Year 9 · Unit 1 Real numbers and indices · Lesson 6 of 7

Very large, very small: a number times a power of ten

Slide the decimal point until one non-zero digit sits in front of it, and count the places. The count is the power of ten, positive for large numbers and negative for small ones.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 5) What is 10−3 as a decimal?
    Answer0.001, because 10−3 = 1/1000.
  2. (Year 7, Unit 1, lesson 1) What is 4.5 × 100?
    Answer450. Each digit moves two places to the left.
  3. (Lesson 3) Write 104 × 102 as one power of 10.
    Answer106, because 4 + 2 = 6.
Words used in this lesson
scientific notation
a number written as a times a power of ten, with a from 1 up to 10. For example: 4.5 × 104.
decimal point
the dot that separates whole numbers from parts. For example: the point in 0.0305.
exponent
the power of ten in scientific notation. For example: −4 in 7.2 × 10−4.
significant digit
a digit that is part of the value, not a place-holder zero. For example: 4 and 5 in 45000.
ordinary number
a number written in the usual way, with no power of ten. For example: 45000.
order of magnitude
roughly how many powers of ten a number has. For example: 105 is larger than 103.

Counting the places

The distance from the Earth to the Sun is about 150000000 km. A red blood cell is about 0.000007 m across. Writing those out takes care, and comparing them takes more. Scientific notation writes each as a number between 1 and 10 times a power of ten. Predict first: how would you write 150000000 as a number from 1 up to 10 times a power of 10? Then use the figure to test the method on other numbers.

45000
4.5a, between 1 and 10
4the power of 10
4.5 × 10⁴scientific notation

45000 = 4.5 × 10⁴: the point moves 4 places left.

Show me the steps
  1. 45000 = 4.5 × 10⁴: the point moves 4 places left.
  2. 0.00072 = 7.2 × 10⁻⁴: the point moves 4 places right.
  3. 6300000 = 6.3 × 10⁶: the point moves 6 places left.
  4. 0.0305 = 3.05 × 10⁻²: the point moves 2 places right, and the 0 after the 3 stays in the number.
  5. 120 = 1.2 × 10²: the point moves 2 places left, and the trailing zero is dropped.
  6. 0.000000041 = 4.1 × 10⁻⁸: the point moves 8 places right.

The method is always the same: slide the point so that exactly one non-zero digit sits in front of it, and count the places it moved. A move to the left means the original number was large, so the power of 10 is positive. A move to the right means it was small, so the power is negative. Notice that the exponent counts places moved, not zeros: 3200000 has five zeros but the point moves 6 places, and 0.00071 has three zeros after the point but the point moves 4. A zero inside the digits, as in 0.0305, stays in a: that number has three zeros written, and the point moves only 2 places. For the prediction: 150000000 moves 8 places left to sit after the 1, so it is 1.5 × 108; and 0.000007 moves 6 places right, so it is 7 × 10−6.

Scientific notation writes a number as a × 10n, where 1 ≤ a < 10 and n is an integer. A large number has a positive n, and a number between 0 and 1 has a negative n. To compare two numbers, compare the exponents first; if they are equal, compare the values of a.

Some exam courses call this standard form. It is not the standard form of a straight line in Unit 3.

A common mistake: leaving a outside the range. 52 × 10−4 has the right value, 0.0052, but it is not in scientific notation, because 52 is not below 10. Move the point one place: 5.2 × 10−3. A negative exponent does not make the number negative either; it makes it small.

Worked example: large and small

Write 380000 and 0.0000046 in scientific notation.

  1. 380000: move the point 5 places left, to sit after the 3: 3.8. So 380000 = 3.8 × 105.
  2. 0.0000046: move the point 6 places right, to sit after the 4: 4.6. So 0.0000046 = 4.6 × 10−6.
  3. Check each by moving the point back: 5 places right from 3.8 gives 380000, and 6 places left from 4.6 gives 0.0000046.

Worked example: ordering

Put 3.2 × 10−3, 8.1 × 10−5 and 4.7 × 10−3 in order from smallest to greatest.

  1. Compare the exponents first: −5 is less than −3, so 8.1 × 10−5 is the smallest.
  2. The other two have the same exponent, −3, so compare 3.2 and 4.7: 3.2 is less.
  3. The order is 8.1 × 10−5, 3.2 × 10−3, 4.7 × 10−3, even though 8.1 is the biggest of the three values of a.

Your go. The first step is given. Write 7.1 × 106 as an ordinary number.

  1. The power is positive, so the number is large: move the point 6 places right.
  2. Fill the empty places with zeros, then check.
Check the answer

7.1 has one digit after the point, so 1 of the 6 places is used and 5 zeros are added: 7100000.

Check before you go on

1. Which is 0.0000072 in scientific notation?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Write 54000000 as a × 10n with 1 ≤ a < 10. What is n?

3. Write 0.000032 as a × 10n with 1 ≤ a < 10. What is n?

Level 2 · Building

4. Write 3.7 × 10−3 as an ordinary decimal.

5. How many digits does 3.4 × 109 have when it is written out as an ordinary number?

6. Which list is in order from smallest to greatest?

Level 3 · Stretch

7. 45.6 × 105 is not in scientific notation. Written properly it is a × 10n. What is n?

8. How many of these are in scientific notation? 3.5 × 104, 0.8 × 10−2, 12 × 103, 6 × 100, 9.99 × 10−7, 10 × 105

Something to think about

In calculator notation, a screen shows 6.3E-5 for the answer to a sum. Write this as an ordinary number. Is it more or less than 1?

One way to think about it

6.3E-5 means 6.3 × 10−5 = 0.000063. It is far below 1, because the power of ten is negative. The E stands for “times ten to the power”, so it is the same scientific notation.

Exit ticket

9. Neha writes 0.00052 = 0.52 × 10−3 and says it is scientific notation, because the value is right. Is she correct?