1 / 55

The Switch, Part 2

The Switch, Part 2. Goals. By the end of this unit, you should understand … How to perform binary addition and subtraction How computers represent signed numbers (positive and negative) How to use Twos Complement How to represent a number using Scientific Notation. Review.

ihelms
Download Presentation

The Switch, Part 2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Switch, Part 2

  2. Goals By the end of this unit, you should understand … • How to perform binary addition and subtraction • How computers represent signed numbers (positive and negative) • How to use Twos Complement • How to represent a number using Scientific Notation.

  3. Review • The engineering decision to use simple, two-state switches to build computers forced computers to “think” in binary. • How do we convert to and from binary? • What about octal? • What about hexadecimal?

  4. Binary Arithmetic • Computer processing requires a little more capability in binary math, notably arithmetic. • Let’s consider addition and subtraction …

  5. Binary Addition • Just a few combinations are possible: • 02 + 02 = 02 • 02 + 12 = 12 • 12 + 02 = 12 • What about 12 + 12?

  6. Binary Addition 12 + 12 = 102 • Remember what that 10 means: it means a zero in the right column, and a one in the left column • When you carry in base two, you are bringing a power of two to the left…

  7. 1 1 Binary Addition 1+ 1+ 11102 +10112 =1410 =1110 0 0 1 =2510 2

  8. Binary Subtraction • Subtraction in base two is similar to addition • 112 – 112 = 002 • 112 – 002 = 112 • 102 – 002 = 102 • What about 102 - 12?

  9. Binary Addition 102 - 12 = 12 • Think about borrowing in decimal – you are really borrowing a power of your base … • In Base-10, we borrow 10 from the left column and give it to the next column to the right. So … • In Base-2, we borrow 2 from the left column and give it to the next column to the right.

  10. Binary Addition 1 1 1 001100112 -000101102 0 0 =5110 =2210 0 0 0 1 1 1 0 1 =2910 2

  11. Binary Borrowing: A Little Trick • Many students find it convenient to just mentally convert the binary number to decimal, complete the subtraction, and then convert the answer back to binary. • Consider 102 – 12 = ? • A one in binary is still a one in decimal • When you borrow for the zero, you bring over a power of two, which is equal to two • The decimal conversion for the borrowing is 2-1, which is 1 • And 110 = 12 • Be sure to add your answer back up as a check

  12. Negative Numbers • We have mastered binary math for positive integers • But what about negative numbers? • Handling the negative sign has proven somewhat problematic, as we will see …

  13. Negative Integers: Sign/Magnitude Notation • The first solution for encoding negative numbers in binary form was to dedicate a switch (which we now know is a memory location) to communicate whether a number was positive or negative • The only thing required was to agree, in the bank of switches representing the number, which switch referred to the sign • In one of the more popular encoding notations for negative numbers, the first switch is the sign.

  14. Negative Integers: Sign/Magnitude Notation • A 0 setting means the number is positive, a 1 setting means the number is negative. • This encoding scheme (for signed integers) is called sign/magnitude notation. • The first switch is the sign, the remaining switches represent the binary magnitude of the number. • Which highlights one of our key concepts about binary encoding – the leftmost digit could be a negative sign, or a “one” in magnitude • The computer must be told which interpretation/decoding scheme to use

  15. Sign/Magnitude: The Problem • What was wrong with this scheme? • Great for all numbers except 0 … • This encoding scheme allows you to come up with 2 distinct encoding schemes for 0, one with a positive sign, and one with the negative sign • The one with the negative sign has no mathematical meaning, and the ambiguity of two patterns representing the same number is problematic. • So, most computers use a different scheme than sign/magnitude for representing negative integers …

  16. Twos Complement • Twos Complement solves the ambiguity problem. • The scheme works because we are in binary math, which has only 2 digits. • Offsetting one moves you to the only other possible digit, etc.

