Re: [AlwaysLearning] Digest Number 353
KT
>Okay, my almost 20 yo who went to public school save for 9th grade
>
>or that 2 + 2 doesn't always equal four
>
always likes to start these heated debates with me, and he said
such-and-such was fact just like "2+2=4". Of course, I had to mention
that 2+2 doesn't ALWAYS equal 4 and he called me on it. Not being a
mathhead, I could only vaguely remember Pam S posting about it long ago
in the AOL math folder and that it had something to do with base 8. I
googled but couldn't find anything and we moved on in our silly debate,
he believing that he'd won a point, when he still hasn't figured out
that I'm humoring his youthful arrogance. (And I can't even REMEMBER
what the debate was about in the first place!)
So, when doesn't 2+2 equal 4 again? <sheepish grin>
Tuck,
Fetteroll
on 5/10/02 2:57 PM, KT at Tuck@... wrote:
bases during the course of daily life so the explanations, just as in
school, go in but don't have anything to stick to os eventually fade away.
As far as I know the only people who put bases to practical use are computer
scientists.
It *can* lead to an "aha" moment of why numbers are represented the way they
are for some people. And for others it can just be a lot of seemingly
useless mathematical noise. ;-)
In base 3 2+2 would equal 11.
But calling it base "3" is confusing since there wouldn't be a single digit
for three. It would be represented as 1 followed by a 0 meaning 1 set of
three and 0 sets of units. So counting in base 3 would go:
01
02
10
11
12
20
21
22
100
101 <- that represents 1 set of 3x3 and 1 set of units or 10 in base ten.
and so on.
So 1202 would represent:
1 set of 3x3x3 = 27
2 sets of 3x3 = 18
0 sets of 3 = 0
2 sets of 1's = 2
which all adds up to and is 47 in base 10.
In base 4 2+2 would equal 10. (1 set of 4 and 0 sets of units.)
In base 5 and all bases above 2+2 would equal 4.
In any base the columns represent how many of that power of that number you
have so:
2345 in base 10 is:
2 x 10x10x10 (10 to the 3rd power which I would write as 10 with a 3 up in
the air if I could!)
3 x 10x10 (10 to the 2nd power)
4 x 10 (10 to the 1st power)
5 x 1 (10 to the 0th power)
2345 in base 8 is
2 x 8x8x8 = 2 x 512 = 1024
3 x 8x8 = 3 x 64 = 192
4 x 8 = 4 x 8 = 32
5 x 1 = 5 x 1 = 5
which in base 10 is 1253.
Every time you write down a number you're doing addition, multiplication and
powers :-) In fact when you write down money, you're using negative powers:
$1.67 is
1 x 1 (10 to the 0th power)
6 x .1 (10 to the -1 power)
7 x .01 (10 to the -2 power)
Things get interesting in bases above 10 since we don't have single digits
for numbers above 9. As far as I know, the only base that gets used is base
16 for computer work and computer scientists use letters:
base 8 base 10
1 = 1
2 = 2
3 = 3
4 = 4
5 = 5
6 = 6
7 = 7
8 = 8
9 = 9
0A = 10
0B = 11
0C = 12
0D = 13
0E = 14
0F = 15
10 = 16
11 = 17
12 = 18
13 = 19
14 = 20
15 = 21
16 = 22
17 = 23
18 = 24
19 = 25
1A = 26
1B = 27
and so on.
So in base 16 8+8 is 10. (1 set of 16 and zero units.)
A+A = 14 (10+10 = 1 set of 16 plus 4 sets of units)
D+D = 1A (13+13 = 1 set of 16 plus 10 sets of units)
If anyone is curious why anyone would want to do such nonsense, it's because
computers at one time sent information in chunks of 16 bits. (You may have
heard the term 16 bit addressing. I think they're up to 32 bit addressing
now.) Computers, of course, use base 2 arithmetic. (Each bit (physically a
wire or some channel carrying electricity or, now, light) can only exist in
one of two states, either on or off, represented by 1 and 0). To talk about
a 16 bit number you need a string of 16 1's and 0's. So to talk about a
particular location (address) in a computer, you'd need to write out 16 1's
and 0's. It's a lot easier to compress that into a single digit that not
only represents that string of 16 but can also be easily expanded back into
the string of 1's and 0's.
Which is probably way more than anyone wanted to know ;-)
Joyce
[Non-text portions of this message have been removed]
> So, when doesn't 2+2 equal 4 again? <sheepish grin>No need to be sheepish about it! Not too many people need to think in other
bases during the course of daily life so the explanations, just as in
school, go in but don't have anything to stick to os eventually fade away.
