Chapter 4 of 8

All 8 chapters
  1. 1Laws of powers, readunderstood
  2. 2Zero and negative powers, readunderstood
  3. 3Roots and fractional powers, readunderstood
  4. 4Standard form, readunderstood
  5. 5Calculating in standard form, readunderstood
  6. 6Rational and irrational numbers, readunderstood
  7. 7Surds, readunderstood
  8. 8Accuracy and bounds, readunderstood

M4.1.4 Chapter 4

Standard form

Which of these is standard form for 340 000 000?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Write 340 000 000 as 3.4 × 10⁸
  • Write 0.0005 as 5 × 10⁻⁴
  • Order numbers written in standard form

Ana reads 340 000 000 and 0.0005 in a table. Both are hard to read and hard to compare. Scientists write them in a short, fixed way called standard form.

A number is in standard form when it is written A × 10ⁿ. Here A is at least 1 and less than 10, and n is a whole number, which can be negative.

Draw it first

Slide the decimal point

  1. The digits 340000000 in nine boxes. A black dot is the decimal point at the right end. A ring shows where it moves. Step 1: the ring is 1 place left, with the label 1 place left. Step 2: the ring is 5 places left, with the label 5 places left. Step 3: the ring is 8 places left, after the 3, with the label 8 places left.34000000011 place left

    1 The point of 340 000 000 is at the right end. Slide it one place left.

  2. The digits 340000000 in nine boxes. A black dot is the decimal point at the right end. A ring shows where it moves. Step 1: the ring is 1 place left, with the label 1 place left. Step 2: the ring is 5 places left, with the label 5 places left. Step 3: the ring is 8 places left, after the 3, with the label 8 places left.340000000123455 places left

    2 Slide it five places left. The digits stay where they are.

  3. The digits 340000000 in nine boxes. A black dot is the decimal point at the right end. A ring shows where it moves. Step 1: the ring is 1 place left, with the label 1 place left. Step 2: the ring is 5 places left, with the label 5 places left. Step 3: the ring is 8 places left, after the 3, with the label 8 places left.340000000123456788 places left

    3 Slide it eight places left, until it is after the 3. Now the number reads 3.4.

The point moved 8 places left. The number got smaller by a factor of 10 each place. To keep the same size, multiply by 10 eight times. That is 10⁸, a 1 followed by eight zeros.

The number 10⁸ written as a 1 and eight zeros in nine boxes, with a bracket under the zeros marked 8 zeros.10⁸1000000008 zeros
340 000 000 = 3.4 × 100 000 000 = 3.4 × 10⁸.

Worked example

Write 340 000 000 in standard form.

  1. Put the point after the first digit that is not zero: 3.4.
  2. The point moved 8 places left.
  3. Moving left made the number smaller, so multiply by 10⁸ to put the size back.
  4. So 340 000 000 = 3.4 × 10⁸.

Answer: 340 000 000 = 3.4 × 10⁸

Small numbers work the other way. For 0.0005 the point slides to the right, until it is after the first digit that is not zero.

The digits 00005 in five boxes. A black dot is the decimal point after the first 0. A ring shows it moved to the right end, after the 5, with the label 4 places right.0000512344 places right
0.0005 has its point moved 4 places right. That made it bigger, so the power of ten is negative.

Worked example

Write 0.0005 in standard form.

  1. The first digit that is not zero is 5. Put the point after it: 5.
  2. The point moved 4 places right.
  3. Moving right made the number bigger, so the power of ten is negative: 10⁻⁴.
  4. So 0.0005 = 5 × 10⁻⁴.

Answer: 0.0005 = 5 × 10⁻⁴

So a number of 10 or more has a positive power. A number below 1 has a negative power. A number that is at least 1 and less than 10 has the power 0, because 10⁰ = 1.

A table with the head Number and Standard form. Rows: 6 000, 6 × 10³; 72 000, 7.2 × 10⁴; 0.3, 3 × 10⁻¹; 0.0045, 4.5 × 10⁻³.NumberStandard form6 0006 × 10³72 0007.2 × 10⁴0.33 × 10⁻¹0.00454.5 × 10⁻³
Each first number is at least 1 and less than 10.

To order numbers in standard form, compare the powers of ten first. A bigger power means a bigger number. If the powers match, compare the first numbers.

Worked example

Put these in order, smallest first: 6.1 × 10⁴, 9.2 × 10³, 2.5 × 10⁴.

  1. Look at the powers of ten first. The powers are 4, 3, 4.
  2. The smallest power is 3, so 9.2 × 10³ is the smallest number.
  3. Two numbers share the power 4. Compare 6.1 and 2.5: the smaller is 2.5.
  4. From smallest to largest: 9.2 × 10³, 2.5 × 10⁴, 6.1 × 10⁴.

