Binary & Hex

The number systems ofComputer Science

What you will learn

DecimalBinaryHexadecimalArithmeticConverting between number basesCounting2’s Compliment

Number Systems

PrehistoryUnary, or marks: /

/////// = 7/////// + ////// = /////////////

Babylonian (Mesopotamia) around 2000 BC Used 60 numerical symbols – sexadecimal. Used to measure time5; 25, 30 = 5hr 25min 30 secs. Was positional based by had no concept

of zero. 5,4560 = 5 x 60 + 45 x 1 = 34510

Roman Numerals: (non positional, no concept of zero)VII + V = VVII = XIIMCMXCIX = 1999MCMXCVIIII = 1999MCMLXXXXVIIII = 1999MDCCCCLXXXXVIIII = 1999

Quote from Pierre Simon Laplace

“The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius. “

Arabic/Indian Numerals (Decimal)

Better, Arabic/Indian based Numerals:7 + 5 = 12 = 1 x 10 + 2

345 is really 3 x 100 + 4 x 10 + 5 x 1 3 x 102 + 4 x 101 + 5 x 100

3 is the most significant symbol (carries the most weight)

5 is the least significant symbol (carries the least weight)

Digits (or symbols) allowed: 0-9Base (or radix): 10

•Try multiplication in (non-positional) Roman numerals(!):

XXXIII (33 in decimal) XII (12 in decimal)---------XXXIIIXXXIIICCCXXX-----------CCCXXXXXXXXXIIIIIICCCLXXXXVICCCXCVI = 396

•The Babylonians wouldn’t have had this problem.

•There are many ways to “represent” a number•Representation does not affect computation result

LIX + XXXIII = LXXXXII (Roman)59 + 33 = 92 (Decimal)

•Representation affects difficulty of computing results•No concept of “Zero”•Computers need a representation that works with fastelectronic circuits

What is the base ?

The decimal numbering system is also known as base 10. The values of the positions are calculated by taking 10 to some power.

Why is the base 10 for decimal numbers ? Because we use 10 digits. The digits 0

through 9.

Main Memory

Original computers used holes/no hole to represent two state, 1/0.Modern computers use transistor on/off translates to values 1/0Requires use of Binary number system

Binary: positional numbers work great with 2-state devices

•Digits (symbols) allowed: 0, 1•Binary Digits, or bits

•Base (radix): 2•10012 is really

•1 x 23 + 0 x 22 + 0 X 21 + 1 X 20

•910

•110002 is really•?•?

•Computers usually multiply Arabic numerals by convertingto binary, multiplying and converting back for display

Counting in Binary

Binary 0 1 10 11 100 101 110 111

Decimal equivalent 0 1 2 3 4 5 6 7

Decimal to Binary

•Divide decimal value by 2 until the value is 0E.g.

47 /2 = 23 Remainder 123/2 = 11 Remainder 111/2 = 5 Remainder 15/2 = 2 Remainder 12/2 = 1 Remainder 01/2 = 0 Remainder 1

Read from the bottom up

Ans: 101111

Binary to Decimal

1 0 0 0 0 0 1 1 X 20 = 1

26 25 24 23 22 21 20 0 X 21 = 0

0 X 22 = 0

20 = 1 24 = 16 0 X 23 = 0

21 = 2 25 = 32 0 X 24 = 0

22 = 4 26 = 64 0 X 25 = 0

23 = 8 1 X 26 = 64

65

Binary to Decimal (alternative)

Converting the binary 100101 to 37

Multiply running total by two, add in next digit. Repeat for all digits.

Some buzz words• Single Binary Digit =Bit• 4 bits = nibble• 8 bits = byte• 16 bits = word• 32 bits = longword• 2 nibbles = 1 byte• 2 bytes = 1 word

The word size depends on the processor used, above is shown the word size for the 68000

Addition of Binary Numbers

Examples:

1 0 0 1 0 0 0 1 1 1 0 0 + 0 1 1 0 + 1 0 0 1 + 0 1 0 1 1 1 1 1 1 0 1 0 1 0 0 0 1

Carry a one

Addition (large numbers)

Example showing larger numbers:

1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 1 + 0 1 1 1 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 1 0 1 0

Working with large numbers

0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1

Humans can’t work well with binary numbers. We will make errors.

