Unit 3.1 Scientific Notation
How far is the Sun from Earth?
Astronomers are used to really big numbers. While
the moon is only 406,697 km from earth at its
maximum distance, the sun is much further away (150
million km). Proxima Centauri, the star nearest the
earth, is 39, 900, 000, 000, 000 km away and we have
just started on long distances. On the other end of
the scale, some biologists deal with very small
numbers: a typical fungus could be as small as 30
μmeters (0.000030 meters) in length and a virus might
only be 0.03 μmeters (0.00000003 meters) long.
Scientific Notation
Scientific notation is a way to express numbers as the product of two numbers: a coefficient and the
number 10, raised to a power. It is a very useful tool for working with numbers that are either very
large or very small.
When working with large numbers, a positive exponent is used. The positive exponent indicates that the
coefficient must be multiplied by 10 that many times to return to the original number.
Example: The distance from Earth to the Sun is about 150,000,000,000 meters – a very large distance
indeed. In scientific notation, the distance is written as 1.5 × 1011 m.
The coefficient is the 1.5 and must be a number greater than or equal to
1 and less than 10.
The power of 10, or exponent, is 11 because you would have to multiply
1.5 by 10, 11 times to get the correct number.
When working with small numbers, numbers less than one, a negative exponent is used. The negative
exponent indicates that the coefficient must be divided by 10 that many times to return to the original
number.
Example: The mass of one proton is .00000000000000000000000167 grams, a very small number. It is
expressed as 1.67 x 10-24 grams.
The coefficient is the 1.67 and must be a number greater than or equal
to 1 and less than 10.
The power of 10, or exponent, is -24 because you would have to divide
1.67 by 10 , 24 times to get the correct number.
So 0.1 meters is 1 × 10-1 meters, 0.01 is 1 × 10-2 and so forth. Note the use of the leading zero (the zero
to the left of the decimal point) in the example column of the table. That digit is there to help you see
the decimal point more clearly. The figure 0.01 is less likely to be misunderstood than .01 where you may
not see the decimal.
Scientific notation is sometimes referred to as exponential notation.
Expanding Scientific Notation
If a number is currently in scientific notation, the number can be expanded to return it to the original
number. This is done by multiplying the coefficient by 10, to the exponent times if the exponent is
positive, or dividing by 10, the exponent times if the exponent is negative.
Example: Expand 5.694 x 1013 to a decimal number.
The exponent is positive, so the coefficient is multiplied by 10 the
exponent times.
5.694 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 =
56,940,000,000,000
Example: Expand 2.8054 x 10-8 to a decimal number.
The exponent is negative, so the coefficient is divided by 10 the
exponent times.
__
2.8054______________ = 0.000000028054
[(10) (10) (10) (10) (10) (10) (10) (10)] 
Summary



Scientific notation allows us to express very large or very small numbers in a convenient way.
This notation uses a coefficient (a number between 1 and 10) and a power of ten sufficient for the
actual number.
Scientific notation can be expanded to decimal numbers by multiplying (for positive exponents) by 10
the exponent times, or dividing (for negative exponents) by 10 the exponent times.
Practice
Practice scientific notation using the link below:
http://www.mathsisfun.com/numbers/scientific-notation.html
Review
1.
2.
3.
4.
5.
What is scientific notation?
Express 150,000,000 in scientific notation.
Express 0.000043 in scientific notation.
Expand 6.2 x 1011 to a decimal number.
Expand 2.94 x 10-6 to a decimal number.
Answers
1.
2.
3.
4.
5.
Expressing large or small numbers as the sum of a coefficient and a power of 10. (Positive exponent
for large numbers, negative exponent for numbers smaller than 1)
1.5 x 108
4.3 x 10-5
62,000,000,000
0.00000294