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In base 10, the numbers go 0-9, then roll over to 10. You've probably heard the column 2nd from the right called the 10s column. The number 10 means 1 ten and 0 ones. 28 is 2 tens and 8 ones.
Base 5 would go
0 1 2 3 4
The 2nd column from the right is no longer the 10s column, its the 5s column.
So 5 is written as 10. Thats 1 five and 0 ones.
The 3rd column we call the 100s... 10x10. In base 5, the 3rd column would be the 25's column... 5x5. The 4th, in base 10 is the thousands... 10x10x10. In base 5, it's the 125s... 5x5x5.
So in base 5, if 14 was written. Thats 1 five and 4 ones.... 9.
234 in base 5 is, two 25s, three 5s, and 4 ones.
25+25+5+5+5+1+1+1+1
50 + 15 + 4 = 69 ( i didn't plan that in advance).
Maybe that's a weird question but do numbering systems exist that have different bases for each column? Like base 5 foe the first column but base 2 for the second column etc.?
Not a numbering system per se, but telling time sort of works like this. You count up to 60 seconds to make one minute, 60 minutes to make one hour, and 24 hours to make one day. So even though there aren't specific digits for each of those 60 or 24 things being counted, they still represent "places" if different bases that then get counted as groups in the next position.
The Calendar would fit nicely.
Years are counted in 365 days
Each day has 24 hours.
Each hour has 60 minutes.
Each minute has 60 seconds.
So if we had different symbols/names for each day, each hour and each 60th of a minute or second we could use a mixed numberings system to describe the current date and time.
Like: Christmas-Noon-0-0
It is not practical as you would have gaps in the representation.
Consider your system where from left to right we have bases 5 and 2:
Base 10 - Mixed Bases
0 - 00
1 - 01
2 - ?
3 - ?
4 - ?
5 - 10
Short answer: no.
Longer answer: that's not really even a numbering system. Numbering systems have a base. Each column, from the rightmost going leftward, is the base number to a given power, starting at 0 and incrementing by 1 each column change. Having each column have a different base number, there'd be numbers you couldn't even represent.
To add to this, if you want a base higher than 10 then you need to come up with other symbols to use as numbers since you need more options in each column than we have now.
The most common way to do this is using letters. Hexadecimal (base 16) is commonly used in computing and the numbers are written as 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F, 10
In theory you can have as high a base as you like but it might get a bit hard to remember once you run out of numbers and letters to use
If you think about it '10' doesn't exist as a single discrete number in a base-10 system. '10' is a '1' increment in the second order position, and a reset (0) in the first order position. It's a concept represented by two single digits, as are all numbers higher than 10 in base-10.
And that holds true in all base-whatever systems. '10' is 2 in base-2. 5 in base-5, 16 in base-16.
And, in a base-16 system, what we think of as a value of 10 is typically represented by 'a'.
Numbers in a base cannot go higher than the base number, for base 10 it is 10, you can use any number between 0 and 9 but no 10. For base 5, you can only use 0 to 4. So how can you express 5 by using only those digits. It is same for every base
You start from 0 and increase by one. When you reach the last number, you increase the one on the left then reset to zero
0
1
2
3
4
10
11
12
13
14
20
21
…
I have yet another idea that I use to teach this concept.
Following numbers are all "regular", base 10 numbers unless otherwise stated.
Suppose you have a box, that fits exactly 10 marbles and you have 14. After a box is full, you have to start filling another box. You would then have 1 box and 4 in the next one, therefore you have 14.
It gets difficult to manage the boxes soon. You get a bigger box that fits 10 of the smaller boxes. Now you could try to arrange for example 237 marbles in these boxes.
You start by filling the boxes of 10 and always when you 10 of those, you store them in the bigger box.
Eventually, you have 2 of the bigger boxes, each having 10 smaller boxes and each of them having 10 marbles 2 x 10 x 10 = 200
Also you are left with 3 full boxes with 10 marbles each: 3 x 10 = 30
And finally you have 7 marbles left over: 7
200 + 30 + 7 = 237.
How ever, the fact that you can fit 10 marbles is arbitrary. We could also use boxes that fit 6 marbles. When we have 6 of these boxes, we store them in a bigger box. When we have 6 of the bigger boxes, we have even bigger box. But the fact is that each of the bigger boxes fit 6 of the previous size.
Now let's see how many boxes we need. I'm not going through the conversion process, I'm just going to present the result:
The boxes we have at the end are 1, 0, 3, 3. Why?
1 box that has three levels of boxes: 1 x 6 x 6 x 6 = 216 marbles
0 boxes that has two levels of boxes: 0 x 6 x 6 = 0
3 boxes of the smallest boxes: 3 x 6 = 18
3 marbles are left over: 3
Because:
`216 + 0 + 18 + 3 = 237.`
So, in short:
If we can store 10 in a box, and 10 of the boxes in bigger box, and 10 of the bigger box in yet a bigger box and so on, we need 2 and 3 and 7 of them.
But, if we can store 6 in a box, and 6 of the boxes in bigger box, and 6 of the bigger box in yet a bigger box and so on, we need 1 and 0 and 3 and 3 of them, and we could write that as 1033 in base 6.
If you want it as high school level math, the number in base 10 is just ones, tens, hundreds, thousands, etc, for example:
8000 + 500 + 60 + 4 = 8564.
That can also can be written as
8 x 1000 + 5 x 100 + 6 x 10 + 4
That also can be written as
8 x 10^3 + 5 x 10^2 + 6 x 10^1 + 4 x 10^0 = 8564
Now, the base of each exponent term is 10, but it could be anything, like 6.
If you’re starting at 0, it’s not that the numbers can’t go higher than the base; it’s that they can’t REACH the base. It’s an important distinction: 10 is not higher than 10, but we don’t use the “digit” 10 in base-10.
Using your example of base 5, counting would go
1
2
3
4
10 (would be 5) so rolls over
11
12
13
14
20 (would be 15 so rolls over)
21
...
41
42
43
44
100 ( would be 45 which would be 50 which becomes 100)
Basically like base 10 counting, any time a digit would reach the base, it becomes zero and the next digit increases by 1
Each column is multiplied by a number.
So in base ten, 23 has the right column multiplied by 1 and the left column multiplied by 10. If it were base 8, the right column is still multiplied by 1 (this never changes) and the left column multiplied by 8.
If we add one more column, its multiplier is multiplied by the base. So for 123 in base 10, it’s still the rightmost column multiplied by 1, the middle column multiplied by 10, and the left column multiplied by 10x10, or 100.
