# Bits and Bytes. Analog versus digital

Save this PDF as:

Size: px
Start display at page:

Download "Bits and Bytes. Analog versus digital"

## Transcription

1 Bits and Bytes Computers are used to store and process data. Processed data is called information. You are used to see or hear information processed with the help of a computer: a paper you ve just typed using Microsoft World, a digital photograph, a MP3 song you ve downloaded on the Internet, etc. But how is the data stored in the computer? Whatever the medium in which it is stored, the most fundamental piece of data is always represented by a two-value system : low or high voltage, positive or negative polarity of a magnetic medium, absence or presence of bumps on a CD or DVD These two values are logically represented by the digits 0 and 1. This is called a binary representation. Usually, the digit 0 is interpreted as the low value or absence of some kind of signal (such as no bump) and the digit 1 is interpreted as the high value of presence of some kind of signal. In other word, we encode the physical signal with the logical binary representation. This 0 or 1 digit is called a bit (short for binary digit). Obviously, just one bit cannot give much information since there are only two possible values. Hence, to store more useful data we must combine bits. However, even combining several (even many) bits will only give a finite numbers of possible objects. For example combining four bits gives the sequences 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, These sixteen sequences can be selected to represent any sixteen objects of our choice but only sixteen objects. So if we want to represent integers, we can decide they represent the integers 0 to 15; if we represent letters they could represent the letters a, b, c, d, e, f, g, h, i, j, k, l, m, and n (or any other combination of sixteen letters of our choice); if we represent colors they could represent 16 predefined colors of our choice. We say that the data stored on a computer, and represented logically by combinations of bits, is discrete: it is finite and made of separable units. Analog versus digital Many different kinds of data can be represented in a computer: numbers: whole numbers, positive and negative integers, real numbers; text: made of letters, characters, digits; audio: speech, music, sound tracks; images and graphics; video, etc. Some of these data types, such as text or integers on a bounded range, are discrete and it will be rather straightforward to convert these data to a binary representation that can be stored in the computer. But most data is analog, which means that it is continuous in nature. Let s look at some examples. Pictures Sound Continuous in space Infinite number of colors Continuous in time Infinite number of frequencies 1

2 Real numbers Continuous even if bounded: always a number between two given numbers Because most data is analog, it must be broken in discrete pieces and those pieces must be represented by a binary representation to be stored in the computer. When this process is performed, we say that we digitize the data. Note that because digitizing data transforms continuous infinite data into finite discrete data, some data will be lost. For example we cannot really represent all the colors of the color spectrum, we cannot represent all the real numbers between 0 and 1. In this unit of the course, we will study how some types of data are represented and discuss some of the issues that arise from the conversion from analog to digital data. We will also use some tools to manipulate data. Binary Numbers Because the representation of more complex data such as sound or color is based on the representation of quantities (i.e. numbers), we must start our study with the representation of whole numbers. As seen above the sixteen first whole numbers (0 to 15) can be represented by all possible four-bit sequences. We will generalize this to longer bit sequences, but we will show how to derive a meaningful way to convert from a decimal number (the regular number) to a binary number and vice versa. The number system we use is the number system in base 10. Base 10 number system We use 10 digits, 0 to 9. In a number, the value of each digit depends of its position in the number. Each position corresponds to a value that is a power of 10. Table 1 shows the decomposition of the number 2324 in the number system of base 10 (also called decimal number system) thousands hundreds tens units Table 1 This can also be written as. Base 2 number system The binary number system is the number system in base 2. We use 2 digits, 0 and 1, and each position in a number corresponds to a power of 2. Table 2 shows the decomposition of the binary number 1110 in the number system of base eights fours twos units Table 2 Therefore the value of the binary number is. When necessary, to avoid confusion we will append the subscript 2 to binary numbers. So the equality means that the number with binary representation 1110 is the same as the number with decimal representation 14 (we obviously cannot write 1110 = 14!). 2