Answer: 9.2 × 10³, 2.5 × 10⁴, 6.1 × 10⁴

See it three ways

With things

Write 1 and then three zeros. That is 1 000, which is 10 × 10 × 10.

As a picture

The number 10³ written as a 1 and three zeros in four boxes, with a bracket under the zeros marked 3 zeros.10³10003 zeros

In symbols

6 000 = 6 × 1 000 = 6 × 10³.

Try it

Slide the point on paper

You need: paper and a pencil.

  1. Write 52 000 000 on the paper. Mark the decimal point at the right end.
  2. Draw an arrow from that point to the place after the 5.
  3. Count the places the point moved.

What you will see: The point moved 7 places left. So 52 000 000 = 5.2 × 10⁷.

If you cannot try it, this is what happens.
The digits 52000000 in eight boxes. A black dot is the decimal point at the right end. A ring shows it moved to after the 5, with the label 7 places left.5200000012345677 places left

A wrong idea many people have

34 × 10⁷ is standard form for 340 000 000, because it equals it.

Why it feels right: It has a power of ten, and the sum 34 × 10 000 000 does give 340 000 000.

The test that breaks it: Standard form has one rule for the first number: at least 1 and less than 10. The number 34 breaks it. So does 0.34 × 10⁹, which is also equal to 340 000 000.

What is true: 3.4 × 10⁸ is the one standard form. Many forms can equal a number, but only one has A at least 1 and less than 10.

A wrong idea many people have

0.0005 = 5 × 10⁴, because the point moves four places.

Why it feels right: The point does move four places, and 4 is the count.

The test that breaks it: Work out 5 × 10⁴ = 50 000. That is far bigger than 0.0005.

What is true: A number below 1 needs a negative power. 0.0005 = 5 × 10⁻⁴.

Key words

standard form
a number written as A × 10ⁿ, with A at least 1 and less than 10, and n a whole number.

Back to your guess

The answer is A: 3.4 × 10⁸, because 3.4 × 100 000 000 is 340 000 000.

All three equal 340 000 000. Only 3.4 × 10⁸ is standard form, because the first number must be at least 1 and less than 10. One rule means everybody writes a number the same way.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which is 52 000 in standard form?

    Tap your answer.

    1. Not this one. This equals 52 000, but 52 is not between 1 and 10. Standard form needs the first number at least 1 and less than 10. Go to M4.1.4. Try another.

    2. Not this one. You counted the three zeros. Count the places the point moves instead: from the end of 52 000 to after the 5 is 4 places. Go to M4.1.4. Try another.

    3. Yes, that is right. The point moves 4 places left, from the end of 52 000 to after the 5. So 52 000 = 5.2 × 10⁴.

    4. Not this one. This equals 52 000, but 0.52 is below 1. The first number must be at least 1. Go to M4.1.4. Try another.

    See the answer

    5.2 × 10⁴

    The point moves 4 places left, from the end of 52 000 to after the 5. So 52 000 = 5.2 × 10⁴.

    • If you chose A: This equals 52 000, but 52 is not between 1 and 10. Standard form needs the first number at least 1 and less than 10. Go to M4.1.4.
    • If you chose B: You counted the three zeros. Count the places the point moves instead: from the end of 52 000 to after the 5 is 4 places. Go to M4.1.4.
    • If you chose D: This equals 52 000, but 0.52 is below 1. The first number must be at least 1. Go to M4.1.4.

  2. The picture shows the point of 0.0072 sliding. Which is 0.0072 in standard form?

    The digits 00072 in five boxes. A black dot is the decimal point after the first 0. A ring shows it moved to after the 7, with the label 3 places right.000721233 places right
    Tap your answer.

    1. Not this one. The point moved right, so the number got bigger by 10 each place. To undo that the power must be negative, not positive. Go to M4.1.4. Try another.

    2. Not this one. This equals 0.0072, but 72 is not between 1 and 10. Slide the point past the 7, so the first number is 7.2. Go to M4.1.4. Try another.

    3. Not this one. You counted all four places after the point. The point only moves to after the 7, which is 3 places, so the power is −3. Go to M4.1.4. Try another.

    4. Yes, that is right. The point moves 3 places right, to after the 7. The number gets bigger, so the power is negative: 7.2 × 10⁻³.

    See the answer

    7.2 × 10⁻³

    The point moves 3 places right, to after the 7. The number gets bigger, so the power is negative: 7.2 × 10⁻³.

