In our every day lives we encounter a physical world that is detected by our senses. As humans, we tend to mostly rely on our visual sense. So, the scale that we often find most applicable to describe the world around us is one that is closely tied to what we normally see: humans, buildings, mountains. Mathematical description of these sizes just requires simple numbers. For example, a typical person is about 2 meters tall. A typical one-story house is about four meters tall. A mountain can be 1,000 to 10,000 meters tall. And so on. On the smaller side, our hair, which is about one ten thousandth (1/10,000) of one meter thick, is one of the smallest things we normally look at. So, in a way all that we see fits somewhere between the height of a mountain (10,000 m) and thickness of a stand of hair (1/10,000 m = 0.0001 m). So, for all of these just a simple decimal representation suffices.
In reality, however, even our senses have to deal with a much wider range in perception. This is in fact the case with our own hearing system. Our ears respond to soft whispers of leaves moving in a gentle breeze to the jolting sounds of fire engines racing past us with loud sirens. Mathematical description of such a vast range requires the so-called scientific notation. In this notation every factor of 10 is represented as a power.
Power zero is "no factor", so it is just multiplication by one:
100=1
Power one is a factor of ten: 101 = 10
Power two is a factor of 10x10 = 100: 102 = 100
Power three is a factor of 10x10x10=1,000: 103 = 1,000
And so on. So, for example, 108 represents a factor of 100,000,000 (i.e. a "one" followed by eight zeros). Similarly, for smaller presentations we us negative powers. The negative signifies not a factor of multiplication, but a portion, or factor of division. For example,
Power -1 is a tenth: 10-1 = 1/10 = 0.1
Power -2 is a tenth of a tenth; or one hundredth: 10-2 = 1/100 = 0.01
Power -3 is a tenth of a tenth of a tenth, or one thousandth: 10-3 = 1/1,000 = 0.001
And so on.
With this notation we can represent very large or very small numbers with just a power of ten. This presentation is necessary when we try to describe our world in the sciences. Not just sizes of objects, but other physical attributes tend to vary in scales that require the "short-hand" notation. In the case of vision, for example, our eyes tend to respond to light intensities that are significantly varied - by many factors of ten (orders of magnitude).
Below is a list of numbers presented both in the standard decimal and in the scientific notation:
decimal presentation equivalent scientific notation
1.001 1001x10-3
876 8.76x102
0.0025 2.5x10-3
Please note that a number's presentation in the scientific notation is not unique! To get a better feeling for this, fill in the blank:
decimal presentation equivalent scientific notation
1.001 ________x10-9
876 _________x10-3
0.0025 ________x105