Hello. I’m Professor Von Schmohawk

and welcome to Why U. In the previous lectures, we explored some

examples of the earliest number systems which were used primarily for counting objects. These counting numbers are called

“natural numbers”. The natural numbers start at one

and can count to arbitrarily large quantities. As we have seen, Roman numerals are one of

many possible ways to represent natural numbers. The Roman system was eventually replaced with

the modern decimal number system which uses “positional notation”

and only ten numeric symbols. The decimal number system was found to be

superior to the ancient Roman system because of the simple rules it uses

to create numbers. In the decimal system there are ten numeric

symbols, 0 through 9, called “digits”. Depending on the column they occupy these digits represent the quantity of ones tens hundreds thousands and so on which make up the number. In positional notation,

the column occupied by a digit determines the “multiplier” for that digit. For example, in the decimal system the value of the right-most digit

is multiplied by 1. The digit in the next column to the left

is multiplied by 10. The next digit is multiplied by 100

and so on. The value of a number is the sum of

all its digits times their multipliers. For example, the value of

the decimal number 1879 is 1 times 1000 plus 8 times 100 plus 7 times 10 plus 9 times 1. In any positional notation, each column’s

multiplier differs from the adjacent column by a constant multiple called

the “base” of the number system. In the decimal system, each column multiplier

is ten times the previous column. Therefore the decimal system is called a

“base-10” number system. There are an infinite number of columns in

the decimal number system with each column multiplier being ten times

bigger than the column to the right. However, when writing a number,

the zeros in front are normally not written. We can count up to 9 using only the ones column. Once we reach 9 the ones column starts over at 0 and the tens column increments. As we continue counting the tens column counts the number of times that the ones column

has passed from 9 to 0. In other words, the tens column registers

the number of tens which we have counted. This continues until we reach 99. At that point

the ones and tens columns start over at 0 and the hundreds column increments. The positional notation system is simple. Every time a column passes from 9 to 0 the next column to the left increments. How is it that we ended up with a number system

based on multiples of ten? There is not any good reason

for choosing ten over some other number other than the fact that people have ten fingers and probably originally communicated quantities

using their fingers. But what if we were cartoon characters

with four digits on each hand? Is it possible that in cartoon land everyone uses a number system

based on multiples of eight instead of ten? How would a base-8 or “octal”

number system work? In octal there are only eight numeric symbols

instead of ten as in decimal. Instead of 0 through 9

the symbols 0 through 7 are used. The symbols 8 and 9 are not needed. Counting in octal is very similar

to counting in decimal. Since there are no symbols for 8 or 9 the highest quantity which can be represented

in the ones column is 7. Counting an eighth item

causes the ones column to start over at 0 and the next column to increment. So the second column

counts the number of eights. Therefore in octal

the number following 7 is 10 which looks just like the decimal number ten. After octal “10” comes octal “11”, “12”, and so on until we get to octal “17”. At that point, we go to octal “20”. The second column has now counted

two “eights” or sixteen. We continue like this until we get to the

highest number we can represent with two digits octal “77”. At that point, the ones and eights columns

start over at 0 and the third column increments. The 1 in the third column represents

eight “eights” or sixty-four. Each column multiplier is

eight times the previous one. Every number which can be written in decimal

can also be written in octal although after counting to 7 the way the quantities are represented

is completely different. It is easy to convert an octal number to decimal when you consider how positional notation works. Let’s take for example, the octal number “1750”. As in decimal, the value of the octal number

is the sum of all its digits times their multipliers. So the number “1750” represents 1 times 512 plus 7 times 64 plus 5 times 8 plus 0 ones which adds up to the quantity

which in decimal is called one-thousand. You may sometimes see a small subscript 8 or 10

after an octal or decimal number in case there may be some confusion

about which base is being used. Digital computers use electronic circuits

called “flip-flops” to represent numbers. Each flip-flop can store a single bit

which can represent either a 0 or a 1. Multiple bits can be combined to represent

a base-2 or “binary” number. In the binary number system

0 and 1 are the only two numeric symbols. Since binary is base-2 each column multiplier is two times

