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• 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 callednatural 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 usespositional notationand 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, calleddigits”.

• 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 themultiplierfor 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 thebaseof 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 oroctalnumber 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 twoeightsor 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 eighteightsor 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 calledflip-flopsto 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 orbinarynumber.

• 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

• 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.

• 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.