Shorthand for binary that’s easier for us to work with - Hexadecimal

Hexadecimal

Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1

Hex 5 0 9 7

Written: 509716

What is Hexadecimal really ?

Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1

Hex 5 0 9 7

A number expressed in base 16. It’s easy to convert binary to hex and hex to binary because 16 is 24.

Counting in Hex Binary Hex Binary Hex 0 0 0 0 0 1 0 0 0 8 0 0 0 1 1 1 0 0 1 9 0 0 1 0 2 1 0 1 0 A 0 0 1 1 3 1 0 1 1 B 0 1 0 0 4 1 1 0 0 C 0 1 0 1 5 1 1 0 1 D 0 1 1 0 6 1 1 1 0 E 0 1 1 1 7 1 1 1 1 F

Decimal to Hex

Repeat dividing by 16 and noting the remainder until the quotient = 0

E.g. 54401d to Hex54401/16 = 3400 Remainder 13400/16 = 212 Remainder 8212/16 = 16 Remainder 413/16 = 0 Remainder 13

Remainders are expressed Hex numbersAnswer: $D481

Hex to Decimal

Write out the 1st 16 numbersSame as for binary to decimalE.g. F9E116

1x160 = 1Ex161 =14x16 = 2249x162 =9x256= 2304Fx163 =15x4096= 61440

= 63969

Another Binary to Hex Conversion

Binary 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1

Hex 7 C 3 F

7C3F16

Binary Coding Schemes

To represent characters we use Binary Coding SchemesTwo of the most popular are

ASCII (American Standard Code for Information Interchange EBCDIC (Extended Binary Coded Decimal Interchange Code)

Here are some ASCII codes CHARACTER ASCII A 01000001 – 41 in hex B 01000010 - 42 in hex C 01000011 - 43 in hex D 01000100 - 44 in hex BELL 00000111 – 7 in hex ESC 00011101 – 1B in hex

Binary Coded Decimal (BCD)

Each digit is converted individually to it’s binary equivalent 2634 = 0010 0110 0011 01003825 = 0011 1000 0010 0101Disadvantages are that arithmetic is more complicated and storage is inefficent.Used only in simple device where memory is not a problem e.g. Calculators, clocks, etc.

Negative Integers

Most humans precede number with “-” (e.g., -2000) Accountants, however, use

parentheses: (2000)

Sign-magnitudeExample: -1000 in hex? 100010 = 3 x 162 + e x 161 + 8 x 160

-3E816

2’s Compliment

Used for negative binary numbersProcedure:Invert all the bits: 0 becomes 1 and 1 becomes 0Add 1

E.g. To get –1110 = 000000012

Invert all bits 11111110Add 1 11111111Ans: -110 = 111111112

2’s Compliment

Using the TC notation, the Most significant indicates the sign. 1 means a negative numberWith 8 bits, can represent signed numbers in the range:+127 -1 0 -12801111111 11111111 00000000 10000000

With N bits, can represent signed numbers in the range:(2N-1)-1 0 -(2N-1)

TC is widely used because it is easy to manipulate. The sign bit is treated the same as the others

2s Compliment Clock

Notice how when we add one to a negative number it’s values move closer to zero

Stepping clockwise increases the binary number

Stepping anti-clockwise decreases the binary number

Addition & Subtraction

Addition of TC’s numbers same as normal binary additionSubtraction of TC’s numbers can use either A-B or A + TC(B)

E.g.01112 – 00112 = ???

Change second number to TC01111101+0100Answer: 01112 – 00112 = 01002

Decimal fractions to binaryMore info on p158 Clements.

Question: Convert 0.6250 (decimal) to a binary fraction

0.3125 x 2 = 0.6250 yielding 0

0.625 x 2 = 1.2500 yielding 1

0.25 = 0.5000 yielding 0

0.5 x 2 = 1.0000 yielding 1

Ans:0.0101

Binary to Decimal fractions

More info on p158 Clements.

Summary

DecimalBinaryHexadecimalArithmeticConverting between number basesCountingCoding schemes2’s Compliment