90 < 10. 90 is nearer 81. Try 9.5Β² = 90.25, a bit too big. Try 9.4Β² = 88.36, too small. So √90 β‰ˆ 9.5 (calculator: 9.487).

6. A square floor has area 200 mΒ². About how long is one side?

196 = 14Β² and 225 = 15Β², so the side is a little more than 14 m. 14.1Β² = 198.81, 14.2Β² = 201.64. Side β‰ˆ 14.1 m.

Common mistakes

Practice quiz

1. 10⁡ equals:
2. 0.0006 in scientific notation is:
3. √40 lies between:
4. Which is an irrational number?
5. 0.35 as a percent is:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is number sense in maths?

It is a feel for numbers: knowing how big they are, comparing them, estimating answers and choosing a good way to calculate.

How do you estimate a square root?

Find the perfect squares on either side, say the root lies between their roots, then test a decimal by squaring it.

Why do we use scientific notation?

It makes very large or very small numbers short, easy to compare and easy to use in calculations.

Where this is taught

Canada (Ontario)Grade 8B. Number

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