Converting between ordinary numbers and standard form

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.

50,000 can be written as: 5 \times 10,000

10,000 = 10 \times 10 \times 10 \times 10 = 10^4

So: 50,000 = 5 \times 10^4

Question

What is 800,000 written in standard form?

800,000 can be written as 8 \times 100,000.

100,000 = 10 \times 10 \times 10 \times 10 \times 10 = 10^5

So: 800,000 = 8 \times 10^5

34 \times 10^7 is not written in standard form as the first number is not between 1 and 10. To correct this, we need to divide 34 by 10. To balance out the division of 10, we also need to multiply the second part by 10, which gives 108.

34 \times 10^7 and 3.4 \times 10^8 are equal in value, but only the second is written in standard form.

Example

What is 87,000 in standard form?

87,000 can be written as 8.7 \times 10,000.

10,000 = 10 \times 10 \times 10 \times 10 = 10^4

So 87,000 = 8.7 \times 10^4.

Question

What is 135,000 written in standard form?

135,000 can be written as 1.35 \times 100,000.

100,000 = 10 \times 10 \times 10 \times 10 \times 10 = 10^5

So 135,000 = 1.35 \times 10^5.

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 6 places away from the units column.

36,000 = 3.6 \times 10^4 because the 3 is 4 places away from the units column.

Question

What is 103,000,000 in standard form?

1.03 \times 10^8 because the 1 is 8 places away from the units column.

Question

What is 1,230 in standard form?

1.23 \times 10^3 because the 1 is 3 places away from the units column.