    • If you chose A: The point moved right, so the number got bigger by 10 each place. To undo that the power must be negative, not positive. Go to M4.1.4.
    • If you chose B: This equals 0.0072, but 72 is not between 1 and 10. Slide the point past the 7, so the first number is 7.2. Go to M4.1.4.
    • If you chose C: You counted all four places after the point. The point only moves to after the 7, which is 3 places, so the power is −3. Go to M4.1.4.

  3. Which of these is the largest number?

    Tap your answer.

    1. Not this one. You compared the first numbers, 9.5 and 1.2. Compare the powers of ten first. 10⁵ is ten times 10⁴, so 1.2 × 10⁵ = 120 000 beats 9.5 × 10⁴ = 95 000. Go to M4.1.4. Try another.

    2. Yes, that is right. The biggest power of ten is 5, and only one number has it. The other power of 5 has a minus sign, so that number is tiny. So 1.2 × 10⁵ is the largest.

    3. Not this one. The minus sign matters. 10⁻⁵ is 1/100 000, so 1.2 × 10⁻⁵ is a tiny number, 0.000012. Look at the sign of the power as well as its size. Go to M4.1.4. Try another.

    See the answer

    1.2 × 10⁵

    The biggest power of ten is 5, and only one number has it. The other power of 5 has a minus sign, so that number is tiny. So 1.2 × 10⁵ is the largest.

    • If you chose A: You compared the first numbers, 9.5 and 1.2. Compare the powers of ten first. 10⁵ is ten times 10⁴, so 1.2 × 10⁵ = 120 000 beats 9.5 × 10⁴ = 95 000. Go to M4.1.4.
    • If you chose C: The minus sign matters. 10⁻⁵ is 1/100 000, so 1.2 × 10⁻⁵ is a tiny number, 0.000012. Look at the sign of the power as well as its size. Go to M4.1.4.

  4. Which of these is NOT in standard form?

    Tap your answer.

    1. Not this one. This is in standard form. 4.7 is at least 1 and less than 10, and a negative power is allowed. Go to M4.1.4. Try another.

    2. Not this one. This is in standard form. The first number can equal 1, and 10⁰ = 1. Go to M4.1.4. Try another.

    3. Yes, that is right. The first number must be at least 1 and less than 10. 12 is too big. As standard form this number is 1.2 × 10⁴.

    See the answer

    12 × 10³

    The first number must be at least 1 and less than 10. 12 is too big. As standard form this number is 1.2 × 10⁴.

    • If you chose A: This is in standard form. 4.7 is at least 1 and less than 10, and a negative power is allowed. Go to M4.1.4.
    • If you chose B: This is in standard form. The first number can equal 1, and 10⁰ = 1. Go to M4.1.4.

  5. Which number is 3 × 10⁻⁵ written out?

    Tap your answer.

    1. Yes, that is right. The power −5 slides the point 5 places left from after the 3: 0.00003.

    2. Not this one. The 3 is one place too far left. The power −5 puts the 3 in the fifth place after the point. Go to M4.1.4. Try another.

    3. Not this one. A negative power makes a small number, not a big one. This answer is 300 000. Go to M4.1.4. Try another.

    See the answer

    0.00003

    The power −5 slides the point 5 places left from after the 3: 0.00003.

    • If you chose B: The 3 is one place too far left. The power −5 puts the 3 in the fifth place after the point. Go to M4.1.4.
    • If you chose C: A negative power makes a small number, not a big one. This answer is 300 000. Go to M4.1.4.

  6. From an earlier chapter

    Work out 27 to the power 2/3.

    See the answer

    9

    The bottom number 3 says: take the cube root, which is 3. The top number 2 says: square it. 3 × 3 = 9.

  7. Write 6.4 × 10⁻² as a decimal.

    See the answer

    0.064

    The power −2 slides the point 2 places left from after the 6: 0.064.

  8. How many digits does 3.4 × 10⁸ have when it is written out in full?

    See the answer

    9

    The power 8 slides the point 8 places right: the 3 stays first and the number is 340 000 000. That has 9 digits.

  9. A virus particle is about 0.00000012 m across. Write 0.00000012 in standard form. What is the first number, A?

    See the answer

    1.2

    The first digit that is not zero is 1. Put the point after it: 1.2. The point moves 7 places right, so 0.00000012 = 1.2 × 10⁻⁷.

Keep it

Copy this into your exercise book.

Standard form is A × 10ⁿ, with A at least 1 and below 10. Slide the point to make A and count the places. Left is a positive power, right is negative.

Draw: The digits 340000000 with the point sliding 8 places left, and 3.4 × 10⁸ written under it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (755 KB). It opens on any phone.