Counting Systems Mr. Bigham

Warm-up: When Does = = = 10 ?

If the clock reads 9, what time will it be if you add 4 hours? If the day is January 30 th what day will it be if in 5 more days? How do computers count to two? (Binary)

Decimal System

Why base 10? Does it 10 have special mathematical properties that make it a better base? “Counting in tens is a biological accident.” – Professor Zirkel, Dozenal Society of America What other bases are used? What are the advantages and disadvantages of different bases?

Mayan base 20 (Vigesimal) 8000's 400's 20's+= 1's =19874

In dozenal counting we use twelve symbols, adding two digits to represent ten and eleven since 10 still represents the base. In dozenal counting he numeral 342 represents 3H(twelve)^2 + 4H(twelve) + 2. Thus 342 in dozenals represents 482 in the familiar awkward decimal base. Many people use X and # to represent the digits for ten and eleven. They are pronounced dek and el.

Babylonian Counting “Babylonians were the one civilization that intelligently developed a number base – base sixty. If twelve has the advantage of the factors 2, 3, 4, and 6, sixty has 2, 3, 4, 5, 6, 10 12, 15, 20, and 30.” -Professor Zirkel

Binary (base 2 counting)

Hexadecimal (base 16)

Exercise Translate the numeral 10010: 1.From base twenty (Mayan) 2.Base 12 (dozenal) 3.Base 2 (binary) 4.Base 16 (hexidecimal) 5.A base of your choosing Translate the numeral 42 into different bases using radix notation. Example: 42 = = = ?

Coding Coding converts messages to numbers and back. Can you decode the message below? A zero is used for space.

Coding Complications If you have a decimal coded message: 202 does it decode to bob or tb? To avoid confusing when two digits refer to a two letters or one, all letters are coded using 2 digits. i.e. A = 01, B = 02, G = 07, space = 00. Decode: ?

How many digits needed? For base ten coding two digits were needed to make enough numerals to correspond to all the letters of the alphabet. How many total symbols could you code with two base ten digits? How many binary (base two) digits would you need to code the alphabet? How many hexadecimal digits would you need to code 250 symbols?

ASCII Code Most computers use an eight digit (bit) binary code to represent letters and symbols. The first three digits give information about what type of symbol (number, punctuation, capital, lower case etc.) If the first three digits are 010, it is a upper case letter. If it is 011 it is lower case. The last 5 digits can be converted to decimal and associated with the normal alphabet key.

Examples = H The 010 signals capital letter. The equals 8 in binary counting. Eight is associated with the letter h in the standard alphabet key = i The 011 signals lower case letter. The equals 9 in binary counting. Nine is associated with the letter I in the standard alphabet key.

Exercises Decode the following two digit hexadecimal coded message: F15000C E00 130F0D E07? Code a message using a base of your choosing.