3 We can check that the 16 sequences given in page 1 do match the decimal numbers 0 to 15 according to our definition. Binary Number Decomposition in sum of powers of 2 s Decimal Value Table 3 We can notice the pattern:,,,. In general,, where n is the number of zero following the 1. This is, of course, consistent with the definition of the binary number: there are n + 1 digits; each digit in this binary number corresponds to a power of 2, from for the left-most digit to for the right-most digit; all powers of 2 s except the left most ( ) are multiplied by 0. Recreation: Magic cards 1 A magician has four magic cards; each card holds eight numbers between 1 and Blue Green Red Yellow She instructs each member of the audience to choose a number between 1 and 15 and hands out the cards to one person at random. The selected person returns only the cards that contain the number she/he had chosen (and kept secret). The magician guesses the number chosen. Suppose for example the person hands out the blue, green and yellow cards. Then the magician says Eleven! And she is right! How does this work? The cards are based on the binary representation of the numbers 1 to 15. The blue card contains all the numbers that have a 1 as right-most bit (i.e., the bit that corresponds to the units). The green card 1 Adapted from Aha! by Martin Gardner (1) 3

4 contains all the numbers that have a 1 as second digit from the right (i.e., the bit that corresponds to 2 1 ). The red card contains all the numbers that have a 1 as the second digit from the left (i.e., the bit that corresponds to 2 2 = 4). And finally the yellow card contains all the numbers that have a 1 as left-most digit (i.e., the bit that corresponds to 2 3 = 8). So, in our example, when the magician is given the blue, green and yellow cards, he knows that the number is 1011 in binary and therefore = 11 in decimal. Note that the magician does not have to know binary numbers; she just needs a clever aide who made the cards for her! She just has to add the number that is at the top left corner of the cards that the member of the audience gave her. Exercises: all about patterns 1. Table 3 shows clearly that the larger the number the more bits are needed to express a number in its binary representation. For example, 6 can be expressed with only 3 bits: but 12 cannot since 12 is It is permissible to add or remove leading zeros (i.e., zeros with no 1 s to their left), but it is not permissible to remove any other digits from the binary representation without altering its value. This is similar as being the same as 1060 in the decimal number system, but certainly different from 0106 or With one bit we can express just two numbers 0 and 1. a) Observe table 3. How many numbers can be represented with 2 bits? with 3 bits? with 4 bits? b) You should get a pattern in question a). From it, can you deduce the number of numbers that can be represented with 5 bits? with 8 bits? with n bits? 2. Observe table 3. Explain how you can distinguish between even and odd numbers if you are given just the binary representation. 3. If you observe the Binary Number column of table 3 you can imagine that this column is split into 4 columns: the column of the first digit (corresponds to the eights), the columns of the second digit (column of the fours), etc. Explain the pattern of 0 and 1 in each of these four columns, starting from the right-most column to the left-most column. 4. To write all the binary numbers from 0 to 31, you need to add one more column(to the left) to the imaginary columns that make up the Binary Number column of table 3 (see question 3) and continue the pattern you described in question 3. Create the table, similar to table 3, that shows all binary numbers from 0 to Create magic cards with numbers from 0 to 31. Practice the magic trick with those cards. 6. Find the decimal values of the following binary numbers. (Note: to make them more readable we often write the digits of a binary numbers by groups of four.) a) b) c)