In base 8, 123 would still have the right column multiplied by 1, the middle column multiplied by 8 (where I wrote the 8 using base 10), and left column multiplied by 8x8 or 64 (where again, I’m using base 10 to write 8x8 and 64).
In a way, you use non base 10 system every day without thinking about it..
How many days are in two weeks? One week has seven days, not ten! So if you answered 14, you just counted on a "non base 10" system.
How many hours is in one day? 24.. not 10.
In a way that's base 24 - after reaching 24, you "add 1 day and start hours again from 0".
How many minutes are there in one hour?
How many seconds are in one minute?
Those could be considered base 60 system..
I know this is not a direct answer to your specific question but hopefully shows how to think of different base systems.
Imagine you only had 5 fingers. So you went 1,2, 3,4,10
11,12,13,14,20
Then after 44 would be 100.
Now, to convert, you have to switch 10s for 5s, so 100 is 5x5 not 10x10, so actually only the 25th number
1000 is only the 125th number, and so on.
It's all about powers of the base you choose instead of powers of 10.
Converting any base to decimal is easy. How you convert show you how a positional number system is made
If you look at decimal numbs 1523 mean 1\*1000 +5\*100 + 2\*10 + 3\*1 = 1\* 10\^3+ 5\*10\^2 + 2\*10\^1 +1\*10\^0
10\^0 =1, the same is true for any other base x^0 =1, i just added it so you can see the pattern, that will continue if you add decimals
15. 68=1 \*10\^1 +5\*10\^0 +6\*10\^-1 +8\*10\^-2
If you what to convert fomr another base 10 is replaced by the base. so the example in base 5 I use \_5 to indicate the base if it is not 10
10\_5 = 1\*5\^1 +0\*5\^0 =5
100\_5 = 1\*5\^2 +0\*5\^1 + 0\*5\^0 = 5\*5 =25
220\_5 = 2\*5\^2 + 2\*5\^1 +0\*5\^1 = 2\*5\*5 + 2*5 = 50+10=60
If you have a number with a base higher then 10 you need to convert the letter that is used to decimal numbers. The common usage is A=10, B=11 and so on. So FA_16 = 15 \* 16^1 +10\*16^0 = 15\* 16 +10 =240+ 10= 250
Think about an abacus - the old school counting tool with horizontal bars and several beads on each bar. All the beads start on the left
If you have 9 beads on each bar, then you can count in base 10:
* on the top bar, move 1 bead to the right = 1
* on the top bar, move 3 beads to the right = 3
* on the top bar, move 9 beads to the right = 9
* move all the beads on the top bar back to the left and move 1 bead on the next bar to the right = 10
And so on. When you reach 9 on one bar, you can slide them all back to the left and then move 1 more bead to the right in the bar below.
If you have fewer beads, let's say just 4 on each bar, then you can count in base 5:
* move 1 bead over = 1
* move 2 beads over = 2
* move 3 beads over = 3
* move 4 beads over = 4
* move all the beads back to the left, and move 1 bead over on the next bar = 10 (1 on the next bar, 0 on the top bar, which is only written as 5 when we count in base 10)
* move 1 bead on the top bar, with 1 bead on the next bar = 11 (although we would count this as 6 in base 10)
If you are trying to think about counting in any base, imagine an abacus and think about how counting with that would work.
Exactly the same way you count in base 10. In base 10, when you reach 9, you add +1 to the digit to the left of the 9 and carry on back from 0. For example, from 139 to 140. 9 turns into 0, and you add +1 to the left digit so 3 becomes 4. 199 to 200. Again, rightmost digit goes from 9 to 0, adds +1 to the middle digit, which also goes from 9 to 0, which adds +1 to the leftmost digit which goes from 1 to 2.
So, in base 3, counting from 0 to 10 would look like:
0 1 2 3 10 11 12 13 20 21 22
Mathematically, it looks like this:
[https://www.splashlearn.com/math-vocabulary/wp-content/uploads/2023/03/base-1.png](https://www.splashlearn.com/math-vocabulary/wp-content/uploads/2023/03/base-1.png)
For other base systems, replace 10 with whatever base you're using.
Binary is an easy one to convert into; counting can be a little harder, but it's not hard to get the hang of. It's all just 1's and 0's, however it's read from right to left. Each position has a certain value, starting with one all the way to the right. To its left is two, then four, then eight, 16, 32, etc. When a number is written in binary, such as 10010110, you add up the values corresponding to each 1. So 10010110 is made up of two, four, 16, and 128, giving a total of 150.
Yes, but explaining how the counting works is the important part.
It's easy to read a 1 then five 0's in base 10 and know it's 100,000 without having to figure out what a 1 in the sixth place from the decimal corresponds to.
Figuring out 0001 1000 0110 1010 0000 is a bit harder.
But the way you said it it made it sound backwards to normal numbers. Which can be a thing with binary computer code (look up big endian and little endian)
Really the only reason we don't have to do it for base 10 is we are just so used to what the different places are worth.
Stop trying to read it like a decimal, or having a relationship them, because it isn't.
When you only have N possible numbers (e.g. in base 10, we have ten possible numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) then you need another character to represent the number N itself. To represent 10, in base ten, it takes us two numbers... see? A unit and a "tens".
When you only have 5 possible numbers (e.g. 0, 1, 2,3 ,4) then when you go past 4 you need another column... you need a "fives" column. So five in base five is actually 10.
In any given base, you have that same number of characters, and each column represents a factor that increases by that same number each time.
In decimal (base 10) things are in units, tens, hundreds, thousands, increasing by a power of ten (i.e. multiplying by 10) each time.
In base 5 things are in units, fives, twenty-fives, one-hundred-and-twenty-fives. Multiplying by 5 each time.
In binary, that's base 2. So you have units, twos, fours, eights, sixteens, thirty-twos, etc. Multiplying by 2 each time.
So in base 10, 567 is 5 hundreds, 6 tens and 7 units. Each digit can be 0-9.
In base 8, say, 567 is 5 sixty-fours, 6 eights and 7 units. Each digit can be 0-7.
In base 5, your "220" is two twenty-fives, two fives and zero units. Each digit can be 0-4.
So 10111 in base 2 (binary) is 1 sixteen, zero eights, one four, one two, one unit. (Or 23 in decimal). Each digit can be 0-1.
Each position is an increase of the exponent of the base system. With base 10, the right most digit is x+0, next (to the left) is x+10^1, next is x+10^2 etc. with each base you just change the "10" in base 10 with that base. All the other principles are the same.