  17. Twos Complement • In Twos Complement, you encode positive integers normally. • To use Twos Complement for negatives: • Complement every digit in the number (change 1s to 0s and 0s to 1s) • Add 1 to the complemented number • Example:–101 = 010 + 1 = 011

  18. Twos Complement: Math • Twos Complement allows us to add the representation and processing of negative integers to our growing computing capability. • To add a positive and negative number, first perform the twos complement trick on the negative number, and then add normally. • To add two negative numbers, first encode both of them in 2’s complement, and then add normally.

  19. What about fractional values? • Our goal is to learn how to encode all numbers in a way that they can be represented using switch settings • We can encode positive integers • We can encode negative integers • But what about decimal numbers (numbers with fractional values)? • To store decimal numbers, most computers utilize scientific notation …

  20. Binary Math and Scientific Notation • As a civilization, we now work with numbers so big and so small that mathematicians have already had to wrestle with creating a manageable notation. • Scientists developed scientific notation as a standard way to represent numbers.

  21. Binary Math and Scientific Notation • Consider the following example:+/-5.334 * 10+/-23 • The 5.334 part is the mantissa. • The 10 part is the base. • The 23 part is the exponent.

  22. Scientific Notation • Let’s move up a level of abstraction to the more generalized case of scientific notation: +/- M * B+/-E • M is the mantissa. • B is the base. • E is the exponent. • +/- means the number can be positive or negative.

  23. Switch Math: the Holes • If we can figure out how to use scientific notation to encode base ten decimal numbers in a binary form of scientific notation, we can represent any finite number in our computer and our switch math will be complete …

  24. Interpreting Scientific Notation • It turns out that handling scientific notation, or any other encoding scheme, is amazingly simple • We design the computer to know which interpretation scheme to use! • We will just figure out how to program in some logic that says “if a number is tagged as being in scientific notation, just use a look up chart to interpret the number.”

  25. Interpreting Scientific Notation • Given an encoded number, to decode it such that it is in scientific notation, you would need to know several things – • How big is the overall memory allocation (how many switches were used)? • How many switches are dedicated to the mantissa, and which are they? • How many switches are dedicated to the exponent, and which are they?

  26. One Last Thing! • Okay, so we know we will always need to know the encoding scheme to interpret scientific notation. • But, once we know the encoding scheme for any given computer, we need to be able to encode in this scientific notation form …

  27. Fractional Value Numbers • Consider the decimal (base 10) number: +5.75 • How can we encode this is switches? • We already know how to do half of the encoding – the integer part (the numbers to the left of the decimal point). • Let’s break the problem into two parts, the left hand side (of the decimal point) and the right hand side …

  28. Left Side of a Decimal Number • Okay, our number is 5.75 • To convert the 510 to ?2: • Use successive division 510 = 1012

  29. Right Side of a Decimal Number • On the left hand side, we divided successively by 2 to convert • On the right hand side, we multiply successively by 2 to convert • We are multiplying numbers by 2 to “promote” them to the left hand side of the decimal place. • Once promoted, we record them as a digit in our answer. • This process is called extraction….

  30. Extraction STEP ONE: Draw a table with two columns. Label the columns for the expression and the extraction, respectively.

  31. Extraction STEP TWO: Write down your original number and multiply it by 2. Put the whole number part in the Extraction column.

  32. Extraction STEP THREE: Bring down the fractional part (the number to the right of the decimal point) and multiply it by two. Extract the whole number, as before.Repeat until you extract 0.

  33. Extraction STEP FOUR: Read the extraction column, from top to bottom. This number represents your binary equivalent. 0.7510 = 1102

  34. Both Sides Now • 5.7510 = 101.1102 This isn’t in scientific notation yet, but at least we have figured out how to convert it to binary • Recall the steps so far: • Do the problem in halves • For the left side of the decimal, use successive division by 2 and read the remainders from bottom to top • For the right hand side of the decimal, use successive multiplication by 2 and read the extractions from top to bottom

  35. Let’s Try One More … • Let’s try another problem. This time, with a number that doesn’t convert to binary as easily … • Let’s try this number: 2.5410 = ?2

  36. Converting • Remember our algorithm: left side by successive division, right side by successive multiplication

  37. Left Side of 2.54 • 2 by successive division Reading the remainders from bottom to top, 210 = 102

  38. Right Side of 2.54 • Reading the extraction column, top to bottom: .5410 = 1000101002 • What would have if we kept extracting, past .24 * 2? • Let’s convert our number back to Base-10 to see what happens …

