Assignment 2a: Reading B Decimal to Sexagesimal Base-10 to Base-60 B10 to B60 by Newsome

Our Decimal Number System: How We Construct Our Numbers [The positional notation system in Base-10] Modern Decimal Decimal Parts Number 103 + 102 + 101 + 100 Decimal equivalent [base-10] 1000 + 100 + 10 + 1

1523 1(1000) + 5(100) + 2(10) + 3(1) 1000 + 500 + 20 + 3

4085 4(1000) + 0(100) + 8(10) + 5(1) 4000 + 0 + 80 + 5

Let's count. 1 2 3 4 5 6 7 8 9 10

What we did was count up to 9 and then start over again at 1, which was moved one place to the left, to make 10. One in the tens position and 0 in the ones position. Because our system is Base-10, we run out of unique symbols after the 10th symbol.

Implied in our system is zero. To be consistent we should really start counting with 0. 0 1 2 3 4 5 6 7 8 9 ...then continue... 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 ... etc...

By counting this way (starting with 0) we list all ten unique symbols of our base-10 system. [Technically the "10" is using two of the unique symbols: the "1" and the "0." It is a compound, not an element, per se.]

Historically, zero is a tricky fellow. People didn't work with a symbol denoting nothing for most of written human history. It's only in the last several hundred years that a symbol for zero has been consistantly used.

The historical introduction of zero gets a lot of play in the history of mathematics. It's not that premodern people didn't understand the idea of nothing, but they, like us, naturally started counting with 1. Nobody counts how many Tootsie Rolls they have by saying (or thinking), "Zero Tootsie Rolls, one Tootsie Rolls, two Tootsie Rolls, etc." To count nothing is not counting. It's..... well.... nothing.

We (modern human beings) typically utilize a decimal number system, Base-10. [Latin: Decimus = tenth] The Roman Numerals are also base-10... more or less. But their system doesn't have a consistant place system... meaning that the position of a symbol of a Roman Numeral doesn't tell you much.

E.g. MMCMLVIII vs. 2958 If I asked, "What does the fourth symbol from the right mean in either of the numbers written above?" For the Roman Numeral you'd answer, "The fourth symbol from the right means five." "Five what?" ..... "I dunno. Just five." In the Hindu-Arabic number you'd say, "The fourth symbol from the right means two thousand."

The Hindu-Arabic numeral includes more information encoded into its position within the number, whereas the Roman Numeral doesn't have this feature. A particular position in a Roman Numeral doesn't tell you much.

Now count up 1 from the previous example.

MMCMLIX vs. 2959

"The fourth Roman Numeral from the right is now one thousand." That's a radical change in that fourth position. It was five, now it's one thousand. The fourth Hindu-Arabic numeral is still 2000.

One big benefit of the positional notation system is that big numbers are big and small numbers are small. 88 is physically much smaller than 23,457,341. It is 2 symbols compared with 8 symbols. At a glance we can see which number is bigger.

In Roman Numerals you have to put some thought into determining which of the following two numbers is greater than the other: LXXXVIII vs. M.

We can easily modify the Roman Numerals to be more like ours.

We just need to define a notational system to indicate a place, a position. Let's propose that the colon, ":", indicates a base-10 positional system... like so...

Here's how it would look imposed onto our number system. Modern With ":" to indicate place. What the place system does.

1(1000) + 5(100) + 2(10) + 3(1) = 1523 1:5:2:3 1000 + 500 + 20 + 3

Similarly, we could easily fit Roman Numerals into this system. Roman With ":" to indicate place. What the place system does.

1(1000) + 5(100) + 2(10) + 3(1) = MDXXIII M:D:XX:III 1000 + 500 + 20 + 3

It's just punctuation. We're using punctuation to group the numbers together into powers of 10 (1s, 10s, 100s, 1000s)

By imposing this colon system, Roman and Hindu-Arabic numerals can function the same way. Their positions between colons indicate what powers of 10 they modify.

It would be no problem to write any number in this way. E.g. MMCMLIX would become MM:CM:L:IX = 2:9:5:9

All we've done is impose an additional layer of organization onto the Roman Numerals. This organization allows us to identify specific positions in exactly the same manner as the Hindu-Arabic numbers.