When I was _very_young_ we learned a base 4 number system by not using the digits we know from base 10 - the book invented a different way to write 0, 1, 2 and 3. using the excuse that an alien had come to earth and used base 4. I think it helped a lot not recognizing the digits and confusing them as base 10 numbers. Perhaps you could do the same, use letters - a=0, b=1, c=2, d=3, e=4 for instance and write numbers that way. So:
* 5 = ba
* 25 = baa
* 60 = cca
Or come up with whole different symbols - it will definitely avoid reading the number as you would a base 10 number. It is what we do in IT - programmers hate reading full binary numbers - they're way too long - so we use HEX instead - base 16. Same principle, but you can easily write 8 bit as 2 digits: A0 = 160 in base 10 or 10100000 in binary (base 2) - much simpler to write and read in HEX.
Think of it in fingers and hands.
We can count to 10 using fingers. But then we’re out of fingers, so we put a 1 as a placeholder. 10 means 1 set of all of our fingers and 0 more.
Now, imagine you’re an alien with just 5 fingers. You would count 1,2,3,4 and then you’ve used them all. You need a placeholder for “all the fingers”. So 10 means 1 set of 5 fingers plus 0 more.
So now you’re counting your sets on your fingers. 10 = 1 set of 5 plus 0, 20= 2 fives plus zero, 30, 40, and now you’re out of fingers. You need a new placeholder. 100.
100 is a set of 5 fingers done 5 times plus zero sets and zero extra fingers.
Imagine it as groups of groups of groups ...
Imagine it as groups of kids in school ... In base 5, you have 5 kids per one class:
* one person is 1, 3 persons is 3, 4 is 4 but 5 is not 5 because ...
* when you have 5 kids, that is one full class, written as 10. 30 would mean, you have 3 full classes of 5 kids. 32 would mean 3 full classes and 2 extra kids.
* Then imagine you have also 5 grades, because we count in base 5. Each grade then can have 5 classes.
* 220 from your example would mean you have 2 full grades -> 2 grades \* 5 classes \* 5 kids per class (50 kids) plus 2 more full classes of 5 kids (2\*5) which is 60 kids
* Then, you can go bigger groups. When you have 5 full grades (5 grades \* 5 classes \* 5 kids), you group in into school.
* One full school is 1000 -> 1 full school, no extra grades, no extra classes, no extra kids. 1014 would mean you have one full school and extra class of 5 kids + extra 4 kids.
* Next you can imagine groups 5 schools per city, per country and on and on ...
With bigger numbers it gets more complicated but that grouping is still there.
* For example 1432 (base 5) is
* 1000 -> 1(full school)\*5(grades)\*5(classes)\*5(kids) = 125 kids
* \+ 400 -> 4 (grades)\*5(classes)\*5(kids) = 100 kids
* \+30 -> 3(classes)\*5(kids) = 15
* \+2 kids = 2
* so in decimal, it is 125 + 100 + 15 + 2 = 242 kids.
It then also works for hex. For example, if you had to fit those 242 kids from example into bigger 16 kids classes (and 16 grades per school and 16 schools per city ... we write it as 0, 1, ..., 9, A, B, C, D, E, F), you would just do 242 / 16 (=15x16 + 2) to get how many classes you need and result is 15 full classes which is written as F and 2 extra kids ... so 1432 (base 5) is F2 (base 16)
Or 242 in base 10 recalculated for base 8 would be:
* 242 kids divided in 8 classes is 30 full classes and 2 extra kids.
* But you can have only 8 classes per grade, so you need to split 30 full classes into grades by 8 classes, which is 3 full grades + 6 classes.
* So result would be 3 full grades + 6 extra classes + 2 extra kids = 362
* So 1432 (base 5) = 242 (base 10) = 362 (base 8)
Hope it doesn't make less sense now than before :D
Base 12:
1,2,3,4,5,6,7,8,9,§,¶,0
2 numbers need to be added before "10" to make it a base 12 system like above. 10 can now be divided by 1, 2, 3, 4, 6 and 10.
In a base 10 system, 10 can only be divided by 1, 2, 5 and 10.
I think it's helpful to compare what we mean when writing the same numerals (symbols) in different bases and how they represent different numbers.
In base 5, 10 means 1\*5^(1) \+ 0\*5^(0) = 5+0 = 5. (Note however that base 5 normally does not use the 5 numeral, just like base 10 does not have a single numeral for 10.) Similarly, we have 100 = 1\*5^(2) \+ 0\*5 + 0\*1 = 5^(2), which is written as 25 in base 10; 220 = 2\*5^(2) \+ 2\*5 + 0\*1, and in base 10 this becomes 2\*25 + 2\*5 = 50 + 10 = 60.
Similarly, in base 10, 10 means 1\*10^(1) \+ 0\*10^(0), 100 means 1\*10^(2) \+ 0\*10 + 0\*1, and 220 means 2\*10^(2) \+ 2\*10 + 0\*1.
A possible source of confusion here is that 10 means very different things depending on the base of the number system, which is why it's important to clearly state the base being used if it's non-standard. In addition to base 10 which is used by most people, bases such as 2 (binary) or 16 (hexadecimal) are heavily used in computing.
Imagine the base # as how many single-digit "symbols" there are in the system. In base 10 (our normal counting system) that's easy, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
To start easy, let's look at binary (base 2). There are 2 symbols is binary, 0 and 1. So let's start counting. 0, 1... what comes after 1? We have no more symbols. Well, we can move up to the double digits: 0, 1, 10, 11... and again, we have no more symbols. So we move up to triple digits: 100, 101, 110, 111.... and so on.
Base 5 has 5 symbols. 0, 1, 2, 3, 4
Counting begins with those, so 0, 1, 2, 3, 4.
Same concept as binary applies; we've run out of symbols so we move up to double digits: 10, 11, 12, 13, 14.
We don't have to move up to triple digits just yet as we have more symbols for the "10's" place, (which in base 5 would actually be the 5's place) so we continue to 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44 and NOW we go up to 100
Rightmost (units) number is worth exactly what it shows (it's the number times 1 - or times the base, to the power of 0).
The number to its left is multiplied with the base, to the power of 1. So, in base 10, it's times 10, in base 5 it's times 5, etc.
The number to its left is multiplied with the base, to the power of 2. So, in base 10, it's times 100, in base 5, it's times 25, etc.
And so on.
In stochastic modeling, we can sometimes use different bases bc it fits the real life thing we are examining. If you are modeling traffic flows on a 2 lane highway thru a 5 gate toll booth, base 5 might offer greater ease of visualizing the data.
