Key to understanding numeric bases

[sprinkles]

I was looking over old threads again and decided to write this up.

How does one understand other numeral bases besides 10? It is pretty simple actually. All bases work the same way that base 10 works.

With base 10 you only have the numerals 0 through 9 right? This means that a 10 is actually composed of separate digits, a 1 and a 0. Since there's no numeral in base 10 which is larger than 9, we use two numerals to indicate a number that is one more than 9. This denotes a 'place' or carry.

So I'm sure everyone who knows base 10 knows that in the numeral 10, the left digit represents the 'tens' place and the right represents the 'ones' place. e.g. when you have 20, the left digit represents 'two tens' and the right represents 'zero ones' How does this translate into other bases? Well other bases are exactly the same, but the bases are obviously a different multiple.

So for example with ternary numerals, you only have three numerals, 0, 1, and 2. What this means is that when you write 10 in ternary, it is not 'one ten and zero ones'. The left digit in 10 is the threes place not the tens place! It is actually 'one three and zero ones'. 20 in ternary is 'two threes and zero ones', or 'six'.

What about when there's more digits? Well, when you have more than two digits, the multiple goes up by powers. For example in base 10, in the numeral 100 you have a ones place, a tens place, and a hundreds place. What's a hundred? It is 10 to the second power. 10^2 or 10 squared.

Similarly, in ternary, with the numeral 100 the left digit is the base squared. It is 3^2 or 3 to the second power, 3 squared, or 3x3 which is 'nine'. So in ternary you don't have a ones, tens, and hundreds place. Instead you have a ones, threes, and nines place. So 20 would be 'six' and 200 would be 'eighteen'. 2x3 and 2x3 squared respectively.

Further digits in ternary, or any base, simply increase the power that the base is multiplied by 1. Therefore after 3^4 you have 3^5, 3^6, 3^7, etc.

This still works with bases larger than ten, such as hexadecimal, where the base is 16 and you have 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, and f

So in hexadecimal when you write a 10, you have a digit in the sixteens place, and a digit in the ones place. So it goes up by powers of 16 instead of 10. Therefore 20 in hexadecimal is 32 in base 10. It is two sixteens.

And that's how bases work. Pretty much all of them.

 

[just me]

How would you apply that to something you have learned through it?

 

[magister343]

Don't forget that the base does not necessarily even have to be a positive integer. Base 0 and base 1 do not make much sense, but using negative or complex bases is completely legitimate. The resulting systems can seem pretty crazy, but supposedly do have their uses.


When I was a Math minor at the Governor's Honors Program during my last summer in high school, there was a day when we were suppose to do our work in base -2.

1 = 1
10 = -2
11 = -1
100 = 4
101 = 5
110 = 2
111 = 3
1000 = -8
1001 = 7
1010 = -10
1011 = -9
1100 = -4
1101 = -3
1111 = -5
10000 = 16
10001 = 17
10010 = 14
10011 = 15
10100 = 20
10101 = 21
10111 = 19
11000 = 8
11001 = 9
11010 = 6
11100 = 12
11101 = 13
11110 = 10
11111 = 11
100000 = -32
etc


We also touched a little on base 2(-1)^(1/2)

1 = 1
10 = 2i
11 = 1 + 2i
100 = -4
101 = -3
110 = -4 + 2i
111 = -3 + 2i
1000 = -8i
1001 = 1 -8i
1010 = -6i
1011 = 1 -6i
1100 = -4 -8i
1101 = -3 -6i
1110 = -4 -6i
1111 = -3 + 6i
10000 = 16
10001 = 17
10010 = 16 + 2i
10011 = 17 + 2i
10100 = 12
10101 = 13
10110 = 12 + 2i
10111 = 13 + 2i
11000 = 16 - 8i
11001 = 17 - 8i
11010 = 16 - 6i
11011 = 17 - 6i
11100 = 12 - 8i
11101 = 13 - 8i
11110 = 12 - 6i
11111 = 13 - 6i
100000 = -32i
etc

 

[sprinkles]

How would you apply that to something you have learned through it?

It's applied in a lot of ways. Converting from metric measurements to other standards for example, e.g. every 1 meter is 3.28084 feet.

Time is also standardized as base 60. It's a hybrid format because it's still expressed in base 10 numerals, but you have 60 seconds in a minute and 60 minutes in an hour.

So let's look at it this way. If the time is 12:42:56, this works exactly like digits if you were to presume that each segment is a quantity of seconds. Therefore you can figure out how many seconds this is. Since hours, minutes and seconds work the same as three digits in base 60 you can imply that 12 hours is 12x60^2 seconds, or 43200 seconds. 42 minutes is 42x60^1 seconds, or 2520 seconds. 56 seconds is 56x60^0 seconds, or 56 seconds. Add up all of these and you get:

43200 + 2520 + 56 = 45776 seconds

 

[sprinkles]

Don't forget that the base does not necessarily even have to be a positive integer. Base 0 and base 1 do not make much sense, but using negative or complex bases is completely legitimate. The resulting systems can seem pretty crazy, but supposedly do have their uses.


When I was a Math minor at the Governor's Honors Program during my last summer in high school, there was a day when we were suppose to do our work in base -2.

Yeah I gave that some consideration but I didn't want to scare people off when a lot of people can hardly get around positive integers for bases.

