Welcome to the Number Bases video,
in this video I will be comparing 3 different number bases,
base 10 which is known as decimal,
base 12 which is known as Duodecimal and base 16 which is known as Hexadecimal.
The numbering system you are probably used to is called decimal and uses 10 symbols
, 0 to 9.
However there is actually nothing special about the number 10,
you could use any number you want as a base.
We probably only use it because we have 10 fingers.
Base 12 and base 16 would actually be better than base 10.
Base 12 is good because it has lots of factors and base 16 is good because it is a power of 2.
When using bases larger than 10 it is common practice to use the letters of the alphabet to represent digits larger than 9.
In this video all decimal numbers will be white,
all Duodecimal numbers will be cyan,
and all Hexadecimal numbers will be yellow to avoid confusion since the same combination of digits will mean something different depending on what base you are using.
Let's have a look at how many factors each number has excluding 1 or itself.
A factor is something that the number can be evenly divided into.
10 has 2 and 5,
12 has 2, 3, 4 and 6 and 16 has 2, 4 and 8.
Here 10 has the least amount of factors, also 5 is not a very useful factor to have.
3, 4, 6 and 8 are all more useful factors to have.
Let's have a look at what some fractions look like in each base.
In decimal half is 0.5,
a third is 0.3 reoccurring which is messy,
a quarter is 0.25 which is also a bit messy,
a fifth is 0.2,
a sixth is 0.16 recurring which is messy,
a seventh, is 0.142857 and an eighth is 0.125 which is a bit messy.
However in duodecimal, a half, third, quarter and sixth are all nice round numbers,
this would make basic maths a lot easier.
The only fraction that is round in decimal but not in duodecimal is a fifth,
however a third, a quarter and a sixth are all more common fractions than a fifth.
The only reason why we think of 5 as a round number is because it is half of 10 which we use as our base,
6 is actually more useful because it can be divided into 2 and 3 which 5 cannot.
In hexadecimal, a half, quarter and eighth are round numbers but a third, fifth, sixth and seventh are not.
So for fractions duodecimal is the best followed by hexadecimal,
and decimal which is the system we use is the worst.
Now let's have a look at what the times tables look like in Decimal, Duodecimal and Hexadecimal.
In decimal it is easy to remember the 1 times table because the last digit counts up in ones and the first digit counts up 1 every time the last digit changes from 9 to 0,
in decimal it is easy to remember the 2 times table because the last digit goes in a repeating pattern of 2 4 6 8 0 and every time the last digit changes from 8 to 0 the first digit increases by 1.
In decimal there is no easy way to remember the 3 times table.
In decimal there is no easy way to remember the 4 times table.
In decimal it is easy to remember the 5 times table because the last digit goes in a repeating pattern of 5 0 and the first digit increases by 1 every time 5 changes to 0.
In decimal there is no easy way to remember the 6 times table.
In decimal there is no easy way to remember the 7 times table.
In decimal it is easy to remember the 8 times table because the last digit goes in a repeating pattern of 8 6 3 2 0 and the first digit counts up by 1 except from when 0 changes to 8 where it stays the same.
In decimal it is easy to remember the 9 times table because the last digit counts down and the first digit counts up.
In decimal it is easy to remember the 10 times table because the last digit is always 0 and the first digit counts up in ones.
In duodecimal it is easy to remember the 1 times table because the last digit counts up in ones and the first digit counts up 1 every time the last digit changes from B to 0. In duo decimal it is easy to remember the 2 times table because the last digit goes in a repeating pattern of 2 4 6 8 A 0 and the first digit goes up 1 every time the last digit changes from A to 0.
In duodecimal it is easy to remember the 3 times table because the last digit goes in a repeating pattern of 3 6 9 0 and the first digit increases by 1 every time 9 changes to 0.
In duodecimal it is easy to remember the 4 times table because the last digit goes in a repeating pattern of 4 8 0 and the first digit increases by 1 every time 8 changes to 0.
In duodecimal there is no easy way to remember the 5 times table.
In duodecimal it is easy to remember the 6 times table because the last digit goes in a repeating pattern of 6 0 and the first digit goes up by 1 every time 6 changes to 0.
In duodecimal there is no easy way to remember the 7 times table.
In duodecimal it is easy to remember the 8 times table because the last digit goes in a repeating pattern of 8 4 0 and the first digit goes up by 1 except from when 0 changes to 8 when it stays the same.
In duodecimal it is easy to remember the 9 times table because the last digit goes in a repeating pattern of 9 6 3 0 and the first digit counts up except from when 0 changes to 9 where it stays the same.
