1. Numeration Systems Introduction to Binary, Octal, and Hexadecimal

2. Before There Were Numbers Quantities represented by metaphors Analog Easy to Understand Comparisons to Real World Imprecise – can’t really count http://www.allaboutcircuits.com/vol_4/chpt_1/1.htm l

3. Let’s Count to 10 WordsCounting LinesRoman NumeralsDecimal Zero??0 One|I1 Two||II2 Three|||III3 Four||||IV4 Five|||||V5 Six||||| |VI6 Seven||||| ||VII7 Eight||||| |||VIII8 Nine||||| ||||IX9 Ten||||| X10 Numbers are digital: finite, precise and countable.

4. Why we use numbers Ancient number systems are pretty bad Decimal System is place-weighted Tens place, Hundreds place, etc. Or, if you want to use math notations ThousandsHundredsTensOnes.TenthsHundredths 10 3 10 2 10 1 10 0. 10 -1 10 -2

5. How the decimal system works Let’s take a number: 2074 Final Value for each column: Place value times digit value Final numerical value: add the column values (2000 + 70 + 4) Digits2074 Place Value10 3 (1000)10 2 (100)10 1 (10)10 0 (1) Column Value2 x 10 3 0 x 10 2 7 x 10 1 4 x 10 0

6. About digits For decimal, we need ten digits: 0 – 9 It because it’s a Base 10 system each “place” needs 10 possible values (including 0)

7. Binary Number System Place-weighted, like Decimal, but Base 2 Bit = binary digit Group of 8 bits is a byte 2 10 (1024) bytes = 1 Kilobyte (KB) 2 10 KB = 1 Megabyte (MB) Sixty- Fours Thirty- Twos SixteensEightsFoursTwosOnes 2626 2525 2424 23232 2121 2020 Place values are powers of 2 instead of 10 Decision of early computer makers.

8. Why Binary? Modern Computers store information as bits, and perform all their calculations on binary numbers. Modern computers are basically millions of tiny switches (called transistors) that can be on or off (1 or 0) They work most efficiently in binary Less errors

9. Let’s Count Again WordsCounting Lines Roman NumeralsDecimalBinary Zero??00 One|I11 Two||II210 Three|||III311 Four||||IV4100 Five|||||V5101 Six||||| |VI6110 Seven||||| ||VII7111 Eight||||| |||VIII81000 Nine||||| ||||IX91001 Ten||||| X101010 Binary needs more “places” to represent the same numbers

10. Binary to Decimal conversions Multiply each bit by its place value and add Example: 1001 Binary1001 Place Values8421 Column Values8001 8 + 1 = 9

11. Octal and Hexadecimal Base 8 (Octal) and Base 16 (Hex) Hex – need 16 digits in one “place” Hex Digits: 0123456789ABCDEF 1 Hex Digit = 4 bits It works because 16 is a powers of 2 Hex is most used, since a byte is represented nicely by 2 Hex Digits

12. Notes on Notation Octal numbers use the digits 0-7 Octal numbers are preceded by oct Hex Digits use CAPITAL LETTERS A-F If we mix hex and decimal, hex numbers will have a lower-case ‘h’ after them Examples: 29h, A7h, 10h Generic math notation uses subscripts: 1101011 2, 733 8, AE 16

13. Hex  Decimal Base 16 place values Let’s try 1B9 1 x 256 + 11 x 16 + 9 = 441 Hex Number 1B(11)9 Place Values16 2 (256)16 1 (16)16 0 (1)

14. Hex Digit  Binary Chart Hex DigitBinaryHex DigitBinary 0000081000 1000191001 20010A1010 30011B1011 40100C1100 50101D1101 60110E1110 70111F1111 Notice that one Hex digit can represent every possible binary number for 4 bits. (Decimal numbers 0 – 15)

15. Hex to Binary is easy! Convert single hex digit to binary using the chart or by counting String the binary digits in order Example: F5 Answer: 11110101 F5 11110101 Don’t forget that zero!

16. Binary to Hex is easy! Reverse what we did for Hex to Binary Divide the Binary number in groups of four bits from right to left Example: 1011101 01011101 5D Extra zero added on end

17. Decimal-To Conversions Trial and Fit method: Keep finding the highest digit that can “fit” into the decimal number, then subtract from the original # Prob: Convert 87 into Binary: We’d start by noticing that a ‘1’ in the 128s place is too much Try a ‘1’ in the 64s place to get a total of 64 Try a 1 in the 32s place gives us a total of 96 That’s >87 so it must be a 0 Trying a 1 in the 16s place gives a total of 80 1 in the 8s place gives 88. It’s too much, so 0 1 in 4s place, 1 in 2s place, 1 in 1s place, you get 87 View this slide in a slide show to see the animation

18. Decimal-To Conversions Is there an easier way? Yes! Division Remainder Method Take your number, divide by 2 for binary, 8 for octal, 16 for hex The Remainder of the division is a digit in your answer Repeat process using the Quotient, until the Quotient is zero Put the remainders together backwards for your answer

19. Examples 137  Binary 137 ÷ 2 = 68 r1 68 ÷ 2 = 34 r0 34 ÷ 2 = 17 r0 17 ÷ 2 = 8 r1 8 ÷ 2 = 4 r0 4 ÷ 2 = 2 r0 2 ÷ 2 = 1 r0 1 ÷ 2 = 0 r1 1 0 0 0 1 0 0 1 The LAST remainder you get becomes the FIRST bit of your answer.

20. Example 137  Hex 137 ÷ 16 = 8 r9 8 ÷ 16 = 0 r8 8 9 Watch out! Hex numbers can look like decimal numbers. In this case, 137 decimal is equal to 89 hex. If this idea confuses you, just remember that the place values are different. In this example, the ‘8’ from the answer is in the “sixteens” place, NOT the ‘tens’ place like it would be normally.

21. Why Hex? You’ve seen that the Hex Binary conversions are very simple. Hex is often used as shorthand for binary, since two hex digits easily represents a byte. Computer Forensics – when you look at “raw” computer data, it’s usually Hex. Hex-editing lets you “hack” and cheat at video games.