Algebra Know-It-ALL

• One times eight gives you 8


• Zero times sixteen gives you 0


• One times thirty-two gives you 32

The decimal-numeral equivalent is therefore 1 + 2 + 0 + 8 + 0 + 32 = 43. This is the

quantity you commonly imagine as forty-three.

9. There are at least three ways to solve this problem. You can “cheat” and use Table 1-2.
   To do it manually, note that forty-three, the decimal equivalent as derived in the
   solution to Prob. 8, is equal to five times eight, plus three more. That means you can
   define quantity forty-three as the octal numeral 53. You can also add up the values in
   octal arithmetic. That’s tricky and hasn’t been covered in this chapter, so we won’t show
   it here.


10. As with the decimal-to-octal conversion, there are at least three ways to solve this
    problem. You can “cheat” and use Table 1-2. To do it manually, note that forty-three
    is equal to two times sixteen, plus eleven more. In hexadecimal notation, eleven is
    represented by B. That means you can define the quantity forty-three in hexadecimal
    form as the numeral 2B. The last method is to add up the values in hexadecimal
    arithmetic. Again, that’s tricky and hasn’t been discussed, so we won’t do it here.

Chapter 2

1. The null set is a subset of any set. The null set lacks elements, like an empty bank
   account lacks money. You can say that it contains nothing as an element! If you have
   a certain set A with known elements, you can add nothing, and you always end up
   with the same set A. The null set is a subset of itself, although not a proper subset of
   itself. Look at an example. If you let the written word “nothing” actually stand for
   nothing, then

∅= { } = {nothing}

and

{nothing}⊆ {nothing, 1, 2, 3}

so therefore

∅ ⊆ {1, 2, 3}

Keep in mind that a subset is not the same thing as a set element. The null set contains

nothing, but the null set is not itself nothing. An empty bank account is a perfectly valid

account unless somebody closes it.

2. You can build up an infinite number of sets if you start out with nothing. First, take
   nothing and make it an element of a set. That gives you the null set. Then consider the
   set containing the null set, that is, {∅}. This is a legitimate mathematical thing, but

Chapter 2 589