The reason we need the colons is because many Roman Numerals require multiple symbols to represent what Hindu-Arabic numerals do with one symbol. E.g. IX = 9 or III = 3.

Roman Numerals are annoyingly inconsistant in length. I II III IV V VI VII VIII IX X 1 symbol 2 symbols 3 symbols 2 symbols 1 symbol 2 symbols 3 symbols 4 symbols 2 symbols 1 symbol

Pros and Cons To count to 10 the Hindu-Arabic system requires that you learn 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. ...whereas the Roman Numerals only require that you learn 3 symbols: I, V, and X. The Roman symbols are much easier to memorize.

Once you learn all 10 Hindu-Arabic symbols the manipulation of numbers is relatively easy. The Roman Numeral symbols are fewer, easier to memorize, but manipulating them is harder.

You may have noticed a slight discrepancy.

Q: How do you write 308 in Roman Numerals? A: CCCVIII

Q: How do you write 308 in our colon-positional notation system? A: CCC: ? :VIII.

There is no zero in the Roman Numerals. In order to make a positional system work with the Roman Numerals we need to invent a new symbol. We need to invent a zero. We need something to put inbetween the colons to show that there are no 10s, just ones and hundreds. How about we invent a symbol for Roman Numeral zero. How about @.

Now 308 can be written as III:@:VIII. The 10s position has nothing in it– zero 10s. Problem solved.

This problem was similarly solved when the cifre, or 0, was introduced. If people were going to use a positional notatation system, they needed a symbol for nothing. That symbol is 0.

Dance Time This song couldn't have existed without this radical addition to the numerical lexicon. No 0, no song. or another link with different graphics: PE-logo.

Now listen to this episode of In Our Time [BBC] A brief history of zero. IOT_ Zero.mp3 [42']

After listening, return to this reading.

Recall our LobsterPeople exercise from class 1b. Count up from zero, in a base-4 system. Read it out loud a couple of times.

0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 100, 101, 102, etc.

After a couple of times, you start to feel the rhythm... every 4 counts things change.

If you recall, doing mathematical operations in B4 (Base-4) was hard. Remember trying to multiply ∆ × π = ? It's not technically any harder than B10, but it is really hard to think in B4. We have been using B10 for so long it seems totally natural... innate. It's not. It's learned. It's a social construction. The Maya used B20. Most premodern astronomers used B6, B60 and/or B12. Computers scientists use B2, and B4, and B8, and B16, and B32, and B64, etc.

Here is a multiplication table for B10. Here is a multiplication table for B4.

It covers what is outlined in green on the B10 table, but it's written in the symbolic form of B4.

Your assingment is to make a base-6 (B6) system. This is an assignment without a provided worksheet. It will be up to you to design and produce it. You could freehand it, or do it in a word processor, or in a spreadsheet. [Then upload it to your folder with the proper file name: "LastName.1b.Base-6.xxx".] The result should be about one page long and should include the following:

1) Introduce what you are doing and provide commentary all along the way. Write some prose.

2) Include a count up to the equivalent of 3610 in B6, starting with 0. E.g. 0, 1, 2, 3, 4, 5, 10, 11, 12, etc. Above your B6 count, put the equivalent in B10.

Here's what I mean in the form of an example using B4...

If you recall, counting up to 1610 in B4 looked like this: B10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B4 0 1 2 3 10 11 12 13 20 21 22 23 30 31 32 33 100

Your job is to do the same within a B6 system. It's a lot more numbers, so figure out a way to present this information in an attractive and useful manner.

3) Make a multiplication table for B6 like the ones shown above for B10 and B4. Notice that 0 is not included in these tables. Set it up on a grid like this:

*Remember, there is no 6 in B6. It will be a 10.

4) Solve the following equations in B6. Answer them in B6. Write these out in full on your sheet and comment on at least 5 problems. Meaning... for 5 of the problems, make an observation of some sort. Several of these problems are really easy. Remember to think of how arithmetic works in general. E.g. Zero times anything equals....

E.g. Example of commentary on a problem. [Not all commentary needs to be this elaborate.]

