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To convert a number into standard form, split the number into two parts - a number multiplied by a power of 10.

Powers of 10

Standard form uses the fact that the decimal place value system is based on powers of 10:

• \(10^{0} = 1\)

• \(10^{1} = 10\)

• \(10^{2} = 100\)

• \(10^{3} = 1000\)

• \(10^{4} = 10000\)

• \(10^{5} = 100000\)

• \(10^{6} = 1000000\)

Large numbers

Example

Write 50,000 in standard form.

Solution

50,000 can be written as \(5 \times 100000\)

\(10000 = 10 \times 10 \times 10 \times 10 = 10^{4}\)

So, \(500000 = 5 \times 10^{4}\)

It's important to remember that a number written in standard form is presented as \(A \times 10^{n}\), where \(A\) is a number bigger than or equal to 1 and less than 10. It can be any positive or negative whole number.

So, \(34 \times 10^{7}\) is not in standard form as the first number is not between 1 and 10.

To correct this, divide 34 by 10. To balance out the division of 10, multiply the second part by 10, which gives \(10^{8}\).

\(34 \times 10^{7}\) and \(3.4 \times 10^{8}\) are equivalent, but only the second is written in standard form.

Example

What is 87, 000 in standard form?

Solution

87,000 can be written as \(8.7 \times 10000\).

\(10000 = 10 \times 10 \times 10 \times 10 = 10^{4}\)

So, \(87000 = 8.7 \times 10^{4}\).

This process can be simplified by considering where the first digit is compared to the units column.

Example

• 3, 000,000 = \(3 \times 10^{6}\) because the 3 is six places away from the units column.

• 36, 000 = \(3.6 \times 10^{4}\) because the 3 is four places away from the units column.

It is useful to look at patterns to try and understand negative indices.

Standard form uses the fact that the decimal place value system is based on powers of 10:

Standard form uses the fact that the decimal place value system is based on powers of 10:

• \(10^{0} = 1\)

• \(10^{–1} = 0.1\)

• \(10^{–2} = 0.01\)

• \(10^{–3} = 0.001\)

• \(10^{–4} = 0.0001\)

• \(10^{–5} = 0.00001\)

• \(10^{–6} = 0.000001\)

Notice that a negative power does not mean that the number is negative. It means that we have gone from multiplying by 10 to dividing by 10.

Example

Write 0.0005 in standard form.

Solution

0.0005 can be written as \(5 \times 0.0001\).

\(0.0001 = 10^{–4}\)

So, \(0.0005 = 5 \times 10^{–4}\)

This process can also be simplified by considering where the first digit is compared to the units column.

Example

• 0.03 = \(3 \times 10^{–2}\) because the 3 is two places away from the units column.

• 0.000039 = \(3.9 \times 10^{–5}\) because the 3 is five places away from the units column.

To convert a number in standard form to an ordinary number, simply do the multiplication.

Examples

• \(1.34 \times 10^{3}\) is 1,340 since \(1.34 \times 10 \times 10 \times 10 = 1340\)

• \(4.78 \times 10^{–3}\) is 0.00478 as \(4.78 \times 0.001 = 0.00478\)

This process can also be sped up by considering where the first digit is compared to the units column.

Examples

• \(3.51 \times 10^{5} =\) 351,000 because the 3 moves five places away from the units column. Two places are filled by 5 and 1. Put zeros in the other three places.

• \(3.08 \times 10^{–4} =\) 0.000308 because the 3 moves four places away from the units column. Put zeros in the other three places. Focus on the 3, not the 8.

Numbers written in standard form can be ordered by first considering the power of 10, which tells you the size of the numbers. If two or more numbers have the same power of 10, use the number at the front to decide on the order.

Example

Write these numbers in ascending order:

\(7 \times 10^{–2}, 3.2 \times 10^{2}, 7 \times 10^{–2} 4.1 \times 10^{4}, 3.81 \times 10^{–5}, 5.6 \times 10^{3}, 2 \times 10^{4}\)

The numbers \(4.1 \times 10^{4}\) and \(2 \times 10^{4}\) have the same power of 10, so looking at the number at the front we can see that \(2 \times 10^{4}\) is less than \(4.1 \times 10^{4}\).

Using the powers of 10, we can write the other numbers in order, smallest to biggest:

\(3.81 \times 10^{–5}, 7 \times 10^{–2}, 3.2 \times 10^{2}, 5.6 \times 10^{3}, 2 \times 10^{4}, 4.1 \times 10^{4}\)

When adding and subtracting standard form numbers you have to:

1. convert the numbers from standard form into decimal form or ordinary numbers
2. complete the calculation
3. convert the number back into standard form

Example

Calculate \((4.5 \times 10^{4}) + (6.45 \times 10^{6})\). Give your answer in standard form.

Solution

\(=45, 000 + 6, 450, 000\)

\(= 6, 495, 000\)

\(6.495 \times 10^{6}\)

When multiplying and dividing you can use index laws:

1. multiply or divide the first numbers
2. apply the index laws to the power of 10

To multiply, add the powers together, eg \(10^6 \times 10^4 = 10^6 + 4 = 1010\). To divide, subtract the powers, eg \(10^7 ÷ 10^2 = 10^7 - 2 = 10^5\). The numbers must have the same base (the number that is raised to the power, in this case 10).

Example 1

Calculate \((3 \times 10^{3}) \times (3 \times 10^{9})\)

Solution

Multiply the first numbers, which in this case is \(3 \times 3 = 9\)

Apply the index law on the powers of 10:

• \(10^{3} \times 10^{9} = 10^{3+9} = 10^{12}\)

• \((3 \times 10^{3}) \times (3 \times 10^{9}) = 9 \times 10^{12}\)

Take care that the answer is in standard form. It is common to have to readjust the answer.

Example 2

Calculate \((4 \times 10^{9}) \times (7 \times 10^{–3})\)

Solution

Multiply the first numbers \(4 \times 7 = 28\)

Apply the index law on the exponents:

• \(10^{9} \times 10^{–3} = 10^{9+–3} = 10^{6}\)

• \((4 \times 10^{9}) \times (7 \times 10^{–3}) = 28 \times 10^{6}\)

However, \(28 \times 10^{6}\) is not in standard form, as the first number is not between 1 and 10. To correct this, divide 28 by 10 so that it is a number between 1 and 10. To balance out that division of 10, multiply the second part by 10 which gives \(10^{7}\).

\(28 \times 10^{6}\) and \(2.8 \times 10^{7}\) are equivalent, but nly the second is written in standard form.

So, \((4 \times 10^{9}) \times (7 \times 10^{–3}) = 2.8 \times 10^{7} \)