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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

  • otherwise known ashexadecimalorhexinstead 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!

Hello. I'm Professor Von Schmohawk and welcome to Why U.

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Pre-Algebra 3 - Decimal, Binary, Octal & Hexadecimal

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    Chun Sang Suen posted on 2018/08/10
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