5 d) From decimal to binary We know how to convert a binary number into a decimal number; but how do we do the reverse? From what we have already learned, some numbers are easy to convert: 1 is of course = 2 1 so 2 is = 2 2 so 4 is = 2 4 so 16 is In general,, where 1 is followed by n zeros. For other positive integers we will use the fact that if we can write them as a sum of powers of 2 s then we can write their binary representations. For example, since 57 = we can write. But how do we find that 57 = ? First, let us review the powers of 2 s: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 (How do you go from one power of 2 to the next one? 2 ) Now let s start with Find the largest power of 2 no larger than 57: it is 32. Subtract 32 from 57: = 25. Find the largest power of 2 no larger than 25: it is 16. Subtract 16 from 25: = 9. Find the largest power of 2 no larger than 9: it is 8. Subtract 8 from 9: 9 8 = 1. When we reach 0 or 1, we stop. 57 = = = Repeated subtraction algorithm Input: A non-negative integer (for practical purpose no greater than 1023) Output: The binary representation of the input Step 1: Let A be the input. Find the largest power of 2 no greater than A, say 2 n. Step 2: Draw a table with 2 rows and n + 1 columns. The cells in the first row should contains the powers of 2 s from 2 n (found in step 1) to 2 0 = 1, written from left to right. Place a 1 in the left most column in row 2. Step 3: Subtract 2 n from A and make this value the new value of A. Step 4: repeat steps 4.a and 4.b until A is equal to 0 or 1 Step 4.a: Find the largest power of 2 no greater than A. Place a 1 in row 2 in the column that contains this power of 2. Step 4.b: Subtract the power of 2 found in the previous step from A and make this value the new value of A. 2 Multiply by 2! 5

6 Step 5: Place one zero in each empty cell of row 2 in the table. Step 6: Write row 2 of the table as one number. It is the binary number representation of the input. Let see how it works on the number 108. Step 1: A = 108. The largest power of 2 no greater than 108 is 64 = 2 6. Step 2: Step 3: A becomes = 44. Step 4: Step 4.a: 32 is the largest power of 2 no greater than Step 4.b: A becomes = 12 Step 4.a: 8 is the largest power of 2 no greater than Step 4.b: A becomes 12 8 = 4 Step 4.a: 4 is the largest power of 2 no greater than Step 4.b: A becomes 4 4 = 0. Since A is equal to 0 we exit step 4. Step 5: Step 6: 108 = Check: = 108. Exercises 7. Find the binary representation of the following numbers. a) 64 b) 65 c) 43 d) The standard unit of computer memory is 8 bits, called a byte. How many nonnegative integers can be represented with 8 bits and what is the largest integer that can be represented with 8 bits(i.e. a byte)? Hexadecimal numbers Binary numbers are easy to handle for the computer because they are made up with only two digits, but they have two big disadvantages for humans: They get quickly very long. For example 256, a 3-digit numbers in our regular decimal number system is , a 9-digit number, in the binary system. 6

8 To convince us that this work, we can convert each of the two numbers to its decimal equivalent = = CC 16 = = 9420 Converting from Hexadecimal to Binary To convert a hexadecimal number to binary, we complete the reverse process. We replace each hexadecimal digit in the number by its 4-bit binary representation. For example the number A39 16 will be We usually leave the space between the 4-bit groups to make the number more legible. Caution You can apply the methods described above only between the binary and the hexadecimal number systems. To convert between the decimal number system and the binary numbers system you must apply the methods describe earlier. The conversions between binary and hexadecimal numbers are easier because the base 16 is a power of base 2. Hence the 16 hexadecimal digits are all digits that can be represented with exactly 4 bits. Such a relationship does not exist between the base 2 and 10. Exercises 9. Convert the following hexadecimal numbers to decimals. a) 3A 16 b) c) FEB Convert the following binary numbers to hexadecimals. a) b) Convert the hexadecimal numbers from question 9. to binary. Representing Graphics Representing images and graphics on the computer is much more challenging than representing nonnegative integers because images and graphics are analog. Graphics can involve an infinite number of colors and a 2D-image is continuous rather than discrete. The most common way to digitize an image is to use raster graphics. In this process, the rectangle that includes the image is divided into a grid where each cell of the grid is called a pixel. Each pixel is assigned a color. If the grid has few rows and columns then the picture would be blurry as shown on Picture 1 below. The same picture (as seen in picture 2) with more rows and columns give a sharp picture. Increasing the number of pixel improves the resolution. 8