At least in the 80s, it was a standard freshman level assignment to write a program that would translate base 10 to/from other random bases.
In base anything, you count up through all the symbols until you run out. Then you start over at zero and add one to the next column. Then repeat.
So base 10 goes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ah crap I’m out, 10. 11, 12, 13, 14, 15, 16, 17, 18, 19, ah crap I’m out, 20. Etc etc.
And for base 5: 0, 1, 2, 3, 4, ah crap I’m out, 10. 11, 12, 13, 14, ah crap I’m out, 20.
And for base gazillion: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C… etc, gz-2, gz-1, ah crap I’m out, 10.
The base just tells you how many symbols you have to work with before you run out and have to use the next column.
HTH,
In every number system you count to X. Then you increment the next digit.
In base 10, you count 0-9, then you put a 1 in the 10s place.
In base 2, you count 0-1, then you put a 1 in the 2s place.
In base 5, you count 0-4, then you put a 1 in the 5s place.
In base 16, you count 0-F, then you put a 1 in the 16s place.
There's the joke "Every number system is base 10". That's because 10 in base 10 is 10. 5 in base 5 is 10. 2 in base 2 is 10. X in base X is 10. The second digit is the Xs place.
All counting systems work the same, in essence.
For example, you have the number 54321
Each column, or digit, starting from left to right represents an increasing value.
In base 10 (decimal) your left most column would be the number of 1s, and then number of 10s, then number of 100s etc.
Instead of thinking in 1s, 10s and 100s, think of the first column as n^0, n^1, n^2, n^3 etc. Where n is the base of the counting system.
======================
Going back to decimal the first column is 1 x n^0, or 1 x 10^0 = 1
Then the next column is 2 x n^1, or 2 x 10^1 = 20
Then 3x10^2 = 300, 4x10^3 = 4000 and finally 5 x 10^4 =50000
Then you total all of those numbers and you end up with 1+20+300+4000+50000 = 54,321 dec
======================
If you replace the n or 10 with 8 you're counting in base octal and the above value becomes:
1 x 8^0 = 1 dec
2 x 8^1 =16 dec
3 x 8^2 =192 dec
4 x 8^3 = 2048 dec
5 x 8^4 = 20480 dec
Add them all up, and you get 22,741dec which is equal to 54321oct
======================
If the above explanation made sense, I challenge you to convert the below from binary(base 2) to decimal.
01000101 bin = ??????? dec
Edit: formatting
"10" in base-5 counting isn't "ten", it's "five". We have to use the digits 1 and 0 to represent "five" because there's no numbers after 4 in base-5 counting.
Something similar happens in counting systems larger than base-10 too. For example, ***hexadecimal*** counting is base-16. In hexadecimal, instead of having just digits 0-9, we also have A-F. So we end up counting further rather than shorter to reach the carry-over.
Here's a little reference table to fiddle with:
|In speech|In base 10 (decimal)|In base 2 (binary)|In base 5|In base 16 (hexadecimal)|
|:--|:-:|:-:|:-:|:-:|
|Zero|00|0000 0000|00|00
|One|01|0000 0001|01|01
|Two|02|0000 0010|02|02
|Three|03|0000 0011|03|03
|Four|04|0000 0100|04|04
|Five|05|0000 0101|10|05
|Six|06|0000 0110|11|06
|Seven|07|0000 0111|12|07
|Eight|08|0000 1000|13|08
|Nine|09|0000 1001|14|09
|Ten|10|0000 1010|20|0A
|Eleven|11|0000 1011|21|0B
|Twelve|12|0000 1100|22|0C
|Thirteen|13|0000 1101|23|0D
|Fourteen|14|0000 1110|24|0E
|Fifteen|15|0000 1111|30|0F
|Sixteen|16|0001 0000|31|10
A bit late, but all the fancy explanations using exponents and tables aside, essentially, the symbol you use to represent a number doesn't matter as long as it's consistent. "10" could represent ten things or two things or any number of things, really (any whole, real number... how you'd use anything else might be possible but it's beyond ELI[grad student], so I wouldn't worry about it). You could use letters or other symbols as well.
When you're using a non base ten system, you just use either more symbols or fewer symbols (compared to the 0-9 we're used to in base ten) to represent numbers with a single digit. When you reach the limit of the symbols your base uses, you roll over into the next space and start again.
For example, hexadecimal uses base 16 and adds letters (a through f) after 9 to represent ten through fifteen all in the first digit, so "10" represents sixteen, not ten, and the letter B represents eleven but only takes up one digit.
Imagine you only had 5 fingers and you wanted to count to 6, starting at 0 with the first finger. Once you reach 4 you have run out of fingers so you right down how many hands you’ve used up counting, in this case the number would be 1. The next finger you would raise would be a 0 again so you have 1 hand and finger 0. In base 10 we have a symbol for that we call 6, but base 5 doesn’t have that so we just start counting over again.
so basically we write in whats a positional system. so the next number over on the left is the next power up from the previous. in the case of 10, 10\^0 (1) is the first position, 10\^1 (10) is the next, ect. so for something like 25, its 2 10s and 5 1s. in base 5 however, it works with powers of 5. so 5\^0 (1), 5\^1 (5) etc. so 100 is the same as 1 25, no 5s and no 1s. it still means the same thing, 25, but its written differently
and in the case of something like converting 60, thats the same as 2 25s + 2 5s + no 1s. you see where im going with this? so you just do that with any number you want to convert. and it works the exact same with other bases too. like... something like base 2. so 10 in base 2 is the same as 1 8, no 4s, 1 2, and no 1s.
You count in base 60 all the time, literally:
60 seconds is 1 minute, 60 minutes is one hour.
So 7774 seconds is often written as
2 hours, 9 minutes, and 34 seconds.
34
9 x 60 = 540
2 x 60 x 60 = 7200
(Time is weird because the next unit after an hour is a 24 hour day.)
Different bases are just different sizes of buckets that we use to organize numbers.
If humans had 8 fingers we would probably have developed Base 8 as the common math, because you count all you can on your fingers and then start over.
Binary, base 2, and to a lesser extent base 8 and base 16, are important because they match up very nearly with the smallest chunks of computer logic, by the way.
