CS 100 (Learn) — CS 100 (Web) — Module 01
In this section we want to do a quick review of how exponents work.
When you have an expression like "two to the power of three", or simply "two to the three" [23], it is the equivalent to two times two times two. We multiply two by itself three times, which equals eight [23 = 8].
This is not the same as "three to the power of two" [32], which is three times three which equals nine [32 = 9].
In general, if you have a number "n to the k" [nk] it is n multiplied together by itself k times.
In this course, we are only going to worry about exponents that are integers. You will not see "two to the one half" [21/2].
There are a few trickier cases that are often forgotten.
Any number to the exponent of one is just itself.
21 = 2
101 = 10
Any number to the exponent of zero is always one.
20 = 1
100 = 1
If the exponent is negative, then the result is the inverse, or "one over" the result.
2-3 = 1 / 23 = 1/8
The exponents (or powers) of ten are especially important and easy to remember:
103 = 1000
102 = 100
101 = 10
100 = 1
10-1 = .1
10-2 = .01
10-3 = .001
For the non-negative exponents, the exponent corresponds to how many zeros are after the one, or in other words, what "position" the one is in.
The exponents (or powers) of two are very important when working with computers:
20 = 1
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256
29 = 512
210 = 1024
In this course, we will provide for you on an exam the first ten powers of two as shown here.
One final issue we want to address is what it means for a number to have a subscript.
As we have seen, two raised to the three [23] is equivalent to two times two times two, which equals eight [23 = 8].
Two lowered to three [23] does not mean anything in traditional mathematics. It has nothing to do with exponents.
You may have seen variables with subscripts to distinguish between variables like "x" one, "x" two, "x" "n" [x1, x2 ... xn] but in traditional mathematics, a simple value (like "two") with a subscript does not mean anything.
In this course, we will use subscripts to distinguish between number systems with different bases, which we will see later.