9 Picture 1 Picture 2 The grid deals with the problem of the continuous space; remains the problem of representing an infinite number of colors. Different models can be used to encode colors. The most used and the one we will study is the RGB model. This model is based on how we perceive color. Our retinas have receptors that respond to the light frequencies corresponding to red, green and blue. Mixing those three primary colors and varying the intensity of each of the primary color create all the other colors. Black is an absence of all three colors, white is a combination of all three colors with maximum intensity, and the varying shades of gray are mixtures of all three primary color with the same intensity. In the RGB model, a color will be represented by three numbers: the first one indicates the amount of red, the second one the amount of green and the third one the amount of blue in the color. The amount for each color is measured on a scale from 0 to 255 where 0 indicates an absence of the color and 255 indicate the color present at full intensity. Number of colors represented by the RGB model You should immediately notice that the RGB model does not represent an infinite number of colors because the number 255 is finite! But if we use all 256 possible values for each of the colors we can represent = 16,777,216 colors Now where does the number 255 come from? As we have seen in exercise 8 page 6, 256 nonnegative integers can be represented with a byte: the integers 0 to 255. So reserving one byte to store each color value allows the values 0 to 255 for each color. This 24-bits representation of a color is called TrueColor. The number of bits used to express the color is called the color depth. The larger the color depth the more color can be represented. HiColor is a color depth of only 15 or 16 bits. There are now color depths greater than True Color such as 30 or 32-bit colors. An issue of size Have you ever downloaded a website and waited, waited, waited because a picture took forever to load. Do not be too quick to blame your computer or your internet connection. Let us do some computations. Let s look at picture 3 below. 9

10 Picture 3 The size of this picture is , which means the grid has 3456 columns and 2304 rows. Hence there are 7,962,624 pixels (i.e., almost 8 millions pixels). If the color for each pixel is stored as a 3-byte data, the amount of data stored is 7,962,624 3 = 23,887,872 bytes or almost 24 Mega Bytes (MB). (A megabyte is a million bytes.) This is very large: you could not send a file of that size as an attachment with Hotmail or Gmail for example. Hotmail has a 10 MB limit on message (attachment included) and Gmail has a 20 MB limit. However, the file size for the picture would be about 24 MB only if the 24- bit TrueColor information of all the pixels were actually saved. This is done in a bitmap file (extension bmp). Most pictures are saved in the JPEG format (file extension jpg and more rarely jpeg). This format applies an algorithm that compresses the data resulting in a smaller file size. The compression is said to be lossy because some data is lost in the process and cannot be recovered. Different amount of compression can be applied resulting in images of varying quality: the higher the compression, the lower the quality. The compression technique used to create a JPEG file is based on averaging colors of neighboring pixels. It works well with image with complex colors and/or subtle color variations. It does not work as well with graphics with very few colors and sharp color contrast such as in line drawings. You will learn more about graphic and pictures in the hands-on lab activities. 10

11 Back to numbers Binary arithmetic We will only talk about adding binary numbers. Good news! There are very few addition facts to learn. Here they are Let us apply the addition facts to the addition of multiple digits numbers. First without carry: Now with some carries: 1 1 Carry Note that = 11 Exercises 12. Perform the following additions. a) b) Perform the following additions. a) b) c) Representing Integers We know how to represent non-negative integers: we use their binary representation. The only limitation is that we always will limit the number of bits available for representing an integer, which in turn limit the number of positive integers that can be represented. We have seen that with n bits we can represent 2 n integers. To represent negative as well as positive integers, we must be able to represent the sign (+ or - ). Assume that we consider 8-bit integers. We can let the first bit represent the sign and the remaining 7 bits represent the number magnitude. The convention is usually to assign + to 0 and to 1. With this pattern would represent the number = 25, while would represent -25. But what would and represent? For both these numbers the magnitude would be 0, therefore both represent 0. Because having two representations for 0 in a computer causes complications, an alternative way of representing signed integers is used. Two s complements Signed integers are represented with a fixed numbers of bits, 8 (rarely), 16 or 32 most often. For simplicity we will describe the technique first with only 4 bits. As described above the first bit will 11