26 ∙ 116 =

Here are the problems/equations to be answered for part 4:

a) 06 ∙ 32436 = f) 32436 + 16 = l) 26 ∙ 26 ∙ 26 =

b) 16 ∙ 32436 = g) 32436 + 36 = m) 36 ∙ 36 ∙ 36 =

32436 c) = h) 32436 − 36 = n) 36 ∙ 36 ∙ 46 = 32436

0 i) 36 ∙ 106 = o) 56 ∙ 126 = [Hard One] d) 6 = 32436 j) 56 ∙ 106 = 3243 e) 6 = 0 6 k) 106 ∙ 106 =

Make your sheet attractive. Make it stand alone. Make it as an art project or a graphic novel or comic book. Make it as if it were for publication. If you are not artsy, tough. Become artsy. Presentation is everything.... well.... almost everything.

When you have completed the above written assignment, continue reading below.....

THE SEXAGESIMAL SYSTEM

The sexagesimal system is based on the number 60 instead of 10. Above is a proposed set of symbols for the sexagesimal system. Sixty distinct symbols. Can you imagine having to learn all 60 symbols? No. Of course you can't. Ten is hard enough. So, in order to simplify the use of the sexagesimal system we tend to use a hybrid system. We use base-10 numbers (0-59) in each positon of a sexagesimal number, instead of a solitary symbol for each of the 60 numbers.

For example, instead of writing , [circled above] we write it like this: 11:30:29. This way we don't have to memorize all these crazy symbols, nor do we have to draw them. And keep in mind, we draw 0-9. We learned how to draw them at some point in our childhood. Eventually we got so good at drawing them that we called it writing, instead of drawing. But if you think about it, we're drawing.

This way of writing in sexagesimal (Base-60) is like the colon-position system we imposed onto the Roman Numerals, only in this one, we are using decimal (Base-10) numbers in a sexagesimal system.

We still use the sexagesimal system for time: hours:minutes:seconds. We count to 59 seconds and the next second gives us 1 minute. ...57, 58, 59, 1:00, 1:01, 1:02...

Then we count to 59 minutes and the next minute gives us 1 hour.

We write 1 hour, 25 minutes, and 15 seconds as

1:25:15. (Like a digital clock.)

[It would look like this - -using the symbols in the graphic at the top of the page.]

1:25:15 = This single sexagesimal number tells you how many seconds you have. We call it hours:minutes:seconds, but it is actually just seconds. We've given names to different collections of seconds.

This is much like how we have given names to collections of 10s, 100s, and 1000s in our regular decimal system, with the words tens, hundreds, and thousands. They are just words telling you how many ones you have.

60 seconds, we call a minute. 60 minutes we call an hour, but those 60 minutes are made up of 60 seconds each. An hour is 3600 seconds. [602 seconds]

So 1:25:15 translates to the decimal system like so...

15 seconds = 15 seconds = 15 seconds 25 minutes = 25(60 seconds) = 1500 seconds + 1 hour = 60 minutes = 60(60 seconds) = 3600 seconds = 5115 seconds

5 thousands, 1 hundred, 1 ten, and 5 ones.

511510 in the decimal system is the same as 1:25:1560 in sexagesimal system. Both systems tell you how many seconds you have.

You might notice that we never go above the number 59 when writing sexagesimal numbers. There is no 60 or 61 or 75.

It turns over after 59. Because it is base-60. [1:00 or 1:01 or 1:15]

Here's how I tend to think of these conversions. This way of doing it will be useful when we start doing it in a spreadsheet. 3600s 60s 1s 602 = 3600 601 = 60 600 = 1 1:25:1560 = 1 ∙ 360010 25 ∙ 6010 15 ∙ 110 Add up to get B10 3600 1500 15 = 511510

For comparison, here's the same tabular form using our familiar B10 system. 1000s 100s 10s 1s 103 = 1000 102 = 100 101 = 10 100 = 1 511510 = 5 ∙ 100010 1 ∙ 10010 1 ∙ 1010 5 ∙ 110 Add up to get B10 5000 100 10 5 This table is sort of silly. It converts B10 to B10.

That should do it. You've completed assignment 2a.