Scientists commonly deal with
very large or
very small numbers. Both are commonly expressed in
scientific notation, which is a
pre-exponential term multiplied by an exponential term.
130 000 would be expressed as 1.3 × 10
5 1.3 is the pre-exponential term (this is usually a number between 1 and 10).
105 is the exponential term
0.000056 would be written as 5.6 × 10
–5 ?
Exponential terms have a
base (10 in this case)
raised to a power.
What is a power? Powers are exponents. The exponent tells us how many times to multiply the base by itself to obtain the corresponding decimal. 10 is the base used in scientific notation.
Why are some exponents positive and some exponents negative? If the
power of 10 is
positive, the digit term (1 below) is
multiplied by the powers of 10.
1 × 100 = 1
1 × 101 = 1 × 10
1 × 102 = 1 × 10 × 10 = 100
1 × 105 = 1 × 10 × 10 × 10 × 10 × 10 = 100 000
Thus numbers larger than 1 expressed in scientific notation have positive powers of 10.
If the power of 10 is
negative, the digit term is divided by the powers of 10.
| 1 × 10–2 = | 1 | = | 1 | = 0.01 |
| 102 | 10 × 10 |
| 1 | = 0.00001 |
| 10 × 10 × 10 × 10 × 10 |
Thus
numbers smaller than 1 expressed in scientific notation have
negative powers of 10.
To convert from 1 × 10
n to the corresponding number:
If the power of ten is positive, move the decimal point on 1* one place to the right for each power of 10
Example: three places to the right for 1 ×103 which equals 1000
If the power of 10 is negative, the decimal point on 1* is moved one place to the left.
Example: 1 × 10–1 = 0.1
*The decimal point on 1 is assumed to be directly after the 1 (that is 1.)