12 indicate the sign of the integer, 0 for a positive integer and 1 for a negative integer. In the case of a positive integer, then the 3 remaining digits will give the magnitude of the integer. The table below show how integers 0 to 7 are represented with this scheme We now have 8 possible values for negative numbers from 1000 to 1111, so we will represent numbers -1 to -8. The trick here is to restart the count at -8 for 1000 and count forward Why the name two s complement? Add the 2 s complement representations of 1 and -1: = Add the 2 s complement representations of 2 and -2: = Add the 2 s complement representations of 3 and -3: = I guess you get the pattern! If n is positive between 1 and 7, the 2 s complement representation of n is the binary number that need to be added to the binary representation of n in order to complete it to This also gives us a direct way to compute directly the 2 s complement of negative integer. Let illustrate it with -5. Step 1: find the 4-bit binary representation of the absolute value of the number. Here find the binary representation 5: it is Note: it is extremely important to include the leading zeros. Step 2: Switch all 0 s to 1 s and all 1 s to 0 s in the binary number found in step 1. Here: Step 3: Add one to the binary number found in step 2. Here: = Note that this is the number that the table gave us. Two s complement of length four bits were used because it is easy to list them all, but they do not have much practical application. In the computer signed integers are usually represented with 16, 32, or 64 12

13 bits two s complements. With 16 bits (or 2 bytes) we represent the integers -32,768 to 32,767 and with 32 bits the integers -2,147,483,648 to 2,147,483,647. Representing real numbers We have seen how to represent nonnegative integers using their binary representation, but how do we store numbers such as or 3.14 or in the computer? Before exploring this topic let us review a topic you studied in math: scientific notation. Scientific notation In the course of computations on a scientific calculator you may encounter the following results: and E+14. If you do not pay attention you may quickly round both number to if you are working to the closest thousandth. However you would be making a big mistake! The second number is E+14 = = 302,343,413,400,000. When the calculator encounters numbers that are very large or very small it writes them in scientific notation. A number written in scientific notation has three parts: the sign (+ or -), the mantissa which is a decimal number with only one digit between 1 and 9 (inclusive) to the left of the decimal point, and a exponent. In a calculator and in many computers output, the power of 10 is replaced by E followed by the exponent. The table below shows a few numbers written in scientific notation. Number Scientific Notation Scientific Notation in Calculator How to multiply by 10 n E E E E+2 If n > 0 move decimal point n places to the right. If n < 0 move decimal point n places to the left. The scientific notation is an example of a floating-point representation of real numbers. The exponent allows the decimal point to float. Take the mantissa for example. If we follow it by 10 3 then we have the number ; if we follow it by 10 6, we have the number ; and if we follow it by 10-2, we get the number Exercises 14. Write the following numbers in scientific notation. a) 231 b) c) d) Write the following numbers in their decimal form. a) b) c) d) 8.349E+2 13

14 Floating-point representation of real numbers The computer representation of real number, as the scientific notation, is a floating-point representation, but the mantissa is a whole number. A floating-point representation will have a fixed length with a fixed number of digits reserved for the mantissa and fixed number of digits reserved for the exponents. The left most digit of the mantissa must be at least 1. Let see how it works. Suppose we use a 10-digit floating point representation. The first digit is reserved for the sign; we will assume that we have 6 digits for the mantissa and 3 digits for the exponents (including its sign). Then the number 4673 would be represented as Sign Mantissa Exponent and the number 46,730,000,000 would be represented as Sign Mantissa Exponent Note that, in the computer, the floating-point representation involves only binary digits. The mantissa and the exponents are binary numbers and the exponent has to be interpreted as a power of 2. The sign are expressed with a bit where 0 represents a positive sign (+) and 1 represents the negative sign ( ). Suppose that we have a 16-bit floating-point number with one bit for the sign, 11 bits for the mantissa and 4 bits for the exponents. Then the floating point number has sign 1 which means negative sign ( ), the mantissa is and the exponent is We convert the mantissa to, then have to multiply the mantissa by 2 5 = 32. Hence the number is = 42,560. If the exponent had been 1011 instead of 0010, it would have been 5 in decimal (see 2 s complements). We would have multiplied 1330 by 2-5 and found = Exercises 16. Give the 10-digit floating- point representations with a 6-digit mantissa and 3-digit exponent of the following numbers: a) 3892 b) c) 4,231,123 d) In this problem we consider the same10-digit floating- point representation as in question 16. Consider the number a) What is its floating-point representation? b) What is the smallest number larger than that has a floating-point representation different from without loss of precision? (Hint: increase the mantissa you found in a) by 1 and convert the number back to decimal.) 14

