Binary Values and Number Systems Chapter 2 Binary Values and Number Systems - Eswari Manickam Materials are from text book with additions and adaptations by Eswari Manickam

Chapter Goals Distinguish among categories of numbers Describe positional notation Convert numbers in other bases to base 10 Convert base-10 numbers to numbers in other bases Describe the relationship between bases 2 and 16 Explain the importance to computing of bases that are powers of 2 24 6

Binary Decimal is base 10 and has 10 digits: 0,1,2,3,4,5,6,7,8,9 Binary is base 2 and has 2 digits: 0,1 For a number to exist in a given base, it can only contain the digits in that base, which range from 0 up to (but not including) the base. What bases can these numbers be in? 122, 198, 178, G1A4 9

Positional Notation 642 is 600 + 40 + 2 in BASE 10 Continuing with our example… 642 in base 10 positional notation is: 6 x 102 = 6 x 100 = 600 + 4 x 101 = 4 x 10 = 40 + 2 x 10º = 2 x 1 = 2 = 642 in base 10

Converting Binary to Decimal What is the decimal equivalent of the binary number 1101110? 1 x 26 = 1 x 64 = 64 + 1 x 25 = 1 x 32 = 32 + 0 x 24 = 0 x 16 = 0 + 1 x 23 = 1 x 8 = 8 + 1 x 22 = 1 x 4 = 4 + 1 x 21 = 1 x 2 = 2 + 0 x 2º = 0 x 1 = 0 = 110 in base 10 13

Converting Binary to Decimal What is the decimal equivalent of the following binary numbers? a) 11101 b) 1011010 c) 10011100

Bases Higher than 10 How are digits in bases higher than 10 represented? With distinct symbols for 10 and above. Hexadecimal is base 16 and has 16 digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F 10

Converting Hexadecimal to Decimal What is the decimal equivalent of the hexadecimal number 32A? 3 x 162 = 3 x 256 = 768 + 2 x 161 = 3 x 16 = 48 + A x 16º = 10 x 1 = 10 = 826 in base 10 Remember, the digits in base 16 are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

Converting Hexadecimal to Decimal What is the decimal equivalent of the following hexadecimal numbers? 87A b) 34E c) F000

Counting in Binary/Hexadecimal/Decimal 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 10 A 1010 11 B 1011 12 C 1100 13 D 1101 14 E 1110 15 F 1111

Converting Binary to Hexadecimal Mark groups of four (from right) Convert each group 10101011 1010 1011 A B 10101011 is AB in base 16 18

Converting Binary to Hexadecimal What is the hexadecimal equivalent of the following binary numbers? 00001001 10101001 010111011110

Converting Decimal to Other Bases Algorithm for converting number in base 10 to other bases While (the quotient is not zero) Divide the decimal number by the new base Make the remainder the next digit to the left in the answer Replace the original decimal number with the quotient 19

Converting Decimal to Hexadecimal What is the hexadecimal equivalent of (3567)10 ? 222 13 0 16 3567 16 222 16 13 32 16 0 36 62 13 32 48 47 14 32 15 D E F 21

Converting Decimal to Hexadecimal What is 356 (base 10) in base 16? What is 1135 (base 10) in base 16? What is 4759 (base 10) in base 16? Try it! 20

Converting Decimal to Binary Example of converting decimal to binary What is the binary equivalent of the decimal number 35? 17 8 4 2 1 0 2 35 2 17 2 8 4 2 1 34 16 8 4 2 0 1 1 0 0 0 1 Adding digits to the left as we calculate: 100011

Converting Decimal to Binary Easy method for converting decimal to binary What is the binary equivalent of the decimal number 35? 2 35 - 1 2 17 - 1 2 8 - 0 2 4 - 0 2 2 - 0 2 1 - 1 So reading from the bottom – The answer would be 100011

Converting Decimal to Binary What is the binary equivalent of the following decimal integers? A) 64 B) 1066 C) 213 D) 1790

Arithmetic in Binary Remember that there are only 2 digits in binary, 0 and 1 1 + 1 is 0 with a carry Carry Values 1 1 1 1 1 1 1 0 1 0 1 1 1 +1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 14

Arithmetic in Binary Calculate: 10001 + 11101 1110 + 1111 1011001 + 111010

Subtracting Binary Numbers Remember borrowing? Apply that concept here: 1 2 2 0 2 1 0 1 0 1 1 1 - 1 1 1 0 1 1 0 0 1 1 1 0 0 15

Subtracting Binary Numbers Calculate: 1011011 - 10010 1010110 - 101010 1000101 - 101100

Binary Numbers and Computers Computers have storage units called binary digits or bits Low Voltage = 0 High Voltage = 1 all bits have 0 or 1 22

Binary and Computers Byte 8 bits The number of bits in a word determines the word length of the computer, but it is usually a multiple of 8 32-bit machines 64-bit machines etc. 23