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Lecture 2 Bits, Bytes & Number systems

Lecture 2 Bits, Bytes & Number systems. Representation of Numbers. Different ways to say “how many”… Human: decimal number system Radix-10 or base-10 Base-10 means that a digit can have one of ten possible values 0 through 9. Computer: binary number system Radix-2 or base-2 Why binary?

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Lecture 2 Bits, Bytes & Number systems

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  1. Lecture 2Bits, Bytes & Number systems

  2. Representation of Numbers • Different ways to say “how many”… • Human: decimal number system • Radix-10 or base-10 • Base-10 means that a digit can have one of ten possible values • 0 through 9. • Computer: binary number system • Radix-2 or base-2 • Why binary? • Each digit can have one of two values • 0 or 1

  3. Bits and Bytes • A binary digit is a single numeral in a binary number. • Each 1 and 0 in the number below is a binary digit: 1 0 0 1 0 1 0 1 • The term “binary digit” is commonly called a “bit.” • Eight bits grouped together is called a “byte.”

  4. Relationship between Decimal & Binary Background: • Number systems are positional • There are 10 symbols that represent decimal quantities • Multi-digit numbers are interpreted as in the following example • 79310 = 7 x 100 + 9 x 10 + 3 = 7 x 102 + 9 x 101 + 3 x 100 • Each place value in a decimal number is a power of 10. • We can get a general form of this • ABCbase • A x (base)2 + B x (base)1 + C x (base) 0 Remember that the position index starts from 0. Indicate positions

  5. Relationship between Decimal & Binary • Binary numbers are represented using the digits 0 and 1. • Multi-digit numbers are interpreted as in the following example • 101112 = 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20 = 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 1 x 1 • Each place value in a binary number is a power of 2.

  6. Converting binary numbers to decimal Step 1: Starting with the 1’s place, write the binary place value over each digit in the binary number being converted. Step 2: Add up all of the place values that have a “1” in them. Interpret the binary number 101012 in decimal • TRY: Interpret the binary number 010101102 in decimal

  7. Converting decimal numbers to binary • We have now learnt how to convert from binary to decimal • Using positional representation • But how about decimal to binary • Repeated division method • Simply keep dividing it by 2 and record the remainder • Repeat above step as many times as necessary until you get a quotient that can’ t be divided by 2 • Remainders give the binary digits, starting from the last remainder • Let’s look at some examples…

  8. Converting decimal numbers to binary • Let’s convert decimal 23 to binary. Step 1: 23/2 = 11 remainder 1 Step 2: 11/2 = 5 remainder 1 Step 3: 5/2 = 2 remainder 1 Step 4: 2/2 = 1 remainder 0 • The last quotient “1” cannot be divided by 2 any more. So the process ends. The final binary number is read from the very end including the last quotient: 1 0 1 1 1 • Try: Convert decimal 73, 96, 127, 128 to binary.

  9. Hexadecimal • Computers use binary number system because of the electric voltage (high or low voltage) • Very difficult to express for large number representation • Hexadecimal to rescue • Hexadecimal system is interface between human brain and computer brain • 4 bits from binary are read together and represented using a single digit • Such 4-bits are known as nibble • This gives a total of 16 different options • The hexadecimal number system is a Base-16 number system: • There are 16 symbols that represent quantities: • Represented by the symbols 0-9 and A-F where the letters represent values: A=10, B=11, C=12, D=13, E=14, and F=15

  10. Numbering systems

  11. Hexadecimal • Thus each byte is two hex digits (shorthand representation for human) • EX: Binary: 110010102 • How to represent this in hex representation? 110010102 • Separate them into nibbles 1100 1010 C A Hexadecimal representation: CA16 • Try: convert 11110101101011002 to Hex representation

  12. Converting Hex number to Binary • Converting hexadecimal numbers to binary is just the reverse operation of converting binary to hexadecimal. • Just convert each hexadecimal digit to its four-bit binary pattern. The resulting set of 1s and 0s is the binary equivalent of the hexadecimal number. • Convert A5B916 to Binary.

  13. Conversion: Hex and decimal • Hex to decimal • Exact similar to binary to decimal • Use base 16 • Try CA16 • Decimal to Hex • Exactly similar to decimal to binary • Divide by 16 • Try 20210

  14. Notes on Bases • Subscript is mandatory at least for a while. • We use all three number bases. • When a number is written, you should include the correct subscript. • Pronunciation • Binary and hexadecimal numbers are spoken by naming the digits followed by “binary” or “hexadecimal.”

  15. Ranges of Number Systems Ranges of Unsigned Number Systems

  16. Electronic Prefixes • There is a set of terms used in electronics to represent very large values and very small values. • Kilo, Mega, Giga, Tera – used for representing very large values • E.g., KiloByte, MegaByte etc. • milli, micro, nano, pico – used for representing very small values • E.g., milliseconds, microseconds etc.

  17. Reading assignment • In Blackboard: • reading02_08_29_2012.pdf • reading03_08_29_2012.pdf

  18. Practice problems • Let us do some quick example problems …

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