15 c) Give a number that would have same floating-point representation as and a number that would have same floating-point representation as the number you found in b). Choose the 2 numbers so that they are between and the number found in b). Representing Sound To understand how sound is represented on the computer we first have to understand what sound is. Sounds are waves of air pressure. These waves vibrate our ear drums and the signal is transmitted to our brain. A wave can be represented by a two-dimensional curve. The simplest wave is a sinusoidal wave. Three such waves are shown in figure 2 to 4. The pressure is on the vertical axes and the time on the horizontal axes. Figure 2- Wave 1 Figure 3- Wave 2 Two major characteristics of a wave curve are the amplitude and the frequency. The amplitude is half the distance between highest and lowest point in the curve. You notice that the wave repeats itself continuously. The frequency is the number of cycles per a given unit of time, usually per second. We can see wave 1 has same frequency as wave 2 but has amplitude twice as large as wave 2. On the other hand, wave 3 has same amplitude as wave 2 but a frequency twice as large (it repeats twice as fast). Figure 4- Wave 3 15

16 Sound waves are not as simple as the ones above. They are combination of many different simple waves as above and can look very different. They are still characterized by amplitude and frequency but these amplitude and frequency change constantly. Figure 5 shows part of the sound wave generated by a brief whistle. Figure 5 The picture seems to show more an area rather than a curve because the frequency is very high. If we zoom in (notice the change in the horizontal scale), you can see clearly the wave curve. Figure 6 The amplitude of the sound wave corresponds to the volume of the sound: the larger the amplitude the louder the sound. The frequency of the sound wave corresponds to the pitch: the higher the frequency the higher the pitch. The frequency is measure in Hertz (abbreviation: Hz). One hertz is equal to 1 cycle per second. The range of human hearing is from 2 Hz to 20,000 Hz. Now that we have some understanding of sound, let see how it is represented in the computer. As for an image it is clear that sound is analog in nature; hence it needs to be digitized to be stored in the computer. This will be done via a process called sampling. At regular interval, the sound is measured and recorded. This can is illustrated on the diagram of figure 7. The dots on the curve at left represent the sample points. The data recorded is represented by the data points shown in figure 8. Figure 8 Figure 7 16

17 The sampling rate is the number of samples recorded per second. In general it is 44, 100 samples per second. It has been shown that with these many samples our ears cannot perceive the lost information. Each sample is then stored as either a 16-bit integer, sometime 24 bit, or a 32-bit floating point number. The larger size sample allow for a greater range in amplitude recorded which means being able to record very soft or very loud sounds. The device that captures the samples is called the analog-to-digital converter (ADC), which is part of the hardware of your computer. To playback the sounds, that is to recreate the sound waves from the samples the reverse Sound files - MP3 files The data collected for sound presents the same problem than images, a problem of size. This is of concern to anybody who likes to download music! Let s compute the number of bits necessary to record a 3-minute song. Assume that the sampling rate is the standard 44,100 samples per second and each sample is stored as a 16-bit integer. Assume that the song is recorded in stereo (we have to double the data because we record from two sources). Computation: 44, = 254,016,000 bits needed Even with a fast 1.5 megabits per second connection, it will take close to 3 minutes to download just this one song. As for images, instead of storing the raw data, data collected during a recording (as described above) is usually processed and stored in a given format. Different sound file formats exist (e.g., WAV, AU, MP3). The different formats store and retrieve the data in distinct ways and, for most, part of the processing includes compression. One of the most popular sound file formats is the MP3 format. It uses a compression algorithm that utilizes the characteristics of human hearing to select the sounds that it eliminates, with a minimal impact on sound quality. In addition, by selecting the number of bits per second encoded in the MP3 files (the bit rate), we can create MP3 files of different file sizes and sound quality from the same recorded sound data. For more information on the MP3 format, visit the website 17