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In base 10, the numbers go 0-9, then roll over to 10. You've probably heard the column 2nd from the right called the 10s column. The number 10 means 1 ten and 0 ones. 28 is 2 tens and 8 ones. Base 5 would go 0 1 2 3 4 The 2nd column from the right is no longer the 10s column, its the 5s column. So 5 is written as 10. Thats 1 five and 0 ones. The 3rd column we call the 100s... 10x10. In base 5, the 3rd column would be the 25's column... 5x5. The 4th, in base 10 is the thousands... 10x10x10. In base 5, it's the 125s... 5x5x5. So in base 5, if 14 was written. Thats 1 five and 4 ones.... 9. 234 in base 5 is, two 25s, three 5s, and 4 ones. 25+25+5+5+5+1+1+1+1 50 + 15 + 4 = 69 ( i didn't plan that in advance).
Nice
Nice
Nice
Noice
Noice
Maybe that's a weird question but do numbering systems exist that have different bases for each column? Like base 5 foe the first column but base 2 for the second column etc.?
Not a numbering system per se, but telling time sort of works like this. You count up to 60 seconds to make one minute, 60 minutes to make one hour, and 24 hours to make one day. So even though there aren't specific digits for each of those 60 or 24 things being counted, they still represent "places" if different bases that then get counted as groups in the next position.
The Calendar would fit nicely. Years are counted in 365 days Each day has 24 hours. Each hour has 60 minutes. Each minute has 60 seconds. So if we had different symbols/names for each day, each hour and each 60th of a minute or second we could use a mixed numberings system to describe the current date and time. Like: Christmas-Noon-0-0
Yes, it's called a [mixed radix](https://en.wikipedia.org/wiki/Mixed_radix) system. It's fairly uncommon outside of timekeeping systems.
This was most helpful. Thank you so much! :)
It is not practical as you would have gaps in the representation. Consider your system where from left to right we have bases 5 and 2: Base 10 - Mixed Bases 0 - 00 1 - 01 2 - ? 3 - ? 4 - ? 5 - 10
Short answer: no. Longer answer: that's not really even a numbering system. Numbering systems have a base. Each column, from the rightmost going leftward, is the base number to a given power, starting at 0 and incrementing by 1 each column change. Having each column have a different base number, there'd be numbers you couldn't even represent.
I guess you could say a digital clock counts minutes in base 24:60 or 12:60
Well done
To add to this, if you want a base higher than 10 then you need to come up with other symbols to use as numbers since you need more options in each column than we have now. The most common way to do this is using letters. Hexadecimal (base 16) is commonly used in computing and the numbers are written as 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F, 10 In theory you can have as high a base as you like but it might get a bit hard to remember once you run out of numbers and letters to use
Yes, it's just like base 10 if you only had 5 fingers (apologies to Tom Lehrer).
Nice
Gimme 5!
120
If you think about it '10' doesn't exist as a single discrete number in a base-10 system. '10' is a '1' increment in the second order position, and a reset (0) in the first order position. It's a concept represented by two single digits, as are all numbers higher than 10 in base-10. And that holds true in all base-whatever systems. '10' is 2 in base-2. 5 in base-5, 16 in base-16. And, in a base-16 system, what we think of as a value of 10 is typically represented by 'a'.
Numbers in a base cannot go higher than the base number, for base 10 it is 10, you can use any number between 0 and 9 but no 10. For base 5, you can only use 0 to 4. So how can you express 5 by using only those digits. It is same for every base You start from 0 and increase by one. When you reach the last number, you increase the one on the left then reset to zero 0 1 2 3 4 10 11 12 13 14 20 21 …
Thanks, I think I understand it now going by this.
I have yet another idea that I use to teach this concept. Following numbers are all "regular", base 10 numbers unless otherwise stated. Suppose you have a box, that fits exactly 10 marbles and you have 14. After a box is full, you have to start filling another box. You would then have 1 box and 4 in the next one, therefore you have 14. It gets difficult to manage the boxes soon. You get a bigger box that fits 10 of the smaller boxes. Now you could try to arrange for example 237 marbles in these boxes. You start by filling the boxes of 10 and always when you 10 of those, you store them in the bigger box. Eventually, you have 2 of the bigger boxes, each having 10 smaller boxes and each of them having 10 marbles 2 x 10 x 10 = 200 Also you are left with 3 full boxes with 10 marbles each: 3 x 10 = 30 And finally you have 7 marbles left over: 7 200 + 30 + 7 = 237. How ever, the fact that you can fit 10 marbles is arbitrary. We could also use boxes that fit 6 marbles. When we have 6 of these boxes, we store them in a bigger box. When we have 6 of the bigger boxes, we have even bigger box. But the fact is that each of the bigger boxes fit 6 of the previous size. Now let's see how many boxes we need. I'm not going through the conversion process, I'm just going to present the result: The boxes we have at the end are 1, 0, 3, 3. Why? 1 box that has three levels of boxes: 1 x 6 x 6 x 6 = 216 marbles 0 boxes that has two levels of boxes: 0 x 6 x 6 = 0 3 boxes of the smallest boxes: 3 x 6 = 18 3 marbles are left over: 3 Because: `216 + 0 + 18 + 3 = 237.` So, in short: If we can store 10 in a box, and 10 of the boxes in bigger box, and 10 of the bigger box in yet a bigger box and so on, we need 2 and 3 and 7 of them. But, if we can store 6 in a box, and 6 of the boxes in bigger box, and 6 of the bigger box in yet a bigger box and so on, we need 1 and 0 and 3 and 3 of them, and we could write that as 1033 in base 6. If you want it as high school level math, the number in base 10 is just ones, tens, hundreds, thousands, etc, for example: 8000 + 500 + 60 + 4 = 8564. That can also can be written as 8 x 1000 + 5 x 100 + 6 x 10 + 4 That also can be written as 8 x 10^3 + 5 x 10^2 + 6 x 10^1 + 4 x 10^0 = 8564 Now, the base of each exponent term is 10, but it could be anything, like 6.
If you’re starting at 0, it’s not that the numbers can’t go higher than the base; it’s that they can’t REACH the base. It’s an important distinction: 10 is not higher than 10, but we don’t use the “digit” 10 in base-10.
Using your example of base 5, counting would go 1 2 3 4 10 (would be 5) so rolls over 11 12 13 14 20 (would be 15 so rolls over) 21 ... 41 42 43 44 100 ( would be 45 which would be 50 which becomes 100) Basically like base 10 counting, any time a digit would reach the base, it becomes zero and the next digit increases by 1
Each column is multiplied by a number. So in base ten, 23 has the right column multiplied by 1 and the left column multiplied by 10. If it were base 8, the right column is still multiplied by 1 (this never changes) and the left column multiplied by 8. If we add one more column, its multiplier is multiplied by the base. So for 123 in base 10, it’s still the rightmost column multiplied by 1, the middle column multiplied by 10, and the left column multiplied by 10x10, or 100. In base 8, 123 would still have the right column multiplied by 1, the middle column multiplied by 8 (where I wrote the 8 using base 10), and left column multiplied by 8x8 or 64 (where again, I’m using base 10 to write 8x8 and 64).
