Scientific Notation
Standard Form

Standard form is a convenient way to write very large or very small numbers. It is particularly useful when working with a scientific calculator.

In standard form a number is written as

a × 10n where 1 ≤ a < 10 and n is an integer

That is, a is a number greater than or equal to 1 and less than 10, and n is a positive or negative whole number.

Note

Because of the way that the powers of 10 are used in standard form, it is important to remember that:
104 = 10 000 and 10−4 = 0.0001
103 = 100010−3 = 0.001
102 = 10010−2 = 0.01
101 = 1010−1 = 0.1
100 = 1

Worked Examples

1

Write each of the following numbers in standard form.

(a)
346 000 000 Show me
346 000 000 = 3.46 × 100 000 000
= 3.46 × 108
(b)
2710 Show me
2710 = 2.71 × 1000
= 2.71 × 103
(c)
0.000 543 Show me
0.000 543 = 5.43 × 0.0001
= 5.43 × 10−4
2

Find

(a)
4.21 × 102 Show me
4.21 × 102 = 4.21 × 100
= 421
(b)
3.1 × 105 Show me
3.1 × 105 = 3.1 × 100 000
= 310 000
(c)
3.6 × 10−2 Show me
3.6 × 10−2 = 3.6 × 0.01
= 0.036
(d)
4.7 × 10−3 Show me
4.7 × 10−3 = 4.7 × 0.001
= 0.0047
3

Write each of the following numbers in normal decimal form.

(a)
3.217 × 103 Show me
3.217 × 103 = 3.217 × 1000
= 3217
(b)
3.68 × 105 Show me
3.68 × 105 = 3.68 × 100 000
= 368000
(c)
4.7 × 10−4 Show me
4.7 × 10−4 = 4.7 × 0.0 001
= 0.00 047

Most calculators can work with numbers in standard form. On a calculator display 3.01 × 1017 would look like

4

Calculate 366 ÷ 0.0012 and write the answer

(a)
exactly, Show me
366 ÷ 0.0012 =
=
= 305 000
(b)
in standard form. Show me 305 000 = 3.05 × 105 in standard form.

Exercises

Write each of the following numbers in standard form.

(a)
47 000 × 10
(b)
52 100 × 10
(c)
32 000 000 × 10
(d)
324 100 × 10
(e)
420 × 10
(f)
81 000 × 10
(g)
5000 × 10
(h)
47 000 000 000 × 10
(i)
3 200 000 000 × 10
(j)
0.00 062 × 10
(k)
0.0571 × 10
(l)
0.0 000 002 × 10
(m)
0.124 × 10
(n)
0.0371 × 10
(o)
0.00 021 × 10
(p)
0.00 007 × 10
(q)
0.471 × 10
(r)
0.0003 × 10

Write each of these numbers in standard form.

(a)
1 million × 10
(b)
15 thousand × 10
(c)
6.4 million × 10
(d)
30.4 million × 10
(e)
4 million × 10
(f)
0.4 million × 10

Write each of the following numbers using normal decimal notation.

(a)
6 × 105
(b)
4.31 × 102
(c)
5.86 × 107
(d)
8.3 × 10−4
(e)
4.172 × 103
(f)
6.42 × 10−5
(g)
4.7 × 101
(h)
3.2 × 10−1
(i)
8.47 × 10−4
(j)
3.34 × 108
(k)
3.471 × 10−4
(l)
8.421 × 102
(m)
1.675 × 101
(n)
8.4 × 10−6
(o)
7.12 × 10−4

For each of the numbers below check whether or not it is in standard form. If it is not in standard form, write it in standard form. Leave blank if not.

(a)
3.2 × 108 × 10
(b)
43.2 × 102 × 10
(c)
15.6 × 10−8 × 10
(d)
0.4 × 103 × 10
(e)
1.3 × 10−8 × 10
(f)
0.7 × 10−4 × 10
(g)
5.471 × 102 × 10
(h)
54.71 × 103 × 10
(i)
8.21 × 106 × 10

Give the answers to the following calculations in standard form.

(a)
2000 × 30 = × 10
(b)
40002 = × 10
(c)
503 = × 10
(d)
= × 10
(e)
= × 10
(f)
0.042 = × 10
(g)
0.004 × 0.7 = × 10
(h)
22 × 400 = × 10
(i)
= × 10
(j)
302 = × 10
(k)
0.022 = × 10
(l)
1002 = × 10

The mass of the earth is 5.9 × 1024 kg. Write this as a decimal number.

kg

The width of a thin strip of metal is mm. Write this in standard form.

× 10 mm