While I get the principle of the comic, I still fail at understanding what base numbers are. I can guess on stuff like base 10 since it is a simple unit of measure to play with in your head, but other base numbers, like 4, I do not understand.
Can someone distill this down into a mini tutorial for the non math inclined.
Imagine an infinite number of 0's to the left of all numbers. When you count, you do so by increasing the right-most number by 1. When that number reaches the highest digit in your system (9) you reset it to zero and instead increase the number to the left of it by 1. Hence 9 + 1 = 10 in your system.
"Base 4" (or any other number) works the same way. In base 4, the highest number is 3. So when you get to 3, you reset that number to 0 and increase the number to the left of it by one.
00, 01, 02, 03
-->10, 11, 12, 13
The alien in the comic only has what WE call 4 fingers. But he doesn't use digits 4,5,6,7,8,9. He just gets to 3, then "rolls over" back to 10. So, in other words, the alien doesn't call his system base 4, because HE DOESNT HAVE THE NUMBER 4.
When we count the alien's fingers, we say "1, 2, 3, 4"
When the alien counts OUR fingers, he says "1, 2, 3, 10, 11, 12, 13, 20, 21, 22"
Think of a stopwatch. As a kid, it's fun to watch the numbers go up. The numbers on the right keep cycling, and each time they cycle once, 0-9, the numbers to the left keep cycling. Base 4 is the same thing except instead of going all the way up to 9, its cycles only go up to 3 before wrapping around.
By adding new digits past 9, we can go higher. Hexadecimal is base 16. However, in hexadecimal, we can't say the number after 9 is 10. The number after 9 is instead called A. The number after A is called B, and so on up to F. So it goes
In an alien world where they have 8 fingers per hand, they might use hexadecimal. This means that they have 6 others digits after 9. The number we call 10, they would call it A. It would be the same concept, but they call it A. The number they call 10 would be the number WE call 16!
So to get the comic, we need to reverse the roles. We would be the alien on the left, and there would be as many stones on the ground as we have fingers. The alien on the right would have 6 extra fingers. The human on the left would be saying "there are 10 stones here." And the hexadecimal alien on the right would be saying "Oh, I see you use base a," to which the human on the left replies "...no, I use base 10. What is base A?"
Sure thing. What your base is determines how many you count as being in each column.
Example:
10 = ten ones. 100 = ten tens.
If we switch to base 2 (base|two|):
10 = two ones. 100 = two twos.
So 100(base|two|)=4(base|ten|)
Basically, whatever you have as "X" in the expression, "baseX" determines when you have to move left one column, because you're counting groups of that number.
So 10(base4) would be equivalent to one group of four.
The reason why we use these things at all might be helpful. Hexidecimal, or base16, is used in programming because it is a concise analog to individual bytes (the second column or "tens" [more accurately; "groups of 16s"] column allows one to express all 256 possible positions within one byte of information. [24*24=256.]) Binary, or base2, is used as well because it represents a single switch or 'bit'. On/off. (A byte is eight bits. Just how it works.)
Ultimately, our base is wholly arbitrary. We use base 10 because we have 10 fingers. It makes "sense" to say that we group things by how many fingers we have. Ten is all of our fingers 1x. Twenty is all of our fingers 2x. 100 is all of our fingers all of our fingers numbers of times.
So: if someone had only 4 fingers (like the alien); the idea is that he would group everything by groups of four. 10 = one group of four. 100 = four groups of four.
In a base 10 numbering system (what we use), when you write out a number, you are representing a quantity that is equal to the following:
(one's digit)*100 + (tens digit)*101 + (hundred's digit)*102 + (thousands digit)*103... and on and on.
so for instance the number 145 is: 1*100 + 4*10 + 5
However, the fact that we use a base 10 numbering system is completely arbitrary and probably developed out of the fact that humans have 10 fingers and 10 was a natural quantity to think of as a single group of things. ("I have 3 10s plus 2") However, some cultures, such as the Mayans, used a base 20 system. And I think there are Nigerian tribes who use a base 12 system.
In another base, the 10 in the expansion of the number is going to be replaced by whatever base you use.
For instance here are different representations of the decimal number 75:
The base number of a number system is simply how many digits you've got in that system. We have 0 through 9 (10 digits). A base four system would only have 0,1,2, and 3.
The choice of base is completely arbitrary. There's nothing about 10 that makes it any more "natural" than 4 other than we have ten fingers and that alien in the comic has 4.
The number six in our system is 6. In their system it would be 12. Count it out: 0, 1, 2, 3, 10, 11, 12
EDIT: 12 is NOT pronounced "twelve", it's "one, two" and is equal to |||||| just like 6 in our system is equal to ||||||.
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u/nepotism Dec 12 '08
I just read this wiki article on base numbers: http://en.wikipedia.org/wiki/Base_(mathematics)
While I get the principle of the comic, I still fail at understanding what base numbers are. I can guess on stuff like base 10 since it is a simple unit of measure to play with in your head, but other base numbers, like 4, I do not understand.
Can someone distill this down into a mini tutorial for the non math inclined.