ECE 233 Intro to Digital Systems Topics Number Systems Converting Between Them ECE 233 Intro to Digital Systems 1 Administrative Find a lab partner Sign Student Responsibility Form Pick up the FPGA board ECE Parts Room C 114 No lab on Wednesday 11 29 23 Will have a normal lecture and lab demonstration Install Quartus software Bring FPGA board Lab 1 Checkpoint 1 due Wednesday 11 29 23 Homework 1 due Monday 12 04 23 ECE 233 Intro to Digital Systems 2 Number Systems All numbers are represented using bases Bases OR radix r is the number of unique digits in the system Each digit D is restricted to 0 D r DN 1 D1D0 DN 1 rN 1 D1 r1 D0 r0 N number of integers or digits Decimal is Base 10 D2D1D0 D2 102 D1 101 D0 100 0 Di 9 Unique digits 0 9 Example 13710 1 102 3 101 7 100 ECE 233 Intro to Digital Systems 3 The Binary Number System What is binary Electrical signals can be represented as a 0 or a 1 o Any voltage within one range represents a 1 o Any voltage within another range represents a 0 Why is binary important Computers use binary numbers not decimal o Why not use decimal o Harder to easily and quickly tell the difference between ten or more levels o Each individual binary digit is called a bit so 110 is made up of three bits Binary representation is Base 2 radix 2 where the only two digits are 1 and 0 B2B1B0 B2 22 B1 21 B0 20 0 Bi 1 Example 510 in binary is 1012 1 22 0 21 1 20 ECE 233 Intro to Digital Systems 4 Unsigned Binary Examples Examples What is decimal 14 in binary 1410 11102 1 23 1 22 1 21 0 20 1 8 1 4 1 2 0 1 8 4 2 0 14 What is binary 1011 in decimal 10112 1110 1 23 0 22 1 21 1 20 1 8 0 4 1 2 1 1 8 0 2 1 11 ECE 233 Intro to Digital Systems 5 In class Activity Unsigned Binary What is decimal 29 in binary 2910 111012 1 24 1 23 1 22 0 21 1 20 1 16 1 8 1 4 0 2 1 1 16 8 4 0 1 2910 111012 What is decimal 57 in binary 5710 1110012 1 25 1 24 1 23 0 22 0 21 1 20 1 32 1 16 1 8 0 4 0 2 1 1 32 16 8 0 0 1 5710 1110012 What is decimal 75 in binary 7510 10010112 1x26 0 25 0 24 1 23 0 22 1 21 1 20 1 64 0 32 0 16 1 8 0 4 1 2 1x1 64 0 0 8 0 2 1 7510 10010112 ECE 233 Intro to Digital Systems 6 In class Activity Unsigned Binary What is binary 10101 in decimal 101012 2110 1 24 0 23 1 22 0 21 1 20 1 16 0 8 1 4 0 2 1 1 2110 What is binary 01001101 in decimal 010011012 7710 0 27 1 26 0 25 0 24 1 23 1 22 0 21 1 20 0 128 1 64 0 32 0 16 1 8 1 4 0 2 1 1 7710 What is binary 101110 in decimal 1011102 4610 1x25 0 24 1 23 1 22 1 21 0 20 1 32 0 16 1 8 1 4 1 2 0x1 4610 ECE 233 Intro to Digital Systems 7 Two Other Number Systems Other common bases you will see in this class 1 Octal Base 8 Radix 8 digits 0 7 o Oct2Oct1Oct0 Oct2 82 Oct1 81 Oct0 80 0 Octi 7 o Example 83 in Octal is 1238 83 1 82 2 81 3 80 2 Hexadecimal Base 16 Radix 16 digits 0 9 A F o H2H1H0 H2 162 H1 161 H0 160 0 Hi 15 o Note that A 10 B 11 C 12 D 13 E 14 and F 15 o Example 92 in Hexadecimal is 5C16 92 0 162 5 161 C 160 o You may also see it written as 0x5C It is usually easier to convert a number to binary and then use these two tricks 1 Octal Starting from the far right look at every three bits and convert that binary value into a decimal equivalent to Octal 2 Hexadecimal Starting from the far right look at every four bits and convert that binary value into a decimal equivalent to Hexadecimal ECE 233 Intro to Digital Systems 8 Octal and Hexadecimal Examples What is decimal 57 in octal 5710 718 7 81 1 80 7 8 1 1 56 1 57 Alternatively 5710 is 1110012 111 0012 111 7 and 001 1 so 718 What is decimal 77 in hexadecimal 7710 4D16 or 0x4D 4 161 13 160 4 16 13 1 64 13 77 Alternatively 7710 is 010011012 0100 11012 0100 4 and 1101 13 or D so 4D16 Can use the same conversion trick to go back to binary i e convert 4 to binary convert D to binary ECE 233 Intro to Digital Systems 9 Handling Fractions Binary Octal and Hexadecimal can also represent fractions For example 110 1102 is 6 7510 1 22 1 21 0 20 1 2 1 1 2 2 0 2 3 1 4 1 2 0 1 1 1 0 4 2 0 0 5 0 25 0 6 75 Same goes for Octal and Hexadecimal i e 8 1 16 1 etc ECE 233 Intro to Digital Systems 10 Handling Fractions In Class Activity What is decimal 29 25 in binary 29 2510 11101 012 1 24 1 23 1 22 0 21 1 20 0x2 1 1x2 2 16 8 4 0 1 0 00 0 25 29 25 What is decimal 57 125 in binary 57 12510 111001 0012 1 25 1 24 1 23 0 22 0 21 1 20 0x2 1 0x2 2 1x2 3 1 32 1 16 1 8 0 4 0 2 1 1 0x0 5 0x0 25 1x0 125 32 16 8 0 0 1 0 0 0 125 57 125 ECE 233 Intro to Digital Systems 11 Handling Fractions In Class Activity What is binary 10101 1 in decimal 10101 12 21 510 1 24 0 23 1 22 0 21 1 20 1x2 1 1 16 0 8 1 4 0 2 1 1 1x0 5 21 510 What is binary 01001101 0001 in decimal 01001101 00012 77 062510 0 27 1 26 0 25 0 24 1 23 1 22 0 21 1 20 0x2 1 0x2 2 0x2 3 1x2 4 0 128 1 64 0 32 0 16 1 8 1 4 0 2 1 1 0x0 5 0x0 25 0x0 125 1x0 0625 77 0625 ECE 233 Intro to Digital Systems 12 Data Sizes in the Digital World A 1 byte is 8 bits ex 01111010 Kilobyte is 1 024 bytes not 1 000 bytes K 210 bytes M 220 G 230 T 240 P 250 …