The Switch Part 2 CSCI 23000 Computing I

The Switch, Part 2 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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? CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Binary Arithmetic • Computer processing requires a little more capability in binary math, notably arithmetic. • Let’s consider addition and subtraction … CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Binary Addition • Just a few combinations are possible: – 02 + 0 2 = 0 2 – 02 + 1 2 = 1 2 – 12 + 0 2 = 1 2 • What about 12 + 12? CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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… CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Binary Addition 1+ 1+ 11102 +10112 =1410 110012 =2510 =1110 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Binary Subtraction • Subtraction in base two is similar to addition – 112 = 002 – 112 – 002 = 112 – 102 – 002 = 102 • What about 102 - 12? CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Binary Addition 0 1 1 00112 -000101102 0 0 0 1 1 1 01 2 =5110 =2210 =2910 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 … CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 … CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Twos Complement • In Twos Complement, you encode positive integers normally. • To use Twos Complement for negatives: – Complement every digit in the number (change 1 s to 0 s and 0 s to 1 s) – Add 1 to the complemented number • Example: – 101 = 010 + 1 = 011 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 … CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 … CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. ” CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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? CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 … CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 … CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Left Side of a Decimal Number • Okay, our number is 5. 75 • To convert the 510 to ? 2: – Use successive division Division Remainder 5/2=2 2/2=1 1/2 =0 1 510 = 1012 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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…. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Extraction STEP ONE: Draw a table with two columns. Label the columns for the expression and the extraction, respectively. Expression Extract. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Extraction STEP TWO: Write down your original number and multiply it by 2. Put the whole number part in the Extraction column. Expression Extract. 0. 75 * 2 = 1. 50 1 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. Expression Extract. 0. 75 * 2 = 1. 50 1 0. 50 * 2 = 1. 00 1 0. 00 * 2 = 0. 00 0 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Extraction STEP FOUR: Read the extraction column, from top to bottom. This number represents your binary equivalent. Expression Extract. 0. 75 * 2 = 1. 50 1 0. 50 * 2 = 1. 00 1 0. 00 * 2 = 0. 00 0 0. 7510 = 1102 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Converting • Remember our algorithm: left side by successive division, right side by successive multiplication CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Left Side of 2. 54 • 2 by successive division Division 2/2 = 1 1/0 = 0 Remainder 0 1 Reading the remainders from bottom to top, 210 = 102 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 … Division Extraction . 54 * 2 = 1. 08 1 . 08 * 2 = 0. 16 0 . 16 * 2 = 0. 32 0 . 32 * 2 = 0. 64 0 . 64 * 2 = 1. 28 1 . 28 * 2 = 0. 56 0 . 56 * 2 = 1. 12 1 . 12 * 2 = 0. 24 0 . 24 * 2 = 0. 48 0 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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… CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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: 2 – 1 1/21 ½ . 5 2 – 2 1/22 ¼ . 25 2 – 3 1/23 1/8 . 125 2 -4 1/24 1/16 . 0625 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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? CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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…. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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! CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 … CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 a few different flavors of 16 bit encoding, with a different allocation of switches between the mantissa and the exponent. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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. CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

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 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Normalizing Examples • A couple more, to make sure we have it: • 1101. 11 * 20 =. 110111 * 24 • 101. 1 * 20 =. 1011 * 23 • . 01 * 20 =. 1 * 2 -1 • . 001 * 20 =. 1 * 2 -2 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Algorithm • 1. Convert the base ten number to base two, one side of the decimal place at a time – Left hand side, successive division – Right hand side, successive multiplication • In our case, 2. 5410 = 10. 1000101002 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Algorithm (con’t) • Put number in normalized form 10. 100010100 * 20 =. 10100010100 * 22 • Determine what the encoding scheme is • For the sake of our problem , assume the following 16 bit encoding: – – 1 = Sign of mantissa 2 -12 = Mantissa 13 = Sign of exponent 14 -16 = Exponent CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Algorithm (con’t) 1 2 3 4 5 6 7 8 9 10 11 12 13 1 4 0101 00 0 1 5 1 16 0 Using the described notation system, the decimal 2. 54 10 would be stored in switch binary math as the following pattern of zeros and ones: 0101000010 CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science

Questions? CSCI 23000: Computing I Copyright © 2015 Department of Computer & Information Science