Binary Numbers
- 1. Recall . . . • The definition of a computer • Analog/Digital • Electric/Mechanical • General Purpose/Special Purpose • The General Purpose Electronic Computer • General Purpose Binary • Binary? 2-1
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- 3. Binary Circuitry • Binary circuitry: – cheap – reliable – able to be extended to very complicated logic • built on only two states » ON (1) » OFF (0) 2-3
- 4. Computers work in Binary • Computers are not only powered by electricity they “compute” with electricity – they shift voltage pulses around internally – circuits allow for electricity to flow or to be blocked depending on the type of circuit Closed Open circuit circuit ON or 1 OFF or 0 2-4
- 5. Representation of Data • So, our binary computer can represent: – 0s and 1s. . . – We need to represent considerably more than that: • Numbers • Characters • Visual Data • Audio Data • Instructions … and we need to do it with only 0’s and 1’s 2-5
- 6. Representation of Numbers • Representing numbers is considerably more than something that looks like the symbol “1” or “2” or “430” – We’re trying to represent numbers; which have conceptual meaning 9 = 3+3+3 = 4+5 = 10-1 = 3*3 = 3^3 = 9 2-6
- 7. Representation of Numbers • Decimal numeration system: (aka base 10) – Uses 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. – The place values of each position are increasing powers of ten. • A number such as 1428 literally means: – Eight Ones – Two Tens 104 103 102 101 100 – Four Hundreds – One (A single) Thousand 10000 1000 100 10 1 = (1 x 1000) + (4 x 100) + (2 x 10) + (8 x 1) 2-7
- 8. Combinations • Imagine we have three light-bulbs in a row, and each bulb can be on (1) or off (0). • How many unique combinations of lights can we have? – (Hint, start with all lights off) 2-8
- 9. Combinations • The number of unique combinations we can have of one light with two states per lights is two: 0 1 • The number of unique combinations we can have of two lights with two states per light is four: 00 01 10 11 • The number of unique combinations we can have of three lights with two states per lights is eight: 000 100 001 101 010 110 011 111 2-9
- 10. Representation of Numbers • Our three-light system – has eight possible combinations of on and off. • With eight unique combinations, we could represent the numbers 0, 1, 2, 3, 4, 5, 6, 7 0 = 000 4 = 100 1 = 001 5 = 101 2 = 010 6 = 110 3 = 011 7 = 111 2-10
- 11. Representation of Numbers • Binary numeration system (aka base 2): – Will use 2 symbols: 0, and 1. (Each is called a bit for binary digit) – The place values of each position are powers of two. – A binary number such as 10110two will be expanded as: • Zero Ones • One Two 24 23 22 21 20 • One Four 16 8 4 2 1 • Zero Eights • One Sixteen 1 0 1 1 0 = (1 x 16) + (0 x 8) + (1 x 4) + (1 x 2) + (0 x 1) = 22 in decimal 2-11
- 12. Binary-to-Decimal Conversion • Convert the following binary number (base two) in decimal (base ten) 1 0 0 0 0 0 1 1 2-12
- 13. Binary Conversion 1 0 0 0 0 0 1 1 • Step 1: Make a table with the same number of columns as places in the binary string and copy the string into the table 1 0 0 0 0 0 1 1 2-13
- 14. Binary Conversion • Step 2: Write out the powers of two corresponding to each position in the binary number: 27 26 25 24 23 22 21 20 1 0 0 0 0 0 1 1 2-14
- 15. Binary Conversion • Step 3: Write out the powers of two corresponding to each position in the binary number in decimal: 5 27 26 2 24 23 22 21 20 128 64 32 16 8 4 2 1 1 0 0 0 0 0 1 1 2-15
- 16. Binary Conversion • Step 4: multiply the second and third rows and put the result in the fourth row: 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1 1 0 0 0 0 0 1 1 128 0 0 0 0 0 2 1 2-16
- 17. Binary Conversion • Step 5: (final step) – Add up all the numbers in the fourth row 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1 1 0 0 0 0 0 1 1 128 0 0 0 0 0 2 1 128+0+0+0+0+0+2+1 = 131 2-17
