Mathematics

[nairi]

Seaphan, where are you? I'm growing increasingly more and more impatient. See above your last post.

 

Nairi

[Azat]

Harut jan, yes etpes el gitey vor qez xapel chi lini.

[Sip]

Sorry!!! I didn't even see it. I spent ALL day today trying to get my car fixed. I have been back and forth at the dealer and they weren't able to fix it ... they replaced several parts in the past week trying to find my a 4x4 problems ... FINALLY just a few hours ago they found it!!! ... and you guessed it ... it's a COMPUTER problem !!!! ahahahahahaha ... I love computers. Apprarently, one of the "modules" which controls the 4x4 thing had died so everything was going nuts ... all kinds of strange codes, things not working, flashing lights ... it was a mess.

 

I am glad we got it fixed since I am leaving for another offroading trip in just a few hours. I would hate for something like that to brake down in the middle of no where. Mechanical things are easy to fix (relatively)

 

I will take a look at what you posted in more detail as soon as possible although I may not have any internet for a couple of days From the brief look I got, looks like you are on to something. Glad someone else is enjoying this as much as I do!

 

[ August 09, 2002, 11:38 PM: Message edited by: Sip ]

[Sip]

... Harut, I think you should check out THIS SITE .... or maybe even THIS ONE.

 

[ August 09, 2002, 11:42 PM: Message edited by: Sip ]

[Harut]

i want our spherical globe back.

[nairi]

Anytime. It's only that I saw you back on and figured that you hadn't seen my post as I was editing it when you posted your last.

 

Have a nice trip.

 

Nairi

[nairi]

Seaphan????? Where are you????

 

My vacation is nearing its end. I won't have much time anymore to do math sequences!

 

Nairi

[Sip]

Oh no! end of vacation? that sucks!

 

Actually, I was trying to make heads or tails out of your post. You are definitely on to something but I think at the end it just comes to the actual pattern.

 

So as I said before, the real pattern is: "All binary integers, assumed to be in base 3, converted to decimal."

 

Quick review:

 

With binary, we have 1, 10, 11, 100, 101, 110, 111, 1000, ... We are counting 1,2,3,4,5,... but only with digits 1 and 0. After 1, you have to go to 10 (which is 2 in decimal) and then 11 (which is 3 in decimal). In binary, each digit represents a power of 2.

 

With base 3, each digit represents a power of 3. In base n, each digit is a power on n. For our traditional decimal (base 10 system), each digit is a power of 10 (1, 10, 100, 1000 and so forth).

 

So the number 123 in decimal is: 1x100 + 2x10 + 3x1

 

The number 123 in base 5 is: 1x25 + 2x5 + 3x1

 

Now in base 3, a number such as 1011 is: 1x27 + 0x9 + 1x3 + 1x1 = 31.

 

Hopefully this was a short refresher on number bases. If you need more info, let me know and I'll be glad to give more details.

 

------

 

So, now that we have numbers of the form 1, 10, 11, 100, 111 (counting in binary) ... you will notice that the "jumps" occur anytime a number of the form 11111 changes to 100000 (next sequence). In binary, that is simply the "next" number. However, in base 3, we jump by a full power of 3!

 

So 111 (in base 3 is 13 dec) goes to 1000 (in base 3 is 27 dec) thus the difference 14.

 

The next jump comes at 1111 -> 10000 (27+9+3+1 -> 81) giving the difference 41.

 

So essentially: The differences are between the next power of 3 and the sum of all previous powers of 3, just due to the nature of this sequence.

 

0 -> 1 (1)

1+3 -> 9 (5)

1+3+9 -> 27 (14)

1+3+9+27 -> 81 (41)

1+3+9+27+81 -> 243 (122)

...

 

Did I make any sense?

[Sip]

Dear Nairi, since sequences and series seem to be so interesting to you, here's another one. It is definitely one of my favorites!

 

1, 1 1, 2 1, 1 2 1 1, 1 1 1 2 2 1, 3 1 2 2 1 1, ...

[Sip]

It'll only take 5 minutes. Believe me, it's MUCH easier than you think. Actually, you already know it. You have been using base-10 numbers which is much more complicated!!! In any case, I'll just post this in case, but feel free to ignore it.

 

Take a number like 5421.