I'd never really worked with non positive integer bases before but I considered the possibility of it, and since you mentioned it I decided to look it up and learn negabinary. The same principles seem to apply in negabinary from what I can tell, but what can make it tricky is the multiplication of positive vs negative numbers causing the sign to flip back and forth between even and odd powers.

I figured out that this is why 110 in negabinary is 2 in decimal, because 100 in negabinary is 4 in decimal, and 10 in negabinary is -2 in decimal.

 

[just me]

16 ounces is 1 pound, so 16= 1 pound 6 ounces
Do they put WT in front of the numbers?
WT16= 1 pound 6 ounces

Why is binary so magical? 0b will be changed.

It's there, but it is still one pound six ounces to me.

A kilo could be a form of measuring, but what are all the codes, or first two figures for each type?

 

[sprinkles]

16 ounces is 1 pound, so 16= 1 pound 6 ounces
Do they put WT in front of the numbers?
WT16= 1 pound 6 ounces

Why is binary so magical? 0b will be changed.

It's there, but it is still one pound six ounces to me.

A kilo could be a form of measuring, but what are all the codes, or first two figures for each type?

Well the idea is that if you understand this theory then you can understand why any base works. You won't just have to remember how every system works, you can figure it out even if it hasn't been used by you before.

For example with ounces and pounds, this is also sort of base 16. It's written in hybrid format like time is, but if you look at it so that ounces are 0 through 15 and when you have 1 more than 15 it becomes 1 pound then that is also kind of how base 16 works.

Measuring systems and such may appear different because they typically do not track empty places. This means when you have 1 pound, you just write 1 pound or 1lb. So with measuring weights it some times appears to be simple multiplication because if you have only 1lb, you write 1lb and not 1lb 0oz

But yes when it is ambiguous you can write a symbol or abbreviation to tell what the number represents, as with pounds you can write lb. Other bases also have a a mark to distinguish them but it is not always written with them.

Binary can be special because it is actually mathematically simple. The only numerals to do operations on are 0 and 1 which makes arithmetic quite trivial, since you're always dealing with those two numerals. This also means that binary arithmetic is very simple for machines to use, which is why computers use binary. Since there's only two numerals and they trivially match to the state of circuits - off or on - then you can make circuits that do math for you.

Binary was also used in Taoism and the Bagua. Just look here at the dashes representing Yin and Yang - they are binary numbers!

 

[Cornerstone]

This thread made me dizzy

 

[muir]

This thread made me dizzy

Yeah what language is that they're speakin?

 

[Cornerstone]

Freedom is the freedom to say that 10010 equals 16 + 2i. If that is granted, all else follows.

 

[sprinkles]

Freedom is the freedom to say that 10010 equals 16 + 2i. If that is granted, all else follows.

It's all symbols. If 5 represents ***** then why can't 101? What's the difference?

5 isn't really ***** is it? It's just defined as such.

That is the difference between numbers and numerals. Numbers have an underlying fundamental theory.

The nun Wu Jincang asked the Sixth Patriach Huineng, “I have studied the Mahaparinirvana sutra for many years, yet there are many areas i do not quite understand. Please enlighten me.”
The patriach responded, “I am illiterate. Please read out the characters to me and perhaps I will be able to explain the meaning.”

Said the nun, “You cannot even recognize the characters. How are you able then to understand the meaning?”

“Truth has nothing to do with words. Truth can be likened to the bright moon in the sky. Words, in this case, can be likened to a finger. The finger can point to the moon’s location. However, the finger is not the moon. To look at the moon, it is necessary to gaze beyond the finger, right?”

If you're only looking at the 5 then you're looking at the finger, not the 'moon' it is pointing to.

 

[Cornerstone]

I get the theory, I just go dizzy when you actually start using the symbols.

 

[sprinkles]

I get the theory, I just go dizzy when you actually start using the symbols.

It's probably the way math is taught that causes this. You're probably using different bases more than you know but it's hybridized as base 10, but really another base in disguise.

Normally a base has b-1 numerals, but base 10 has been used in a lot of places that aren't actually base 10 which leads to confusion.

Take ounces and pounds for example. Using base 10 we might write 1lb 12oz which looks disconnected and you may not notice that a carry is taking place here. The base 10 notation disguises it, so that you can't write it as 112 because it wouldn't make any sense.

However if pounds and ounces used b-1 numerals, in this case 16-1, then you could clearly write an amount of pounds and ounces as one numeral with two digits. 1lb 12oz could instead be written as 1C because you have a 1 in the pounds place, and a C in the ounces place. Converted to base 10 this would represent 1 in the sixteens place, and 12 in the ones place - exactly equal to 1lb 12oz.

Edit: actually I wrote that slightly wrong. It's actually b numerals that range between 0 and b-1. It's easy to see what I meant though.

Also a base doesn't need to have b numerals. If a negative base had b numerals it would have less than 1 numeral which makes no sense. But in general cases b numerals is helpful.

 

[muir]

Freedom is the freedom to say that 10010 equals 16 + 2i. If that is granted, all else follows.

Oh my god...they've got to cornerstone! Is there no one left?

 

[just me]

Have you divided the spoils in Texas again?

 

[Tin Man]

That was very interesting. And I agree with your belief that people are unreceptive to math because of the way it was taught.

Well the idea is that if you understand this theory then you can understand why any base works. You won't just have to remember how every system works, you can figure it out even if it hasn't been used by you before.

The same can be said for other areas such as programming. If you understand the theory of say Java, you have the foundation to utilize many others (with exceptions to those deliberately esoteric).

 

[just me]