The Mathematics 11
Competency Test
Scientific Notation
In so-called scientific notation, numbers are written as a product of two parts:
(i) a number between 1 and 9.9999…..
and
(ii) a power of 10.
To convert a number to scientific notation, just move the decimal point from its initial position until
it is just to the right of the first nonzero digit. The number of positions you move the decimal point
gives the numerical value of the power of 10 required. If the decimal point was moved to the left,
the power of 10 is positive. If the decimal point was moved to the right, the power of 10 is
negative.
Examples:
4325.7
4.3257 x 103
=
three positions
to the left
so the exponent of 10 is +3
(Note that 4.3257 x 103 = 4.3257 x 1000 = 4325.7, demonstrating that the scientific
notation form is the same numerical value as the original number.)
0.00593
=
three positions
to the right
5.93 x 10-3
so the exponent of 10 is -3
(Note that 5.93 x 10-3 = 5.93 x 0.001 = 0.00593, demonstrating that the scientific
notation form is the same numerical value as the original number.)
-750,000,000
eight positions
to the left
=
-7.5 x 108
so the exponent of 10 is +8
(Note that -7.5 x 108 = -7.5 x 100,000,000 = -750,000,000, demonstrating that the
scientific notation form is the same numerical value as the original number.)
David W. Sabo (2003)
Scientific Notation
Page 1 of 4
-0.000000036
-3.6 x 10-8
=
eight positions
to the right
so the exponent of 10 is -8
(Note that -3.6 x 10-8 = -3.6 x 0.00000001 = -0.000000036, demonstrating that the
scientific notation form is the same numerical value as the original number.)
Notice that the number itself can be positive or negative and the exponent on the 10 can be
positive or negative, and that these two signs are quite independent of each other.
Obviously, to convert back from scientific notation to ordinary decimal numbers, you just move
the decimal point the number of places left or right as indicated by the power of 10.
Examples:
5.96953 x 104
=
the exponent
of 10 is +4
59695.3
so move the decimal point
four places to the right
(Or, using ordinary arithmetic, 5.96953 x 104 = 5.96953 x 10,000 = 59695.3.)
7.353 x 10-6
=
the exponent
of 10 is -6
0.000007353
so move the decimal point six
places to the left
(Or, using ordinary arithmetic, 7.353 x 10-6 = 7.353 x 0.000001 = 0.000007353.)
-2.3 x 102
the exponent
of 10 is +2
=
-230
so move the decimal point
two places to the right
(Or, using ordinary arithmetic, -2.3 x 102 = -2.3 x 100 = -230.)
-3.592 x 10-3
the exponent
of 10 is -3
=
-0.003592
so move the decimal point
three places to the left
(Or, using ordinary arithmetic, -3.592 x 10-3 = -3.592 x 0.001 = -0.003592.)
David W. Sabo (2003)
Scientific Notation
Page 2 of 4
To help you remember these rules, just keep in mind that when the power of 10 is positive, the
original numerical value was bigger than 1. When the power of 10 is negative, the original
numerical value was a fraction, smaller than 1. The rules are not something new or mysterious.
As shown in brackets following each example above, the rules just reflect simple numerical
properties – they produce the result of ordinary multiplication by powers of 10.
Remark 1:
Scientific notation is particularly useful in situations where very large or very small numbers
occur. For example, it is very much easier to write that the speed of light is 2.9979 x 108 m/s than
to write 299,790,000 m/s. Also, using scientific notation for numbers such as these eliminates the
need to key large numbers of zeros into calculators when we do calculations, and makes it less
likely that an error will occur because we miscounted the number of zeros in a very large or a
very small number.
It is very easy to do arithmetic with numbers in scientific notation. All scientific calculators have
automatic entry of numbers in scientific notation, but even if you don’t have such a calculator, you
can do the arithmetic by making use of the laws of exponents.
For example, since
24 hours
60 minutes
60 seconds
x
x
1 minute
1 day
1 hour
1 year = 365 days x
= 31536000 seconds
= 3.1536 x 107 seconds
and
1 km = 1000 m = 1.000 x 103 m
and, the speed of light is 2.9979 x 108 m/s. Then, the number of kilometres in 1 light year (the
distance light travels in 1 year) is
( 2.9979 × 10
8
m/s
) ( 3.1536 × 10
7
s / yr
)
1.000 × 10 m / km
3
=
( 2.9979 )( 3.1536 ) × 108107
= 9.45417744 × 108 +7−3
= 9.4542 × 1012
km
103
1.000
km
km
yr
yr
yr
(Can you explain why we rounded this result to five significant digits? Hint: which numbers in
this calculation are approximate numbers?)
David W. Sabo (2003)
Scientific Notation
Page 3 of 4
If you have a scientific calculator capable of handling scientific notation, then entry of such
numbers is straightforward. First key in the numerical part. Then press a key (typically it has an
upper case ‘E’ on it) and key the exponent. You could use the above example as a test case for
checking to see that you know how to enter numbers in scientific notation into your calculator.
Remark 2:
The use of scientific notation solves the problem of how to distinguish between trailing zeros in
whole numbers which are significant, and those which are present only to tell us where the
decimal point is located (and so are not significant digits). (To review the nature of this problem,
just re-read the previous document on significant digits.)
When we use scientific notation, there can be no trailing zeros to the left of the decimal point.
The only trailing zeros in a number in scientific notation are to the right of the decimal point, and
therefore are automatically significant digits, according to the rules.
So, for example
32000 is written 3.2 x 104 if none of the three zeros are significant.
32000 is written 3.20 x 104 if just the first of the three zeros is significant, but the
remaining two zeros are present only to locate the decimal point.
32000 is written 3.200 x 104 if just the first two of the three zeros are significant, but the
third zero is present only to locate the decimal point.
32000 is written 3.2000 x 104 if all three zeros are significant.
David W. Sabo (2003)
Scientific Notation
Page 4 of 4