  39. Let’s Go Back the Other Way • Putting 10.1000101002 back to base 10 • Left hand side first, expanded notation • 102 = (0 * 20) + (1 * 21) = 0 + 2 = 210 • Now, to convert the right hand side…

  40. Right Side of a Decimal Number • Remember, the left hand side represents powers of two … positive powers of two • Moving left from the decimal point, we saw the unit values were 20, 21, 22, 23, 24, etc. • Doesn’t it seem reasonable that the units to the right of the decimal represent negative powers of two? • Moving from the right of the decimal point, we will see unit values for 2-1, 2-2, 2-3, 2-4, 2-5, etc.

  41. Okay, But What Does That Mean? • Exactly what is a number to a negative power, such as 2-1, 2-2, 2-3? • These are numbers you already know we are just used to seeing them written in a different format:

  42. Back to Our Conversion … • Remember, we are translating 10.100010100 • We finished the left hand side = 102 = 210 • Now for the right hand side: • .100010100 = • (1 * 2-1) + (0* 2-2) + (0* 2-3) + (0* 2-4) + (1* 2-5) + (0* 2-6) + (1* 2-7) + (0* 2-8) + (0* 2-9) • Which is equal to ½ + 1/32 + 1/128 (we can throw out the zero terms) • Which is equal to 69/128 = .539 • Combining both sides, 10.1000101002 = 2.53910 • But we started with 2.54! Why did we get the error?

  43. The Imprecision of Conversions • We started with 2.5410, took it to switch units in binary, and then back to decimal • Our result was 2.539 • This is a precision error: • 2.539 / 2.540 = .99, so our percentage error is 1%!!! • As you might imagine, in large, multi-step calculations these errors can become significant enough for you to see if you are working in a package such as Excel….

  44. Storing Binary in Scientific Notation • Remember, we said if we could encode a decimal number in switch math, we were good to go. • Decimal numbers are usually stored in a binary version of scientific notation. • If we can take our translation of 2.54, and store it in a binary version of scientific notation, we can quit!

  45. Last Step • There are several switch encoding schemes for scientific notation. • The good news, remember, is that you (and the computer!) will always have to be told which encoding scheme to use. • One common encoding scheme is 16-bit encoding …

  46. 16 Bit Scientific Notation Encoding • In 16 bit encoding, 16 bits (or switches) are allocated to the numeric part of a number expressed in scientific notation. • There are a few different flavors of 16 bit encoding, with a different allocation of switches between the mantissa and the exponent.

  47. One 16 Bit Encoding in Detail • Remember the parts of a number in scientific notation? – the mantissa, base and exponent? • Each is assigned to specific locations in the 16 bit switch pattern. In the variation we are looking at first: • Switch 1 = Sign of the mantissa • (Decimal point assumed to go next, but isn’t stored) • Switches 2-10 – next 9 switches are for the mantissa • Switch 11 – Sign of the exponent • Switches 12-16 – next 5 switches are for the exponent

  48. 16 Bit: More • What happened to the base? We don’t have to store that, because it will always be base two, the binary base from the world of switches • Don’t need to store the decimal point, because the encoding scheme will guarantee it is placed correctly when decoded • There’s only one problem: our number must be “normalized” before encoding • This means that we need to adjust the exponent as required to insure that the first significant digit is the first digit to the right of the decimal point

  49. Normalizing • Take the number 101.112, which is really 101.11 * 20 • Before we can store this number in scientific notation, we must normalize it. • We have to adjust the exponent until the first significant digit, which is the leftmost 1, is the first digit to the right of the decimal point. • In other words, we have to move the decimal place in our example 3 places to the left, so that the number become .10111.

  50. Normalizing • How can we legally do this, without changing the value of the number? • Every time you move the decimal place to the left, you must adjust the exponent by a positive increment. • In our case, we moved 3 decimal points to the left, so our new exponent is 23 • In other words, 101.11 * 20 = .10111 * 23

More Related