In duodecimal it is easy to remember the 10 times table because the last digit goes in a repeating pattern of A 8 6 4 2 0 and the first digit counts up except from where 0 changes to A where it stays the same.
In duodecimal it is easy to remember the 11 times table because the last digit counts down and the first digit counts up.
In duodecimal it is easy to remember the 12 times table because the last digit is always 0 and the first digit counts up.
If we counted in duodecimal instead of decimal it would make learning the times tables a lot easier.
It would also make counting in threes fours and sixes a lot easier.
The only times table that is easier to remember in decimal than it is in duodecimal is the 5 times table,
however the 3, 4 and 6 times tables are all more useful than the 5 times table.
In hexadecimal it is easy to remember the 1 times table because the last digit counts up in ones and the first digit counts up when the last digit changes from F to 0.
In hexadecimal it is easy to remember the 2 times table because the last digits go in a repeating pattern of 2 4 6 8 A C E 0 and when E changes to 0 the first digit counts up.
In hexadecimal there is no easy way to remember the 3 times table.
In hexadecimal it is easy to remember the 4 times table because the last digit goes in a repeating pattern of 4 8 C 0 and when C changes to 0 the first digit goes up by 1.
In hexadecimal there is no easy way to remember the 5 times table.
In hexadecimal there is no easy way to remember the 6 times table.
In hexadecimal there is no easy way to remember the 7 times table.
In hexadecimal it is easy to remember the 8 times table because the last digit goes in a repeating pattern of 8 0 and when 8 changes to 0 the first digit goes up by one.
In hexadecimal there is no easy way to remember the 9 times table.
In hexadecimal there is no easy way to remember the 10 times table.
In hexadecimal there is no easy way to remember the 11 times table.
In hexadecimal it is easy to remember the 12 times table because the last digit goes in a repeating pattern of C 8 4 0 and the first digit counts up except from when 0 changes to C where it stays the same.
In hexadecimal there is no easy way to remember the 13 times table.
In hexadecimal it is easy to remember the 14 times table because the last digit goes in a repeating pattern of E C A 8 6 4 2 0 and the first digit counts up except from when 0 changes to E where it stays the same.
In hexadecimal it is easy to remember the 15 times table because the last digit counts down and the first digit counts up.
In hexadecimal it is easy to remember the 16 times table because the last digit is always 0 and the first digit counts up in ones.
Here are some big numbers written in decimal.
In decimal you can probably tell which of these are even just by if the last digit is even.
In decimal you can also tell which are multiples of 5 by weather the last digit is a multiple of 5,
and you can also tell which are multiples of 10 because the last digit will be a 0.
However in decimal you cannot easily tell which of these are multiples of 3 4 6 8 12 or 16.
Here are the same numbers written in duodecimal,
you can also tell which ones are even by weather the last digit is even.
In duodecimal it is also easy to tell which are multiples of 3 by weather the last digit is a multiple of 3,
you can also tell which ones are multiples of 4 by weather the last digit is a multiple of 4,
you can also tell which of them are multiples of 6 by weather the last digit is a multiple of 6 and you can also tell which are multiples of 12 because the last digit will be a 0.
In duodecimal you cannot easily tell which are multiples of 5 8 10 or 16.
Here we have the same numbers again but in hexadecimal.
In hexadecimal you can see which numbers are even by weather the last digit is even.
In hexadecimal you can also tell which are multiples of 4 by if the last digit is a multiple of 4,
it is also easy to tell which ones are multiples of 8 by if the last digit is a multiple of 8 and you can see which are multiples of 16 by if the last digit is a 0.
In hexadecimal you cannot tell which are multiples of 3 5 6 10 or 12.
This is what happens if you keep doubling 1 in decimal, duodecimal and hexadecimal.
You have probably seen some of these numbers in decimal before.
In decimal and duodecimal the numbers are very messy,
however in hexadecimal the first digit goes 1 2 4 8 and just adds and extra 0 each time 8 changes to 1,
so you always get nice round numbers.
In decimal the last digit goes in a repeating pattern of 2 4 8 6.
In duodecimal the last digit goes in a repeating pattern of 4 8.
So for doubling 1, hexadecimal is the best, followed by duodecimal and decimal is the worst.
Here is what happens if you keep halfing 1 in decimal, duodecimal and hexadecimal.
In decimal and duodecimal the numbers are very messy but in hexadecimal the last digit goes 1 8 4 2 adding a 0 to the beginning every time 1 changes to 8.