### Data Storage 3.1. Foundations of Computer Science Cengage Learning

3 Data Storage 3.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List five different data types used in a computer. Describe how

### Data Storage. Chapter 3. Objectives. 3-1 Data Types. Data Inside the Computer. After studying this chapter, students should be able to:

Chapter 3 Data Storage Objectives After studying this chapter, students should be able to: List five different data types used in a computer. Describe how integers are stored in a computer. Describe how

### Bits, bytes, and representation of information

Bits, bytes, and representation of information digital representation means that everything is represented by numbers only the usual sequence: something (sound, pictures, text, instructions,...) is converted

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### The string of digits 101101 in the binary number system represents the quantity

Data Representation Section 3.1 Data Types Registers contain either data or control information Control information is a bit or group of bits used to specify the sequence of command signals needed for

### Computers. Hardware. The Central Processing Unit (CPU) CMPT 125: Lecture 1: Understanding the Computer

Computers CMPT 125: Lecture 1: Understanding the Computer Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 3, 2009 A computer performs 2 basic functions: 1.

### Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals. Introduction

Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals Modified from the lecture slides of Lami Kaya (LKaya@ieee.org) for use CECS 474, Fall 2008. 2009 Pearson Education Inc., Upper

### CHAPTER TWO. 2.1 Unsigned Binary Counting. Numbering Systems

CHAPTER TWO Numbering Systems Chapter one discussed how computers remember numbers using transistors, tiny devices that act like switches with only two positions, on or off. A single transistor, therefore,

### The Essentials of Computer Organization and Architecture. Linda Null and Julia Lobur Jones and Bartlett Publishers, 2003

The Essentials of Computer Organization and Architecture Linda Null and Julia Lobur Jones and Bartlett Publishers, 2003 Chapter 2 Instructor's Manual Chapter Objectives Chapter 2, Data Representation,

### Binary Numbers. Bob Brown Information Technology Department Southern Polytechnic State University

Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University Positional Number Systems The idea of number is a mathematical abstraction. To use numbers, we must represent

### Fundamentals Series Analog vs. Digital. Polycom, Inc. All rights reserved.

Fundamentals Series Analog vs. Digital Polycom, Inc. All rights reserved. Fundamentals Series Signals H.323 Analog vs. Digital SIP Defining Quality Standards Network Communication I Network Communication

### Data Communications Prof. Ajit Pal Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Lecture # 03 Data and Signal

(Refer Slide Time: 00:01:23) Data Communications Prof. Ajit Pal Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Lecture # 03 Data and Signal Hello viewers welcome

### Binary math. Resources and methods for learning about these subjects (list a few here, in preparation for your research):

Binary math This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

### Binary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1

Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1

### 6 3 4 9 = 6 10 + 3 10 + 4 10 + 9 10

Lesson The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you

### Bits and Bytes. Computer Literacy Lecture 4 29/09/2008

Bits and Bytes Computer Literacy Lecture 4 29/09/2008 Lecture Overview Lecture Topics How computers encode information How to quantify information and memory How to represent and communicate binary data

### Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 INTRODUCTION TO DIGITAL LOGIC

Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 1 Number Systems Representation Positive radix, positional number systems A number with radix r is represented by a string of digits: A n

### COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

### Base Conversion written by Cathy Saxton

Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,

### TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

### Computer Science 281 Binary and Hexadecimal Review

Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two

### Trigonometric functions and sound

Trigonometric functions and sound The sounds we hear are caused by vibrations that send pressure waves through the air. Our ears respond to these pressure waves and signal the brain about their amplitude

