1.3 - Scientific notation
Scientific notation: A system, based on powers
of 10, most useful for expressing very large and
very small numbers.
Physicists like to measure the very big, the very small and everything in between. To
express the results efficiently and clearly, they use scientific notation.
Scientific notation expresses a quantity as a number times a power of 10. Why is this
useful? Here’s an example: the Earth is about 149,000,000,000 meters from the Sun.
You could express that distance as we just did, with a long string of zeros, or you could
use scientific notation to write it as 1.49×10^11 meters. The latter method has proven
itself to be clearer and less prone to error.
The value on the left (1.49) is called the leading value. The power of 10 is typically
chosen so the leading value is between one and 10. In the example immediately above,
we multiplied by 10^11 so that we could use 1.49. We also could have written 14.9×10^10
or 0.149×10^12 since all three values are equal, but a useful convention is to use a
number between one and 10.
In case you have forgotten how to use exponents, here’s a quick review. Ten is the
base number. Ten to the first power is 10; 10^2 is ten to the second (ten squared) or 100;
ten to the third is 10 times 10 times 10, or 1000. A positive exponent tells you how many
zeros to add after the one. When the exponent is zero, the value is one: 10^0 equals one.
As mentioned, scientists also measure the very small. For example, a particle known as
a muon has a mean lifespan of about 2.2 millionths of a second. Scientific notation
provides a graceful way to express this number: 2.2×10í^6 (2.2 times 10 to the negative
sixth). To review the mathematics: ten to the minus one is 1/10; ten to the minus two is
1/100; ten to the minus three is 1/1000, and so forth.
You can also write 1.49×10^11 as 1.49e11. The two are equivalent. You may have seen
this notation in computer spreadsheet programs such as Microsoft® Excel. We do not
use this "e" notation in the text of the book, but if you submit answers to homework
problems or interactive checkpoints, you will use it there.
Scientific notation
Number between 1 and 10 (leading
value)
Multiplied by power of 10How do you write the numbers
above in scientific notation?
45 = 4.5 × 10 = 4.5×10^1
0.012 = 1.2 × 1/100 = 1.2×10í^2
1.4 - Standards and constants
Standard: A framework for establishing
measurement units.
Physical constant: An empirically based value.
Physicists establish standards so they can measure things consistently; how they define
a standard can change over time. For example, the length of a meter is now based on
how far light travels in a precise interval of time. This replaces a standard based on the
wavelength of light emitted by krypton-86. Prior to that, the meter was defined as the
distance between enscribed marks on platinum-iridium bars. Advances in technology,
and the requirement for increased precision, cause scientists to change the method
used to define the standard. Scientists strive for precise standards that can be
reproduced as needed and which will not change.
By choosing standards, scientists can achieve consistent results around the globe and compare the results of their experiments. Well-equipped
labs can measure time using atomic clocks like the one shown in Concept 1 on the right. These clocks are based on a characteristic frequency
of cesium atoms. You can access the official time, as maintained by an atomic clock, by clicking here.
You will encounter two types of constants in this textbook. First, there are mathematical constants like ʌ or the number 2. Second, there are
physical constants, such as the gravitational constant, which is represented with a capital G in equations. We show its value in Concept 2 on
the right. Devices such as the torsion balance shown are used to gather data to determine the value of G. This is an active area of research, as
G is the least precisely known of the major physical constants.
Constants such as G are used in many equations. G is used in Sir Isaac Newton’s law of gravitation, an equation that relates the attractive
force between two bodies to their masses and the square of the distance between them. You see this equation on the right.
Standards
Establish benchmarks for measurement