Converting Between Binary and Decimal for Computing Learners
Learn how to convert numbers between binary and decimal, understand powers of 2, do binary arithmetic, and avoid overflow in addition. Explore decimal and binary systems, practice conversions, and master powers and exponents in this comprehensive guide.
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TeachingLondon Computing Topic 2.2: Converting Binary and Decimal William Marsh School of Electronic Engineering and Computer Science Queen Mary University of London
Aims Understand powers of 2 Number of numbers Convert between binary and decimal Using powers of 2 Do binary arithmetic with more interpretation How fixed width leads to overflow in addition Look at some pictures Notes on the syllabus
Teaching Issue So far, used counting now more mathematics. Still, minimise notation and terms
CONVERTING BETWEEN BINARY AND DECIMAL
Decimal base 10 Base 10 10 numerals 0 , 1 , 2 , , 9 What does 123 mean? 123 = 1 x 100 + 2 x 10 + 3 x 1
Base 10 Table 123 = 1 x 100 + 2 x 10 + 3 x 1 100 10 1 1 2 3 most significant digit least significant digit
16 8 4 2 1 Base 2 Table 1 0 1 0 0 least significant bit most significant bit 10100 = 1 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 0 x 1 10100 = 16 + 4 = 20
Conversion to Binary To convert a decimal integer to binary: Odd 1, Even 0 Divide by 2 Stop when result of the division is 0 123 61 30 15 7 3 1 0 Least Most Significant bit Significant 1 1 1 0 1 1 1 bit 12310 = 1 1 1 1 0 1 12
Quiz Convert 00112 to decimal Convert 11112 to decimal
Powers and Exponents 10N Power of 10 N is an exponent 100 = 1 101 = 10 102 = 10 x 10 103 = 10 x 10 x 10 21 = 1 21 = 2 22 = 2 x 2 23 = 2 x 2 x 2 100 = 1 10(X+Y) = 10X x 10Y
24 23 22 21 20 Base 2 Table 1 0 1 0 0 least significant bit most significant bit 10100 = 1 x 24 + 0 x 23 + 1 x 22 + 0 x 21 + 0 x 20 = 1 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 0 x 1 = 16 + 4 = 20
Quiz Write out powers of 2, up to 28 (then 216) Convert 101000112 to decimal Convert 011111112 to decimal What is the next number after (the 10 digit number) 11111111112 in base 10?
K and 210 and 103 210 = 1024, approximately equal to 103 210 abbreviated by K 1 KByte is 1024 Bytes 220 = 210 x 210 103 x 103 = 106 220abbreviated M 1 MByte = 220 Bytes 106 Bytes
Quiz Which is larger 232 or the number of people in the world?
How Many Numbers? How many decimal numbers less that 100? 2-digit numbers : NN 0 .. 99 100 different numbers General rule: 10n n-digits (decimal) numbers 2n n-digit (binary) numbers
How Many Binary Numbers? bits max binary max base10 how many 1 2 3 4 5 6 7 8 1 11 111 1111 11111 111111 1111111 11111111 1 3 7 15 31 63 127 255 2 4 8 16 32 64 128 256
Quiz 161 student in this class How many bits to represent each student with a unique binary number? A computer can execute 9 different machine instructions: ADD, SUB, MUL, DIV, JUMP, LOAD, READ, WRITE, STOP. How many bits do we need to give each instruction a different code? What could these codes be?
Quiz Answers 7 bits?? NO! With 7 bits we can only represent 27 = 128 patterns. We need 8 bits. 8 bits can represent up to 28 = 256 patterns To represent 9 bit patterns we need 4 bits: 24 = 16 0000 ADD 0001 SUB 0010 MUL 0011 DIV 0100 JUMP 0101 LOAD 0110 READ 0111 WRITE 1111 STOP
Overflow Arithmetic with Fixed Number of Digits
Overflow The addition of 8 bit numbers may overflow 8 bits 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 + Computer arithmetic has a limited number of bits
Fixed Bit Arithmetic 0000 1111 0001 4-bit 0010 1110 0 15 1 14 2 0011 1101 13 3 12 4 1100 0100 11 5 0101 1011 6 10 9 7 8 1010 0110 1001 0111 1000
Fixed Bit Arithmetic 0000 1111 0001 4-bit 0010 1110 Add 4 0011 1101 0 0 0 0 0 1 0 0 0 1 0 0 + 1100 0100 0101 1011 1010 0110 1001 0111 1000
Fixed Bit Arithmetic 0000 1111 0001 4-bit 0010 1110 Add 4 0011 1101 0 1 1 1 0 1 0 0 1 0 1 1 + 1100 0100 0101 1011 1010 0110 1001 0111 1000
Fixed Bit Arithmetic 0000 1111 0001 4-bit 0010 1110 Add 4 0011 1101 1 1 1 0 0 1 0 0 1 0 0 1 0 + 1100 0100 0101 1011 1010 0110 1001 0111 1000
Overflow (Unsigned) 0000 1111 0001 When you passed the read line E.g. 14 + 4 = 2 0010 1110 0 15 1 14 2 0011 1101 13 3 12 4 1100 0100 11 5 0101 1011 6 10 9 7 8 1010 0110 1001 0111 1000
Two Ideas Images Pixels and resolution Image is an array of pixels Number of bits per pixel Colour of each pixel is a number
Original Red 8 bits Green 8 bits Blue 8 bits Million colours 508 578 pixels 24 bit RGB colour
Fewer Pixels 100 114 pixels 24 bit RGB colour 50 57 pixels 24 bit RGB colour m
Fewer Colours 508 578 pixels 24 bit RGB colour 508 578 pixels 256 colours (indexed)
Syllabus Binary GCSE (OCR) Conversion between binary and decimal Hexadecimal Binary addition AS/A2 (AQA) (AS) Negative numbers - two s complement (AS) More arithmetic (A2) Real (floating point) numbers
Summary Understand powers of 2 How many bits binary representation Arithmetic with fixed number of bits leads to overflow Images Pixels Bits per pixel Anything can be represented by numbers (i.e. digitally)