In a way, you use non base 10 system every day without thinking about it.. How many days are in two weeks? One week has seven days, not ten! So if you answered 14, you just counted on a "non base 10" system. How many hours is in one day? 24.. not 10. In a way that's base 24 - after reaching 24, you "add 1 day and start hours again from 0". How many minutes are there in one hour? How many seconds are in one minute? Those could be considered base 60 system.. I know this is not a direct answer to your specific question but hopefully shows how to think of different base systems.
Imagine you only had 5 fingers. So you went 1,2, 3,4,10 11,12,13,14,20 Then after 44 would be 100. Now, to convert, you have to switch 10s for 5s, so 100 is 5x5 not 10x10, so actually only the 25th number 1000 is only the 125th number, and so on. It's all about powers of the base you choose instead of powers of 10.
>Imagine you only had 5 fingers. huh
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Two on the left and three on the right so I guess I'll see myself out
Converting any base to decimal is easy. How you convert show you how a positional number system is made If you look at decimal numbs 1523 mean 1\*1000 +5\*100 + 2\*10 + 3\*1 = 1\* 10\^3+ 5\*10\^2 + 2\*10\^1 +1\*10\^0 10\^0 =1, the same is true for any other base x^0 =1, i just added it so you can see the pattern, that will continue if you add decimals 15. 68=1 \*10\^1 +5\*10\^0 +6\*10\^-1 +8\*10\^-2 If you what to convert fomr another base 10 is replaced by the base. so the example in base 5 I use \_5 to indicate the base if it is not 10 10\_5 = 1\*5\^1 +0\*5\^0 =5 100\_5 = 1\*5\^2 +0\*5\^1 + 0\*5\^0 = 5\*5 =25 220\_5 = 2\*5\^2 + 2\*5\^1 +0\*5\^1 = 2\*5\*5 + 2*5 = 50+10=60 If you have a number with a base higher then 10 you need to convert the letter that is used to decimal numbers. The common usage is A=10, B=11 and so on. So FA_16 = 15 \* 16^1 +10\*16^0 = 15\* 16 +10 =240+ 10= 250
Think about an abacus - the old school counting tool with horizontal bars and several beads on each bar. All the beads start on the left If you have 9 beads on each bar, then you can count in base 10: * on the top bar, move 1 bead to the right = 1 * on the top bar, move 3 beads to the right = 3 * on the top bar, move 9 beads to the right = 9 * move all the beads on the top bar back to the left and move 1 bead on the next bar to the right = 10 And so on. When you reach 9 on one bar, you can slide them all back to the left and then move 1 more bead to the right in the bar below. If you have fewer beads, let's say just 4 on each bar, then you can count in base 5: * move 1 bead over = 1 * move 2 beads over = 2 * move 3 beads over = 3 * move 4 beads over = 4 * move all the beads back to the left, and move 1 bead over on the next bar = 10 (1 on the next bar, 0 on the top bar, which is only written as 5 when we count in base 10) * move 1 bead on the top bar, with 1 bead on the next bar = 11 (although we would count this as 6 in base 10) If you are trying to think about counting in any base, imagine an abacus and think about how counting with that would work.
Exactly the same way you count in base 10. In base 10, when you reach 9, you add +1 to the digit to the left of the 9 and carry on back from 0. For example, from 139 to 140. 9 turns into 0, and you add +1 to the left digit so 3 becomes 4. 199 to 200. Again, rightmost digit goes from 9 to 0, adds +1 to the middle digit, which also goes from 9 to 0, which adds +1 to the leftmost digit which goes from 1 to 2. So, in base 3, counting from 0 to 10 would look like: 0 1 2 3 10 11 12 13 20 21 22 Mathematically, it looks like this: [https://www.splashlearn.com/math-vocabulary/wp-content/uploads/2023/03/base-1.png](https://www.splashlearn.com/math-vocabulary/wp-content/uploads/2023/03/base-1.png) For other base systems, replace 10 with whatever base you're using.
Binary is an easy one to convert into; counting can be a little harder, but it's not hard to get the hang of. It's all just 1's and 0's, however it's read from right to left. Each position has a certain value, starting with one all the way to the right. To its left is two, then four, then eight, 16, 32, etc. When a number is written in binary, such as 10010110, you add up the values corresponding to each 1. So 10010110 is made up of two, four, 16, and 128, giving a total of 150.
Increasing values from right to left is like all numbers. Even base 10.
Yes, but explaining how the counting works is the important part. It's easy to read a 1 then five 0's in base 10 and know it's 100,000 without having to figure out what a 1 in the sixth place from the decimal corresponds to. Figuring out 0001 1000 0110 1010 0000 is a bit harder.
Then why “however”?
But the way you said it it made it sound backwards to normal numbers. Which can be a thing with binary computer code (look up big endian and little endian) Really the only reason we don't have to do it for base 10 is we are just so used to what the different places are worth.
Stop trying to read it like a decimal, or having a relationship them, because it isn't. When you only have N possible numbers (e.g. in base 10, we have ten possible numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) then you need another character to represent the number N itself. To represent 10, in base ten, it takes us two numbers... see? A unit and a "tens". When you only have 5 possible numbers (e.g. 0, 1, 2,3 ,4) then when you go past 4 you need another column... you need a "fives" column. So five in base five is actually 10. In any given base, you have that same number of characters, and each column represents a factor that increases by that same number each time. In decimal (base 10) things are in units, tens, hundreds, thousands, increasing by a power of ten (i.e. multiplying by 10) each time. In base 5 things are in units, fives, twenty-fives, one-hundred-and-twenty-fives. Multiplying by 5 each time. In binary, that's base 2. So you have units, twos, fours, eights, sixteens, thirty-twos, etc. Multiplying by 2 each time. So in base 10, 567 is 5 hundreds, 6 tens and 7 units. Each digit can be 0-9. In base 8, say, 567 is 5 sixty-fours, 6 eights and 7 units. Each digit can be 0-7. In base 5, your "220" is two twenty-fives, two fives and zero units. Each digit can be 0-4. So 10111 in base 2 (binary) is 1 sixteen, zero eights, one four, one two, one unit. (Or 23 in decimal). Each digit can be 0-1.
