- Return of your exams, and a key
- The extra credit was essentially either completely right or wrong -- I didn't give partial credit on that one.
- For the others, you could receive partial credit (and often did).
- Curve: add 3 to your score
- Question of the day(s): How is Fraudini's trick related to base 2?
- So far we've been busting up numbers in different ways. The first
way was illustrated by the number 1729, and its base 10 representation.
When we say "Seventeen twenty nine", what do we mean?
What does each of the digits in the number 1729 mean?
- The story of base 10....
- Place value -- each power has its place
- One of my favorite early introductions to bases was Tom Lehrer's New Math.
- What's the story of base 2? Of base 8? Of base 20?
- Let's review Tom Lehrer's problem:
342 - 173 The easiest way to work with these problems is often to translate between the "strange" base, and base 10, which we will do now.
How can we understand this representation?
- A base is a natural number, a counting number, larger than 1.
- Each number is written using only powers of that
base, starting from the power 0 (since anything to the power 0
is 1): for example,
-
- There are as many digits as the base: so base two uses only two digits (0 and 1); base 10 uses only 10 digits (0 through 9).
- The number is written so that each digit falls in a place, which represents a unique power of the base.
- The Great Fraudini writes numbers in base 2 (see p. 11 of our
text) -- that is, using powers of two.
47=32+8+4+2+1 Nothing in the 24=16 place 89=64+16+8+1 Nothing in the 25=32, 22=4, 21=1 place How would we write these numbers in base 2?
47 = 101111 "Nothing" -- zero -- in the 24=16 place 89 = 1011001 "Nothing" in the 25=32, 22=4, 21=1 place - For your homework, you should have read this bases
reference. Other bases are used constantly -- for example, in this web
page. How so?
- This page uses HTML to control its appearance. For example, the
background color for this page is given by the command
<body bgcolor="#FFF8DC">
What does #FFF8DC have to do with color?
- What are some other bases that we use in our lives? Where?
- This page uses HTML to control its appearance. For example, the
background color for this page is given by the command
- Translating between bases
- Write 53 (written here in base 10) in base 2
- Write 100101101 (written here in base 2) in base 10
- Write 53 (written here in base 10) in base 8
- Write 100101101 (written here in base 8) in base 10
- Write 53 (written here in base 10) in base 16
- Write 100101101 (written here in base 16) in base 10
- Let's play Detective in the times of Babylon....