That number means: 5x10^3 + 4x10^2 + 2x10^1 + 1

 

Actually, an arbitrary 7-digit number in decimal let's say "abcdefg" (each letter is a digit) is:

 

ax10^6 + bx10^5 + cx10^4 + dx10^3 + ex10^2 + fx10^1 + gx10^0

 

Right?

 

That's it! For base n, all you have is powers of n in the sum instead of powers of 10. The other thing you have to know is that for base n, you need n digits to count. For base-10, you need 10 digits, for base 3, you need 3 digits. For binary (base-2), you need 2 digits: 1 and 0.

 

So let's say you have a number 11010 in binary. What does that mean in decimal?

 

It is: 1x2^4 + 1x2^3 + 0x2^2 + 1x2^1 + 1x2^0 = 16+8+2 = 26.

 

I fully realize how strange this must all sound. But believe me when I say the concepts are quite simple (otherwise, I would NOT have put it up here). We spend our whole life messing with decimal numbers so it seems very natural to count 1,2,3,4,5,6,7,8,9,10,11

 

But our computers are not that smart so they count 1,10,11,100,101,110,111,1000,1001, ... (only 2 digits). All you have to do is to understand this sequence and you know how to count in binary

 

You got it. No fooling, honest! And I think it is a sequence

 

1 3 1 1 2 2 2 1, ...

 

[ August 17, 2002, 01:17 PM: Message edited by: Sip ]

[Sip]

It is the difference of the left and right side of the '->'symbol I use for jump. For for 1+3 -> 9, we have 4 jumping to 9 and thus the difference 5.

 

For

 

1+3+9+27 -> 81

 

We have 40 "jumping to" 81 and so the difference is 41.

 

Sorry for my confusing notation.

[nairi]

Believe me, it's MUCH more difficult than you think. I'm really not that far in math. Start at the beginning.

 

What does "^" mean? And how do you calculate binary into decimals? For instance, how did you calculate 11010? And where did you get 16, 8 and 2 from?

 

So let's say you have a number 11010 in binary. What does that mean in decimal?

 

It is: 1x2^4 + 1x2^3 + 0x2^2 + 1x2^1 + 1x2^0 = 16+8+2 = 26.

 

Nairi

[nairi]

Hmmm. I guess verbally it is...

 

Oops, missed out on a few numbers there.

[Sip]

I believe you! Most likely it is because I am trying to say too much too quickly. Actually, this is very useful info to me since I will have to teach this stuff probably very soon. I just have to learn to express things clearly.

 

'^' means to the power. 10^5 (read "10 to the 5" or "10 to the 5-th power") means 10x10x10x10x10 (5 times) which is 100000.

 

With binary, you get things like 16,8,4,and 2 because those are the "powers of 2".

 

2^1 = 2 (1-st power of 2)

2^2 = 2*2 = 4 (2-nd power of 2)

2^3 = 2*2*2 = 8 (3-rd power of 2)

2^4 = 2*2*2*2 = 16 (4-th power of 2)

...

 

Note that any number to the 0-th power is 1. So 2^0=1 and 10^0=1.

 

So now take a look at the previous post. Do you understand what I mean when I say for example a decimal number 123 is the same as (decompose): 1x100 + 2x10 + 3 (i.e. 1x10^2 + 2x10^1 + 3x10^0) ? If so, then the above posts should make more sense now.

 

[ August 17, 2002, 03:00 PM: Message edited by: Sip ]

[nairi]

Okay, I'm vaguely beginning to grasp something, but I still don't get how you got from 11010 to 26. How do you calculate that?

 

Nairi

[Sip]

If 11010 were decimal, we would have:

 

1x10^4 + 1x10^3 + 0x10^2 + 1x10^1 + 0x10^0 =

1x10000+ 1x1000 + 0x100 + 1x10 + 0*1 =

10000 + 1000 + 10 =

11010

 

----

 

in binary, it is the exact same way, except the digits correspond to powers of 2:

 

1x2^4 + 1x2^3 + 0x2^2 + 1x2^1 + 0x2^0 =

1x16 + 1x8 + 0x4 + 1x2 + 0x1 =

16 + 8 + 2 =

26

 

----

 

similarly, if the number 11010 were in "base 5" we would convert it to decimal using:

 

1x5^4 + 1x5^3 + 0x5^2 + 1x5^1 + 0x5^0 =

1x625 + 1x125 + 0x25 + 1x5 + 0x1 =

625 + 125 + 5 =

755

 

Nairi, your patience and attitude about this stuff is great! Most people would have just given up the second they realized it is something they have never seen before.