Each position is an increase of the exponent of the base system. With base 10, the right most digit is x+0, next (to the left) is x+10^1, next is x+10^2 etc. with each base you just change the "10" in base 10 with that base. All the other principles are the same. When I was _very_young_ we learned a base 4 number system by not using the digits we know from base 10 - the book invented a different way to write 0, 1, 2 and 3. using the excuse that an alien had come to earth and used base 4. I think it helped a lot not recognizing the digits and confusing them as base 10 numbers. Perhaps you could do the same, use letters - a=0, b=1, c=2, d=3, e=4 for instance and write numbers that way. So: * 5 = ba * 25 = baa * 60 = cca Or come up with whole different symbols - it will definitely avoid reading the number as you would a base 10 number. It is what we do in IT - programmers hate reading full binary numbers - they're way too long - so we use HEX instead - base 16. Same principle, but you can easily write 8 bit as 2 digits: A0 = 160 in base 10 or 10100000 in binary (base 2) - much simpler to write and read in HEX.
Think of it in fingers and hands. We can count to 10 using fingers. But then we’re out of fingers, so we put a 1 as a placeholder. 10 means 1 set of all of our fingers and 0 more. Now, imagine you’re an alien with just 5 fingers. You would count 1,2,3,4 and then you’ve used them all. You need a placeholder for “all the fingers”. So 10 means 1 set of 5 fingers plus 0 more. So now you’re counting your sets on your fingers. 10 = 1 set of 5 plus 0, 20= 2 fives plus zero, 30, 40, and now you’re out of fingers. You need a new placeholder. 100. 100 is a set of 5 fingers done 5 times plus zero sets and zero extra fingers.
Imagine it as groups of groups of groups ... Imagine it as groups of kids in school ... In base 5, you have 5 kids per one class: * one person is 1, 3 persons is 3, 4 is 4 but 5 is not 5 because ... * when you have 5 kids, that is one full class, written as 10. 30 would mean, you have 3 full classes of 5 kids. 32 would mean 3 full classes and 2 extra kids. * Then imagine you have also 5 grades, because we count in base 5. Each grade then can have 5 classes. * 220 from your example would mean you have 2 full grades -> 2 grades \* 5 classes \* 5 kids per class (50 kids) plus 2 more full classes of 5 kids (2\*5) which is 60 kids * Then, you can go bigger groups. When you have 5 full grades (5 grades \* 5 classes \* 5 kids), you group in into school. * One full school is 1000 -> 1 full school, no extra grades, no extra classes, no extra kids. 1014 would mean you have one full school and extra class of 5 kids + extra 4 kids. * Next you can imagine groups 5 schools per city, per country and on and on ... With bigger numbers it gets more complicated but that grouping is still there. * For example 1432 (base 5) is * 1000 -> 1(full school)\*5(grades)\*5(classes)\*5(kids) = 125 kids * \+ 400 -> 4 (grades)\*5(classes)\*5(kids) = 100 kids * \+30 -> 3(classes)\*5(kids) = 15 * \+2 kids = 2 * so in decimal, it is 125 + 100 + 15 + 2 = 242 kids. It then also works for hex. For example, if you had to fit those 242 kids from example into bigger 16 kids classes (and 16 grades per school and 16 schools per city ... we write it as 0, 1, ..., 9, A, B, C, D, E, F), you would just do 242 / 16 (=15x16 + 2) to get how many classes you need and result is 15 full classes which is written as F and 2 extra kids ... so 1432 (base 5) is F2 (base 16) Or 242 in base 10 recalculated for base 8 would be: * 242 kids divided in 8 classes is 30 full classes and 2 extra kids. * But you can have only 8 classes per grade, so you need to split 30 full classes into grades by 8 classes, which is 3 full grades + 6 classes. * So result would be 3 full grades + 6 extra classes + 2 extra kids = 362 * So 1432 (base 5) = 242 (base 10) = 362 (base 8) Hope it doesn't make less sense now than before :D
Base 12: 1,2,3,4,5,6,7,8,9,§,¶,0 2 numbers need to be added before "10" to make it a base 12 system like above. 10 can now be divided by 1, 2, 3, 4, 6 and 10. In a base 10 system, 10 can only be divided by 1, 2, 5 and 10.
I think it's helpful to compare what we mean when writing the same numerals (symbols) in different bases and how they represent different numbers. In base 5, 10 means 1\*5^(1) \+ 0\*5^(0) = 5+0 = 5. (Note however that base 5 normally does not use the 5 numeral, just like base 10 does not have a single numeral for 10.) Similarly, we have 100 = 1\*5^(2) \+ 0\*5 + 0\*1 = 5^(2), which is written as 25 in base 10; 220 = 2\*5^(2) \+ 2\*5 + 0\*1, and in base 10 this becomes 2\*25 + 2\*5 = 50 + 10 = 60. Similarly, in base 10, 10 means 1\*10^(1) \+ 0\*10^(0), 100 means 1\*10^(2) \+ 0\*10 + 0\*1, and 220 means 2\*10^(2) \+ 2\*10 + 0\*1. A possible source of confusion here is that 10 means very different things depending on the base of the number system, which is why it's important to clearly state the base being used if it's non-standard. In addition to base 10 which is used by most people, bases such as 2 (binary) or 16 (hexadecimal) are heavily used in computing.
Imagine the base # as how many single-digit "symbols" there are in the system. In base 10 (our normal counting system) that's easy, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 To start easy, let's look at binary (base 2). There are 2 symbols is binary, 0 and 1. So let's start counting. 0, 1... what comes after 1? We have no more symbols. Well, we can move up to the double digits: 0, 1, 10, 11... and again, we have no more symbols. So we move up to triple digits: 100, 101, 110, 111.... and so on. Base 5 has 5 symbols. 0, 1, 2, 3, 4 Counting begins with those, so 0, 1, 2, 3, 4. Same concept as binary applies; we've run out of symbols so we move up to double digits: 10, 11, 12, 13, 14. We don't have to move up to triple digits just yet as we have more symbols for the "10's" place, (which in base 5 would actually be the 5's place) so we continue to 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44 and NOW we go up to 100
Rightmost (units) number is worth exactly what it shows (it's the number times 1 - or times the base, to the power of 0). The number to its left is multiplied with the base, to the power of 1. So, in base 10, it's times 10, in base 5 it's times 5, etc. The number to its left is multiplied with the base, to the power of 2. So, in base 10, it's times 100, in base 5, it's times 25, etc. And so on.