As far as I know the only people who put bases to practical use are computer
scientists.
It *can* lead to an "aha" moment of why numbers are represented the way they
are for some people. And for others it can just be a lot of seemingly
useless mathematical noise. ;-)
In base 3 2+2 would equal 11.
But calling it base "3" is confusing since there wouldn't be a single digit
for three. It would be represented as 1 followed by a 0 meaning 1 set of
three and 0 sets of units. So counting in base 3 would go:
01
02
10
11
12
20
21
22
100
101 <- that represents 1 set of 3x3 and 1 set of units or 10 in base ten.
and so on.
So 1202 would represent:
1 set of 3x3x3 = 27
2 sets of 3x3 = 18
0 sets of 3 = 0
2 sets of 1's = 2
which all adds up to and is 47 in base 10.
In base 4 2+2 would equal 10. (1 set of 4 and 0 sets of units.)
In base 5 and all bases above 2+2 would equal 4.
In any base the columns represent how many of that power of that number you
have so:
2345 in base 10 is:
2 x 10x10x10 (10 to the 3rd power which I would write as 10 with a 3 up in
the air if I could!)
3 x 10x10 (10 to the 2nd power)
4 x 10 (10 to the 1st power)
5 x 1 (10 to the 0th power)
2345 in base 8 is
2 x 8x8x8 = 2 x 512 = 1024
3 x 8x8 = 3 x 64 = 192
4 x 8 = 4 x 8 = 32
5 x 1 = 5 x 1 = 5
which in base 10 is 1253.
Every time you write down a number you're doing addition, multiplication and
powers :-) In fact when you write down money, you're using negative powers:
$1.67 is
1 x 1 (10 to the 0th power)
6 x .1 (10 to the -1 power)
7 x .01 (10 to the -2 power)
Things get interesting in bases above 10 since we don't have single digits
for numbers above 9. As far as I know, the only base that gets used is base
16 for computer work and computer scientists use letters:
base 8 base 10
1 = 1
2 = 2
3 = 3
4 = 4
5 = 5
6 = 6
7 = 7
8 = 8
9 = 9
0A = 10
0B = 11
0C = 12
0D = 13
0E = 14
0F = 15
10 = 16
11 = 17
12 = 18
13 = 19
14 = 20
15 = 21
16 = 22
17 = 23
18 = 24
19 = 25
1A = 26
1B = 27
and so on.
So in base 16 8+8 is 10. (1 set of 16 and zero units.)
A+A = 14 (10+10 = 1 set of 16 plus 4 sets of units)
D+D = 1A (13+13 = 1 set of 16 plus 10 sets of units)
If anyone is curious why anyone would want to do such nonsense, it's because
computers at one time sent information in chunks of 16 bits. (You may have
heard the term 16 bit addressing. I think they're up to 32 bit addressing
now.) Computers, of course, use base 2 arithmetic. (Each bit (physically a
wire or some channel carrying electricity or, now, light) can only exist in
one of two states, either on or off, represented by 1 and 0). To talk about
a 16 bit number you need a string of 16 1's and 0's. So to talk about a
particular location (address) in a computer, you'd need to write out 16 1's
and 0's. It's a lot easier to compress that into a single digit that not
only represents that string of 16 but can also be easily expanded back into
the string of 1's and 0's.
Which is probably way more than anyone wanted to know ;-)
Joyce
[Non-text portions of this message have been removed]
[email protected]
In a message dated 5/11/2002 7:21:39 AM Eastern Daylight Time,
fetteroll@... writes:
Nevertheless, it WAS interesting (when I wasn't glazed over).
At least I know where to send my children when they ask! <G>
Kelly
[Non-text portions of this message have been removed]
fetteroll@... writes:
> Which is probably way more than anyone wanted to know ;-)Yup.
>
Nevertheless, it WAS interesting (when I wasn't glazed over).
At least I know where to send my children when they ask! <G>
Kelly
[Non-text portions of this message have been removed]
Nancy Wooton
on 5/11/02 4:18 AM, Fetteroll at fetteroll@... wrote:
Nancy
(Like I read it!!! LOL My eyes glaze over at Digest Number whatevers)
> Which is probably way more than anyone wanted to know ;-)Yes.
>
> Joyce
Nancy
(Like I read it!!! LOL My eyes glaze over at Digest Number whatevers)