Number Systems

1. Number Systems • Denary Base 10 • Binary Base 2 • Hexadecimal Base 16 • Conversion between number systems. • Signed numbers, 2’s complements binary and binary fractions. • Floating point formats. • Simple arithmetic operations using binary number. • Bits, bytes and words

2. Denary Base 10 • The base-10 system (decimal) is the most commonly used today ( 0 1 2 3 4 5 6 7 8 9) • To represent the number four thousand three hundred and twenty one (4321)

3. Binary Base 2 • The base-2 system (Binary) the binary system is used internally by virtually all modern computers. ( ‘0’ or ‘1’ )

4. Hexadecimal Base 16 • The base-16 system (Hexadecimal) uses the numerals “0 1 2 3 4 5 6 7 8 9 B C D E F” • Easy to represent binary code (form of shorthand) in which one hexadecimal digit stands in place of four binary bits. • Hex f a • Binary ,1111 1010

5. Converting from denary to binary • 1) Start with the number you want (44) • 2) Divide by the base you want to convert to (2) • 3) Calculate the remainder from the sum • 4) Place result of the sum on the next row • 44 denary = 101100 binary

6. Converting from denary to hex • 1) Start with the number you want (4321) • 2) Divide by the base you want to convert to (16) • 3) Calculate the remainder from the sum • 4) Place result of the sum on the next row • 4321 Denary = 10E1 Hexadecimal

7. Denary, Hex & Binary

8. Signed numbers • In mathematics negative numbers can be prefixed by the '-' symbol. • On a computer system several methods cane be used • Use of a sign bit • Often Most Significant bit is used for the sign • 4 Bits gives a range -7 to +7 • Two representations of '0'

9. 1's complement • To obtain the negative number the positive binary value is complemented • '0' replaced with '1' and '1' replaced with '0' • For example binary 00001010 (decimal 10) complemented becomes 11110101

10. 2's complement • To obtain the 2's complement of a number • First the 1's complement is obtained • Then 1 is added • i.e. decimal 10 = 00001010, complemented = 11110101 add 1 = 11110110. • Advantage of 2's complement is that no separate addition and subtraction circuits are required

11. Binary fractions • So far only integer numbers have been considered Whole Number Fraction • Decimal 10.75 would be 1010.11 • Limited by the number of bits assigned to whole number • Accuracy is lost smallest value is 0.00390625

12. Floating point numbers • In scientific applications we often need very small or large numbers • In the floating point system number is represented by a signed fractional component called the mantissa and a signed exponent. mantissa × 10exponent • The value is normalised i.e. 1230000.0 would be become 0.123×107 • In a typical system where a float is 32 bits 24 bits hold the signed mantissa and 8bits for the exponent • Floating point calculations often carried out in a separate co processor

13. Binary arithmetic • Binary addition

14. Binary arithmetic • Binary subtraction * * * * (starred columns are borrowed from) 1 1 0 1 1 1 0 − 1 0 1 1 1 ---------------- = 1 0 1 0 1 1 1

15. Binary arithmetic • Multiplication - Two numbers A and B can be multiplied by partial products: for each digit in B, the product of that digit in A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used. The sum of all these partial products gives the final result. 1 0 1 1 (A) (decimal 11) × 1 0 1 0 (B) (decimal 10) --------- 0 0 0 0 ← Corresponds to a zero in B + 1 0 1 1 ← Corresponds to a one in B + 0 0 0 0 + 1 0 1 1 --------------- = 1 1 0 1 1 1 0 (decimal 110)

16. Binary arithmetic • Division i.e. divide 0110101 (53) by 0101 (5). ___ 1010 (10 answer) 0101| 0110101 0101 110 0101 11 (3 remainder)

17. Bits Bytes & Words • Bit either a '1' or a '0' • Byte comprises 8 bits ' 10110011' • Nibble - Half a byte (4 bits) '1011' • Word Normally the bus width of the microprocessor – in this case 32 bits • Hex numbers in the keil compiler are prefixed by '0x' (0xa7) • Floating point numbers – exponent is represented by the letter 'e' (1.25e-3)

18. ASCII character codes • Integer numbers are easily represented in binary. This is not the case for characters and other punctuation marks. • The standard ASCII code defines 128 character codes (from 0 to 127), of which, the first 32 are control codes (non-printable), and the other 96 are represent printable characters. • The ASCII code requires a 7 bit binary code. Memory is usually a 8-bit wide which leaves 1 unused bit – use to provide additional characters called Extended ASCII. Not standardised!

19. The Standard ASCII Table * This table is organized to be easily read in hexadecimal: row numbers represent the first digit and the column numbers represent the second one. For example, the A character is located at the 4throw and the 1st column, for that it would be represented in hexadecimal as 0x41 (6510).

20. Unicode • ASCII only supports 128 characters • ASCII was extended to 255 characters using the iso 8859 method, where 15  regions/language groups were defined. • Unicode method contains all the worlds characters (well almost) in one coding list. • Uses multibyte encoding of 1,2 or 4 bytes • Common coding is UTF-8 • first 128 codes same as ASCII so provides backwards compatibility See http://www.unicode.org