[nairi]

This one doesn't make sense to me. Where does (5) come from?

[nairi]

Nice one. This isn't a real sequence! Don't try to fool me!

 

1 3 2 2 2 1, 1 1 1 3 3 2 1 1, 3 1 2 3 1 2 2 1, 1 3 1 1 1 2 1 3 1 1 2 2 1 1, 1 1 1 3 3 1 1 2 1 1 1 3 2 1 2 2 2 1, 3 1 2 3 2 1 1 2 3 1 1 3 1 2 1 1 3 2 1 1, ...

 

Nairi

[nairi]

Actually none of this is making sense to me. I never studied binary numbers, so I don't know how to use or calculate them either. It's a pity, because I was having fun. But this is all getting a tad bit too complex for me. (If only I worked a little harder during my math classes, I may have succeeded now...)

 

Thanks anyway for taking the time to try and explain. I really enjoyed it.

 

Nairi

[nairi]

Okay, let me try this:

 

10110

 

1x2^4=16

0x2^3=0

1x2^2=4

1x2^1=2

0x2^0=0

 

16+0+4+2+0=22

 

Did this make sense?

 

Next, how do you convert decimals into binary?

 

Nairi

[Sip]

Oh one thing I just remembered ... if you are using windows, just start the calculator, choose "scientific" from the view menu, and now you can convert numbers to and from binary!

 

Just type a number and either click on BIN or choose "binary" from the view menu. To go back to decimal mode, click on DEC or choose "decimal" from the view menu. That calculator is a GREAT little tool that comes very handy sometimes.

[nairi]

110110000

 

1x2^8=256

1x2^7=128

0x2^6=0

1x2^5=32

1x2^4=16

 

256+128+32+16=432

 

Now this:

 

248 in binary:

 

2, 4, 8, 16, 32, 64, 128, 256

 

248-128=120

120-64=56

56-32=24

24-16=8

8-8=0

 

120+56+24+8=208

 

What went wrong? Because if this was right then binary should be:

 

128 (2^7), 64 (2^6), 32 (2^5), 16 (2^4) = 11110000

 

Or not?

 

Btw, I don't have that calculator on my computer unfortunately. I do have a scientific one, but it doesn't have BIN or DEC on it.

 

Nairi

[Sip]

Very close but not quite. You did the conversion correctly. The only thing is that at the end, you forgot about the 8. So the binary is actually 11111000.

 

About your "check" 120+56+24+8=208 ... that is incorrect. Those are the left overs after the subtractions. They really don't mean much other than to help us figure out in the conversion process. In the end, you should check to see that the powers of 2 which you picked add up to 248: i.e. 128+64+32+16+8

 

It basically comes down to splitting the number up into a sum of powers of 2

 

If you have any MS Windows, you probably have the calculator. If you have win95 and later, click start, select run, type "calc", click on OK. It should come up. Again, assuming you have a PC with Windows. Don't know about MACs But HERE is a page which does just that + it has "explanations"

 

[ August 19, 2002, 03:48 AM: Message edited by: Sip ]

[nairi]

Okay, let me try one more:

 

674 in binary:

 

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

 

674-512=162

162-128=34

34-32=2

2-2=0

 

512+128+32+2=674

 

512 (2^9), 128 (2^7), 32 (2^5), 2 (2^1) = 1010100010

 

Thanks for the converter on that site! Very helpful to check answers.

 

Nairi

[Sip]

Yup! You got it.

 

Going the other way is a bit more interesting. The most straight forward way is to repeatedly try to find, and subtract, the largest powers of 2 that go into the number that you want to convert to binary. Instead of trying to explain it using long sentences, I'll just do an example and it should become clear.

 

Example: 432

For reference, powers of 2 are: 1,2,4,8,16,32,64,128,256,512,1024,...

 

We see 256 is the largest one which is still less than 432. Let's take it out: 432-256=176.

 

Now we see 128 is the largest one which is still less than 176. Let's take it out: 176-128=48.

 

Now we see 32 is next: 48-32=16.

Now we see 16 is next: 16-16=0.

We are done!

 

We used 256 (2^8), 128 (2^7), 32 (2^5), and 16 (2^4). So digits 8, 7, 5, and 4 should be 1 and the rest should be 0.

 

So 432 in binary is 110110000

 

You may want to try to convert it back to check to see if I did it correctly.