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- 19. Decimal to Binary 2(number of places) = number of unique combinations we can achieve with some number of places in binary; but we have to use one of the mappings for zero so. . . 2(number of places) - 1 = largest binary number we can store with that many places 1 21 – 1 = 1 with 1 place we can store numbers from 0 to 1 5 25 – 1 = 31 with 5 places we can store numbers from 0 to 31 7 27 – 1 = 127 with 7 places we can store numbers from 0 to 127 8 28 – 1 = 255 with 8 places we can store numbers between 0 and 255 2-19
- 20. Binary Conversion Step 2: Write out a binary conversion table with n many places, and fill in the values of the first two rows: 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1 2-20
- 21. Binary Conversion • Step 3: Subtract out the decimal powers of two from left to right. – If you can subtract that amount and the result is non-negative, write a 1 in the binary string and continue with the result – If the subtraction results in a negative number, write a 0 and continue with the last positive number 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1 245 245- 117 - 53 - 21 - 5 - 8 5 - 4 1 - 2 1 - 1 128 = 64 = 32 = 16 = = -3 = 1 = -1 = 0 117 53 21 5 error error 1 1 1 1 0 1 0 1 2-21
- 22. Addition/Subtraction • Addition in any number system. . . – We add in the places from right to left – If the sum of the two numbers exceeds the symbols we have at our disposal we “carry” some amount. . . 4 + 8 = “twelve” ; in base ten, we have no single numeric symbol that equals twelve • We don’t need to express “twelve” in a single symbol, because we can do so by breaking the number down and incrementing the base • Instead we write 12 which is: One Ten and Two Ones. • Logically, we subtract 10 (the base amount) from our sum, and write the result in that place, and we carry a one into the next place, which represents One additional ten (our base number) 2-22
- 23. Binary Addition • Addition of binary numbers: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 2ten How do we write this? So we’ll carry the symbol 1 into the next place (which represents two in decimal), and we’ll write down the sum minus our base amount (2) 1 + 1 = 10two Example: adding two binary numbers 2-23
- 24. Binary Subtraction • Subtraction of binary numbers: 0 - 0 = 0 0 - 1 = -1 problem! We need to borrow … 1 - 0 = 1 1 - 1 = 0 Remember when borrowing in base 10: – We decrease the symbol to the left by one – Remember when we borrow 1 from the symbol to our left, it has a value that’s “ten more” (or the base amount) 2-24
- 25. Binary Subtraction • Let’s subtract the following numbers: 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 2-25
- 26. Binary Subtraction • We need to borrow for the third term, but the eighth is the closest term with something to borrow from! 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1 2-26
- 27. Binary Subtraction • So, we borrow 2ten from the first place we can 0 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 2-27
- 28. Binary Subtraction • Keep borrowing two 1 0 2 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1 2-28
- 29. Binary Subtraction • Keep borrowing 2 . . . 1 1 0 2 2 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1 2-29
- 30. Binary Subtraction • Keep borrowing 2 . . . 1 1 1 0 2 2 2 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1 2-30
- 31. Binary Subtraction 1 1 1 1 0 2 2 2 2 2 now, we’re able to subtract here 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1 1 2-31
- 32. Binary Subtraction • The rest of the subtractions are easy . . . 1 1 1 1 0 2 2 2 2 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 0 0 1 1 1 1 1 1 So our answer is -> 00111111 Let’s convert to decimal to check our work 2-32
- 33. Binary Subtraction • converting binary to decimal we get. . . 1 0 0 0 0 0 1 1 -> 128+2+1 = 131 - 0 1 0 0 0 1 0 0 -> 64+4 = 68 ------------------ 0 0 1 1 1 1 1 1 -> 32+16+8+4+2+1 = 67 131 – 68 = 67 2-33
- 34. Binary Numbers • Binary numbers actually have some other neat properties as well . . . – QUESTION: Can you multiply a binary number by two (decimal) and give the result in binary quickly? • Multiply the number 00110 by two (and give the answer in binary) – Why does this work? 25 24 23 22 21 20 32 16 8 4 2 1 0 0 0 1 1 0 2-34