the multiplier of the previous digit. And just like decimal or octal numbers the value of a binary number is sum of

all its digits times their multipliers. Since the digits are either 1 or 0

the calculation is simple. We just add the multipliers

of all the columns which contain ones. For example, the binary number 11010 represents 1 sixteen plus 1 eight plus 1 two which is equal to twenty-six. Even though digital computers

store numbers in binary it can be quite tedious to write down

or remember large binary numbers. For instance, the number one-million

in binary is one one one one zero one zero one one zero zero one one zero one one zero zero zero zero. Early in the history of digital computers engineers found that it was easier

to use octal notation than to deal with long strings

of ones and zeros. Three binary digits can be represented

by a single octal symbol. It is easy to memorize the eight possible

combinations of three binary bits. To convert a multiple-digit octal number

to binary each octal digit in the number

is converted to a 3-bit binary equivalent and the binary digits are all combined

into a single binary number. Any leading zeros can be removed. To convert a binary number to octal

we do the same thing in reverse. To convert this binary number back to octal we split it into 3-bit groups

starting from the right and each 3-bit group is then converted

to its equivalent octal symbol. So the octal equivalent to this binary number

is “3654660” a lot easier to remember

than all those ones and zeros. Today, computer storage is normally organized

into 8-bit groups called “bytes”. Because of this, many computer engineers

prefer to use base-16 otherwise known as “hexadecimal” or “hex”

instead of octal. With hexadecimal, every group of four bits

converts to a single hex symbol. Two hex symbols represent exactly one byte. Even fewer digits than octal are required

to represent a given number and it’s just as easy to convert

back and forth to binary. Hexadecimal numbers

use sixteen numeric symbols. The symbols 0 through 9

are used just as in decimal but six more symbols are needed. Instead of making up new symbols,

the letters A through F are used to represent what we call

ten through fifteen in decimal. Counting in hexadecimal

works the same way as in decimal or octal except that hex uses sixteen symbols per digit. Because each column multiplier is

sixteen times larger than the previous column hexadecimal can represent large numbers with fewer digits than octal or decimal. When counting in hexadecimal after getting to F which is decimal 15 we go to “10” which is decimal 16 then “11”, “12”, and so on. Once we reach 1F we go to “20” which is decimal 32. When we get to the largest number which we

can represent with two hex digits, FF we go to “100” which is decimal 256 and so on. As we mentioned using hex notation, four binary bits

can be represented by a single hex symbol. Each of the sixteen possible combinations

of four bits is equivalent to a single hex digit. Let’s convert the same binary number as before

to hex. Starting from the right,

we group the digits into groups of four. Each group of binary digits is then converted

to its equivalent hex symbol. So we have seen how the same natural number can be represented in base-2

using two numeric symbols base-8 using eight symbols base-10 using ten symbols and base-16 using sixteen symbols. But no matter how we choose to write

this natural number it still represents the same quantity. As you have seen we use the same basic rules for counting in binary octal decimal and hexadecimal. The only difference is that each base has

a different number of numeric symbols. So using positional notation we can create a number system

using any natural base we like. Try creating one of your own. Who knows, it might catch on!

Thanks, great video. đ

Very well explained and very fluid and clear ! Thanks

3:21 I laughed at the sheep hovering in midair. Thanks for discussing these topics in a clear — and sometimes mildly funny — way.

TRY BASE 1

Very easy to understand, thanks!

As a 20 year old male who did horribly in school in math, I must TRUTHFULLY say that..cartoons are better at explaining things than people lol.

base 1 is just plain ol' tally marks

This was quite informative. I now know everything!

This seems more like a course on understanding the 16 base number and converting it to a 10 base number system than understanding "any" base number system.

10=1111111111

I thought 0 was a natural number too

My teacher told me that 0 is not a natural number. He said it was a part of the 'whole set'. Was he wrong? (I'm from Australia)

wow thank you guys so much, absolutely chuffed i found this.

Wow, love the presentation style. Great job and thanks for posting!

I came here because I attended a fail of a tut class in com sci and didn't know how to convert for **** after the 90 minute session. After watching this 14 minute video I now know (and will not forget) how to convert between the different numeral systems.