In decimal it gets 1 digit longer every time you half it,
in duodecimal it gets 1 digit longer every 2 times you half it and in hexadecimal it gets 1 digit longer every 4 times you half it.
I am going to right align these number so you can see some patterns in the last digits,
in decimal the last digit is always 5, the second to last digit is always 2, the third to last digits alternate between 1 and 6 and the fourth to last digit go in a repeating pattern of 0 3 5 8.
In duodecimal the last digit goes in a repeating pattern of 6 3 6 9 and the second to last digit goes in a repeating pattern of 1 0 4 2 1 6 4 8. When it comes to halfing 1,
hexadecimal is the best, followed by duodecimal and decimal is the worst.
Here we have some powers of each base written in binary,
I am using the color magenta to represent numbers written in binary.
In decimal and duodecimal the binary numbers are a messy combination of ones and zeros.
In hexadecimal they are all just a 1 followed by zeros.
This is because hexadecimal translates well to binary.
There is a common misconception that there are 1024 bytes in a kilobyte, 1024 kilobytes in a megabyte, 1024 megabytes in a gigabyte, 1024 gigabytes in a terabyte and 1024 terabytes in a petabyte,
however this is not the case.
There are in fact 1000 bytes in a kilobyte,
1000 kilobytes in a megabyte, 1000 megabytes in a gigabyte, 1000 gigabytes in a terabyte and 1000 terabytes in a petabyte.
But there are 1024 bytes in a kibibyte,
1024 kibibytes in a mebibyte, 1024 mebibytes in a gibibyte, 1024 gibibytes in a tebibyte and 1024 tebibytes in a pebibyte.
The symbol for megabyte is capital M capital B and the symbol for mebibyte is capital M lowercase i capital B.
This misconception probably comes from the fact that Windows uses the wrong units,
effectively lying about the capacity of your hard drive.
Windows uses the term terabyte when it is actually meaning tebibyte because most people haven't heard of tebibytes before.
This is why hard drives appear smaller than advertised.
This makes people feel like they have been ripped off since it looks like they are getting almost a terabyte less than what they have paid for.
There is a common misconception that this is because of the space used up by the file system index,
however this is not the case,
the file system index is actually stored on the partition, it is just so small that you don't notice it,
if you create a 20 megabyte partition then you can see that about 10 megabytes have been used even though there is nothing on it.
This 8 terabyte hard drive appears to be only 7.27 terabytes,
but if you look at the bytes you can see that it is almost exactly 8 terabytes.
This makes sense if you look at how many bytes are in a tebibyte because 8 divided by this number divided by 1 trillion is 7.27.
Because Windows uses to wrong units,
it is now confusing weather terabyte means terabyte or tebibyte.
If you have ever looked at the size of a file you may think it is strange how the first 3 digits of the number of bytes does not match what Windows calls a terabyte.
If you look at Linux you will see that it actually uses the correct units.
All of these problems and confusion I have mentioned about data size units are only a problem because we count in decimal,
if we counted in hexadecimal then none of this would be a problem.
If we used hexadecimal it would also make it easier to convert between bits and bytes because there are 8 bits in a byte.
Hexadecimal is used in some areas of computing because there are 8 bits in a byte and 2 to the power of 8 is 256 and the square root of 256 is 16 which makes it easy to represent 1 bytes worth of data using 2 hexadecimal digits.
0.1 in decimal or 1 tenth is 0.000110011 reoccurring in binary.
This means that a computer cannot precisely store 0.1 in binary since doing so would require an infinite amount of memory,
therefor it has to round it slightly.
This is why when you try and calculate 0.1 + 0.2 on a computer you get 0.30000000000000004 when we all know it should be 0.3,
this would not happen in hexadecimal because 0.1 in hexadecimal or 1 sixteenth is just 0.0001 in binary.
In decimal it is difficult to manage money because you can't go up in round amounts.
To keep it in sync with the base you have to go 1 times 2 then 2 times 2 and a half then 5 times 2 then 10 times 2 then 20 times 2 and a half then 50 times 2.
In duodecimal it is better because you can go 1 times 2 then 2 times 3 then 6 times 2 and so on which only uses whole numbers.
In hexadecimal it is even better because you can just double it every time and still stay in sync with the base.
In hexadecimal it is also possible to times by 4 every time and still stay in sync with the base.
In decimal there is no easy way to arrange all the digits into a keypad,
so you end up having to go 3 by 3 and then put the 0 below.