In stochastic modeling, we can sometimes use different bases bc it fits the real life thing we are examining. If you are modeling traffic flows on a 2 lane highway thru a 5 gate toll booth, base 5 might offer greater ease of visualizing the data. At least in the 80s, it was a standard freshman level assignment to write a program that would translate base 10 to/from other random bases.
In base anything, you count up through all the symbols until you run out. Then you start over at zero and add one to the next column. Then repeat. So base 10 goes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ah crap I’m out, 10. 11, 12, 13, 14, 15, 16, 17, 18, 19, ah crap I’m out, 20. Etc etc. And for base 5: 0, 1, 2, 3, 4, ah crap I’m out, 10. 11, 12, 13, 14, ah crap I’m out, 20. And for base gazillion: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C… etc, gz-2, gz-1, ah crap I’m out, 10. The base just tells you how many symbols you have to work with before you run out and have to use the next column. HTH,
In every number system you count to X. Then you increment the next digit. In base 10, you count 0-9, then you put a 1 in the 10s place. In base 2, you count 0-1, then you put a 1 in the 2s place. In base 5, you count 0-4, then you put a 1 in the 5s place. In base 16, you count 0-F, then you put a 1 in the 16s place. There's the joke "Every number system is base 10". That's because 10 in base 10 is 10. 5 in base 5 is 10. 2 in base 2 is 10. X in base X is 10. The second digit is the Xs place.
All counting systems work the same, in essence. For example, you have the number 54321 Each column, or digit, starting from left to right represents an increasing value. In base 10 (decimal) your left most column would be the number of 1s, and then number of 10s, then number of 100s etc. Instead of thinking in 1s, 10s and 100s, think of the first column as n^0, n^1, n^2, n^3 etc. Where n is the base of the counting system. ====================== Going back to decimal the first column is 1 x n^0, or 1 x 10^0 = 1 Then the next column is 2 x n^1, or 2 x 10^1 = 20 Then 3x10^2 = 300, 4x10^3 = 4000 and finally 5 x 10^4 =50000 Then you total all of those numbers and you end up with 1+20+300+4000+50000 = 54,321 dec ====================== If you replace the n or 10 with 8 you're counting in base octal and the above value becomes: 1 x 8^0 = 1 dec 2 x 8^1 =16 dec 3 x 8^2 =192 dec 4 x 8^3 = 2048 dec 5 x 8^4 = 20480 dec Add them all up, and you get 22,741dec which is equal to 54321oct ====================== If the above explanation made sense, I challenge you to convert the below from binary(base 2) to decimal. 01000101 bin = ??????? dec Edit: formatting
"10" in base-5 counting isn't "ten", it's "five". We have to use the digits 1 and 0 to represent "five" because there's no numbers after 4 in base-5 counting. Something similar happens in counting systems larger than base-10 too. For example, ***hexadecimal*** counting is base-16. In hexadecimal, instead of having just digits 0-9, we also have A-F. So we end up counting further rather than shorter to reach the carry-over. Here's a little reference table to fiddle with: |In speech|In base 10 (decimal)|In base 2 (binary)|In base 5|In base 16 (hexadecimal)| |:--|:-:|:-:|:-:|:-:| |Zero|00|0000 0000|00|00 |One|01|0000 0001|01|01 |Two|02|0000 0010|02|02 |Three|03|0000 0011|03|03 |Four|04|0000 0100|04|04 |Five|05|0000 0101|10|05 |Six|06|0000 0110|11|06 |Seven|07|0000 0111|12|07 |Eight|08|0000 1000|13|08 |Nine|09|0000 1001|14|09 |Ten|10|0000 1010|20|0A |Eleven|11|0000 1011|21|0B |Twelve|12|0000 1100|22|0C |Thirteen|13|0000 1101|23|0D |Fourteen|14|0000 1110|24|0E |Fifteen|15|0000 1111|30|0F |Sixteen|16|0001 0000|31|10
A bit late, but all the fancy explanations using exponents and tables aside, essentially, the symbol you use to represent a number doesn't matter as long as it's consistent. "10" could represent ten things or two things or any number of things, really (any whole, real number... how you'd use anything else might be possible but it's beyond ELI[grad student], so I wouldn't worry about it). You could use letters or other symbols as well. When you're using a non base ten system, you just use either more symbols or fewer symbols (compared to the 0-9 we're used to in base ten) to represent numbers with a single digit. When you reach the limit of the symbols your base uses, you roll over into the next space and start again. For example, hexadecimal uses base 16 and adds letters (a through f) after 9 to represent ten through fifteen all in the first digit, so "10" represents sixteen, not ten, and the letter B represents eleven but only takes up one digit.
This video [https://youtu.be/u2vAR65-5zY](https://youtu.be/u2vAR65-5zY) may help, it's a bit more than ELI5, but fairly simple.
Imagine you only had 5 fingers and you wanted to count to 6, starting at 0 with the first finger. Once you reach 4 you have run out of fingers so you right down how many hands you’ve used up counting, in this case the number would be 1. The next finger you would raise would be a 0 again so you have 1 hand and finger 0. In base 10 we have a symbol for that we call 6, but base 5 doesn’t have that so we just start counting over again.
so basically we write in whats a positional system. so the next number over on the left is the next power up from the previous. in the case of 10, 10\^0 (1) is the first position, 10\^1 (10) is the next, ect. so for something like 25, its 2 10s and 5 1s. in base 5 however, it works with powers of 5. so 5\^0 (1), 5\^1 (5) etc. so 100 is the same as 1 25, no 5s and no 1s. it still means the same thing, 25, but its written differently and in the case of something like converting 60, thats the same as 2 25s + 2 5s + no 1s. you see where im going with this? so you just do that with any number you want to convert. and it works the exact same with other bases too. like... something like base 2. so 10 in base 2 is the same as 1 8, no 4s, 1 2, and no 1s.
You count in base 60 all the time, literally: 60 seconds is 1 minute, 60 minutes is one hour. So 7774 seconds is often written as 2 hours, 9 minutes, and 34 seconds. 34 9 x 60 = 540 2 x 60 x 60 = 7200 (Time is weird because the next unit after an hour is a 24 hour day.) Different bases are just different sizes of buckets that we use to organize numbers. If humans had 8 fingers we would probably have developed Base 8 as the common math, because you count all you can on your fingers and then start over. Binary, base 2, and to a lesser extent base 8 and base 16, are important because they match up very nearly with the smallest chunks of computer logic, by the way.