Mayan system is also vigesimal

THANKS PROFESSOR VON!!!!!!!!!!!!!!!!!!!!!!!!

This video is one of the best for me trying to help others learn hex and binary. The animations are great.

this doesnt tell me what a base 3 number system

great explanaton đ

I can't help but hearing the professor's voice while reading your comments XD.

thanx for such a feed it was awesome

so you can have base n (assuming n doesn't equal 0, 1, or infinity) where n=any number?

i thought there wasn't a 1-base system because wolfram alpha didn't know what i ment.

Lets create the base 30 notation for the numbers the islanders fromīģŋ the Coco-Loco island shown during the 1st movie used! zup, blup, dop, trom, mim, zap, wini, lumbi, bimpy, nichi, fromp, waked, trimble, walakai, pramet, jolo, honzo, kolo, trob, walep, zumi, mombozo, elimi, toremu, palezib, smit, banzulu, ikina, itchina, and back to blup zotan, blup blup, blup dop again! (zotan was degraded from 30 to 0, so it retains its important role – blup zotan zotan = 30 squared.)

I personally think Balanced Ternary is one of the most interesting number systems. I would love to see a video on that. It uses three digits like Base-3 and also increases by powers of three with each position, but the digits used are 0, 1(or +) and -1(or -). So the number 8 in Balanced Ternary would be +0-. 1×9+0x3+(-1)x1=8. And -8 is represented -0+, no need to stuck an extra symbol in front.

Didn't the ancient Mesopotamians used a 360 base number system. That is how we got our 360 degrees in a circle. Also how we got our 60 minuets and 60 seconds as it was 60 seconds in a minute and 60 minutes in a degree as it was used with the sundial.

one I made up out of boredom is a base 35 with 0-9 and A-Z omitting letter O to dispel confusion of zero and O. So the base 35 number "5A7" in decimal would be 6482.

Cool, well explained!

Very nice!

realy nice explanation

thanks

agreed, im going to watch them all!

omg thank you so much! has helped a lot, I'm beginning to self learn computer science and trying to get my head around the binary system etc

Great vid, I'm subscribing xx

it's idiot proof

Thanks for everything man, it's just simple math…

Cheers!

If my calculation is correct, your expression of 1 million in binary is incorrect.

This is just awesome, it was sooo useful

Because our hands would not be symmetrical.

What about Base -10?

How does addition and multiplication work in base -10? For example -19=1, because (-1)

(-10)^1+9(-10)^0=1, so 19+8=7, 18*18=-16 and so on…I'm sorry, that plus sign is actually a minus sign.

(-1) * (-10)^1 + (-9) * (-10)^0 = 1. Now it's correct.

I subsribed your chanel

my math teacher tell me to go here

ĐŅĐĩĐŊŅ Đ¸ĐŊŅĐĩŅĐĩŅĐŊĐž Đ¸ ĐŋĐžĐˇĐŊĐ°Đ˛Đ°ŅĐĩĐģŅĐŊĐž. ĐĄĐŋĐ°ŅĐ¸ĐąĐž!

These vidoe is helpful for beggineers that want to learn networking

this is an awesome utility!! helps with my math drills big time. ThanksÂ

did he say at the endÂ to create my own number system..?Â yeah in time maybe….but this kind of information is, to me, is very complex, and requires drill after drill to absorb. It's highly intimidating, but manageable. When I came into IT I think I was that dude all IT pros hate, the one who knows enough to be cocky, around laymen, but not enough to stand tall in front of real CIS experts. I have since humbled myself and I am realizing the more I learn the less I realize I actually understand. This field of academia, to me is so exciting and cool, and yet overwhelming at the same time. ButÂ practice, practice, practice and the repetitions start formulating the conceptual framework in your mind almost at an osmotic level. Seriously though computers rock!!!! I am so digging this!!!

I don't know bout ya'll but I have multiple nerd-gasms every time I learn something new on hereÂ or in this field in general.

ok Dr. Schmohawk is bitch slapping me inÂ the face……I will get this!!!!! view again!!

I understood everything but felt sleepy (during lecture for same topic) in my college.this is very nice and easy to understand.