In duodecimal you can go 3 by 4 and in hexadecimal you can go 4 by 4.
To divide a circle up into tenths you have to divide it in half then divide each half into 5.
To divide a circle into twelfths you can divide it in half then divide each half in half again then divide each quarter into 3.
To divide a circle into sixteenths you can divide it in half then divide each half in half again then divide each quarter in half again then each eighth in half again.
Twelves are all around us,
eggs are sold in batches of 12,
if we used duodecimal it would make it easy to work out the cost per egg.
There are also 12 months in a year.
We also use 12 in our clocks.
There are 12 inches in a foot,
if we used duodecimal it would make using feet and inches just as easy as using meters and centimetres is in decimal,
only it would actually be easier since a third of a foot is 4 inches and a third of a meter is 33.3 reoccurring centimetres.
The only places where we ever see tens are places like the metric system where 10 has deliberately been used to match our decimal counting system.
4, 8 and 16 are also all around us.
Pizzas always have 4, 8 or 16 slices.
You never see a pizza with 10 slices. T
This is because if you keep slicing a pizza in half you will always get a power of 2.
If we used hexadecimal it would make it easy to work out the cost per slice.
There are also 8 pints in a gallon and 16 ounces in a pound.
If we used hexadecimal it would make using pounds and ounces just as easy as using kilograms and grams is in decimal,
only it would actually be easier because if you keep halfing a pound you get to an ounce and if you keep halfing a kilogram in decimal you get to 62.5 grams.
You may think 10 is good because we have 10 fingers,
but there are other methods of counting that allow you to count to numbers higher than 10 on 1 hand.
It is possible to count to 12 on 1 hand using the segments on each finger because we have 4 fingers and 3 segments per finger which multiply to make 12.
You can use your thumb to point to each segment.
By using 2 hands it is possible to count to 144 by counting on the left hand every time you reach 12 on the right hand.
This would be difficult to do in decimal since the numbers are annoying to work with.
But in duodecimal it would be easy because the number on the left hand would represent the first digit and the number on the right hand would represent the second digit.
In hexadecimal it is possible to count to 31 on 1 hand using positional notation.
You assign a power of 2 to each finger,
then make the number by adding powers of 2 together.
This is basically the same as binary,
at the left I have the decimal number in white so you can see what number it is,
the hexadecimal number in yellow so you can see what it would look like in hexadecimal and the binary number in magenta so you can see that the finger being up is 1 and down is 0.
Using 2 hands it is possible to count all the way up to 1023,
for instance if you wanted to represent 365 you could add 256 + 64 + 32 + 8 + 4 + 1
These numbers are difficult to deal with in decimal,
however in hexadecimal these numbers become a lot easier.
This makes sense because 1 = 1, 4 + 2 = 6 and 8 + 4 + 1 = D.
In decimal the maximum you can count to on 1 hand is 5 and the maximum you can count on 2 hands is 10.
In duodecimal you can count to 12 on 1 hand and 144 on 2 hands and in hexadecimal you can count to 31 on 1 hand and 1023 on 2 hands.
So for counting on your fingers hexadecimal is the best,
followed by duodecimal and decimal is the worst,
even though that is the only reason why we use decimal.
In conclusion, it is debatable which base is better out of 12 and 16.
But one thing is for sure which is that 10 is the worst.
However trying to switch now would be much too difficult because you would have to relearn basic maths,
the metric system is based around base 10,
almost all software and hardware user interfaces now use base 10 and numbers written in base 10 would be confused with other numbers if we were to switch,
so we are now stuck using a suboptimal counting system that is messy, makes everyday maths difficult and does not work well with computers.
Goodbye.
[End Music]
For more infomation >> НОВЫЙ МАТЧМЕЙКИНГ + КОНКУРС! / НОВАЯ СИСТЕМА ПОДБОРА ИГРОКОВ - Duration: 3:06. 
For more infomation >> ▶️ [Bendy and the Ink Machine] - Duration: 2:27:14. 
For more infomation >> [시원스쿨 독일어] 하루에 한 문장 typisch 독일어 #9"그거 비밀이다!" - Duration: 1:49. 
For more infomation >> Коучинг и консультирование. Стоит ли заниматься коучингом и консультированием. - Duration: 5:40. 
For more infomation >> [Vietsub] Peanut&Smeb-chuyện sinh đẻ của Malzahar - Duration: 1:30. 
For more infomation >> Schnelles Geld und leicht Geld verdienen? | Los Angeles - Duration: 7:24. 
Không có nhận xét nào:
Đăng nhận xét