Best video on this for a newbie I've seen so far.

Awesome !!! Thank u Professor !

THIS IS AMAZING!!! I missed the class in compsci where my prof. was explaining what these numbers were. this helped SO MUCH!! also great for conversions :)) any compsci/compeng majors can appreciate this video!!!

I support hexadecimal system as the common counting system of human beings.

cool

ŅĐŋĐ°ŅĐ¸ĐąĐž

thnks a lot !!

How about making numbers to the base of three-quarters.

this video actually helped me understand what binary numbers are

#MyWhyU ==> Perfect explanation , thanks

next time also explain how to convert point values( .212 , .101101 ,…..)

Thank you! Way better than a power point

cool explanation and style đ

Thank you. You've explained so much better than my prof in uni

way easier than school

base 3/base 10

123. ?

This helped a bit in me understanding octal. I seem to get hex better though, actually I feel that I don't, I think it just easier to toggle an early personal (hobby) computer with it. Like the IMSAI 8080. Now I just need to memorize what each equals in binary to toggle the switches. Then I'll memorize 8-bit ASCII by hex.

what about base 4 with including 0,1,2,3

Base 3=2

0,1,2,10,11,12,20,21,22,100

Base 20=I

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G,H,I

Vigesimal goes 1 2 3 4 5 6 7 8 9 A B C D E F G H I J 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1H 1I 1J 20 21 22 23 24 25 26 27 28 29 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 30 ———————————————————– JJ —- 100

2017 is MMXVII. Smart.

MyWhyU vs mathantics.

Amazing

đ

Vigesmal goes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F

I thank you everyday for making these videos.

I'm 11

I hope you know a fifth grader is learning from this

Thank you…Good vedio

This guy who made this video is a genius

one of the best explanation with perfect animation.

This series reminds me of Cyberchase. Man, I loved that show…

3:33 oh my god so many sheep [Falls Asleep]

base 7 (numbers 0, 1, 2, 3, 4, 5, 6)

0 1 2 3 4 5 6 10 11 12 13 14 15 16…………63 64 65 66 100 101

x343 x49 x7 x1

1 4 6 6

1×343 + 4×49 + 6×7 + 6×1

343+196+42+6

587

Numbers : 1-1000

The amount of effort that went into this is actually really appreciated. Like who animated this? 10/10 (or should I say 1010/1010)

Amazing video đđ

The video is superb and impeccable.

Who else is watching this for Computer Science Principles

Ez

this video was about binaryoctalhexadecimalquaguca!!!! Nice video!!!!!!!!!!!!

base 62:

1,2,3,4,5,6,7,8,9,A……..x,y,z ,10,11

1 2 4 S z

62^4 Times 62^3 Times 62^2 Times 62Times

1/1/1/1/1/1/1/1=10(1)

128/64/32/16/8/4/2/1=10(2)

2187/729/243/81/27/9/3/1=10(3)

16384/4096/1024/256/64/16/4/1=10(4)

78125/15625/3125/625/125/25/5/1=10(5)

279936/46656/7776/1296/216/36/6/1=10(6)

823543/117649/16807/2401/343/49/7/1=10(7)

2097152/262144/32768/4096/512/64/8/1=10(8)

4782969/531441/59049/6561/729/81/9/1=10(9)

10000000/1000000/100000/10000/1000/100/10/1=10(10)

Where we can find solutions

Yeah Hexadecimals are killing my test scores man -_-. fine with binary and changing to decimals. Yet, hexadecimals really i don`t get, mostly when i`m working with multiple place holders.

If i`m changing 17 to HEX, i understand its (1*16)+(1*7)=23…. Yet why am i not doing (1*16)+(1*17) to get 33?

what would happen if i worked with 349 or something crazy, how would i solve?

P.S. i am self studying all this and learning from the bottom to hopefully self-learn programming with a solid base. Don`t be to harsh.

These videos are grossly good. Makes me question my public school education.

are you saying that cocolocos number system is a base 30? noice

Can someone convert F4240 from Hex to either Dec, Oct, or Bin? Tell me what you get and see if it's the same as 13:01?