### Decimal Numbers: Base 10 Integer Numbers & Arithmetic

Decimal Numbers: Base 10 Integer Numbers & Arithmetic Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 )+(1x10 0 ) Ward 1 Ward 2 Numbers: positional notation Number

### 4.3 Analog-to-Digital Conversion

4.3 Analog-to-Digital Conversion overview including timing considerations block diagram of a device using a DAC and comparator example of a digitized spectrum number of data points required to describe

### Analog Representations of Sound

Analog Representations of Sound Magnified phonograph grooves, viewed from above: The shape of the grooves encodes the continuously varying audio signal. Analog to Digital Recording Chain ADC Microphone

### Section 1.4 Place Value Systems of Numeration in Other Bases

Section.4 Place Value Systems of Numeration in Other Bases Other Bases The Hindu-Arabic system that is used in most of the world today is a positional value system with a base of ten. The simplest reason

### Chapter 03. After first exam

Chapter 03 After first exam Representing Real Numbers In general, we use sign-mantissa-exponent R = mantissa * 10 exp R = mantissa * 2 exp Depending on the form of the mantissa, we have: Floating-point

### Binary Representation. Number Systems. Base 10, Base 2, Base 16. Positional Notation. Conversion of Any Base to Decimal.

Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must be able to handle more than just values for real world problems

### LSN 2 Number Systems. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology

LSN 2 Number Systems Department of Engineering Technology LSN 2 Decimal Number System Decimal number system has 10 digits (0-9) Base 10 weighting system... 10 5 10 4 10 3 10 2 10 1 10 0. 10-1 10-2 10-3

### 2011, The McGraw-Hill Companies, Inc. Chapter 3

Chapter 3 3.1 Decimal System The radix or base of a number system determines the total number of different symbols or digits used by that system. The decimal system has a base of 10 with the digits 0 through

### To convert an arbitrary power of 2 into its English equivalent, remember the rules of exponential arithmetic:

Binary Numbers In computer science we deal almost exclusively with binary numbers. it will be very helpful to memorize some binary constants and their decimal and English equivalents. By English equivalents

### Data Representation. Data Representation, Storage, and Retrieval. Data Representation. Data Representation. Data Representation. Data Representation

, Storage, and Retrieval ULM/HHIM Summer Program Project 3, Day 3, Part 3 Digital computers convert the data they process into a digital value. Text Audio Images/Graphics Video Digitizing 00000000... 6/8/20

### Maths Module 2. Working with Fractions. This module covers fraction concepts such as:

Maths Module Working with Fractions This module covers fraction concepts such as: identifying different types of fractions converting fractions addition and subtraction of fractions multiplication and

### > 2. Error and Computer Arithmetic

> 2. Error and Computer Arithmetic Numerical analysis is concerned with how to solve a problem numerically, i.e., how to develop a sequence of numerical calculations to get a satisfactory answer. Part

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### CHAPTER 3: DIGITAL IMAGING IN DIAGNOSTIC RADIOLOGY. 3.1 Basic Concepts of Digital Imaging

Physics of Medical X-Ray Imaging (1) Chapter 3 CHAPTER 3: DIGITAL IMAGING IN DIAGNOSTIC RADIOLOGY 3.1 Basic Concepts of Digital Imaging Unlike conventional radiography that generates images on film through

### Bits, Bytes, and Codes

Bits, Bytes, and Codes Telecommunications 1 Peter Mathys Black and White Image Suppose we want to draw a B&W image on a computer screen. We first subdivide the screen into small rectangles or squares called

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### 198:211 Computer Architecture

198:211 Computer Architecture Topics: Lecture 8 (W5) Fall 2012 Data representation 2.1 and 2.2 of the book Floating point 2.4 of the book 1 Computer Architecture What do computers do? Manipulate stored

### Moving Average Filters

CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average

### OA3-10 Patterns in Addition Tables

OA3-10 Patterns in Addition Tables Pages 60 63 Standards: 3.OA.D.9 Goals: Students will identify and describe various patterns in addition tables. Prior Knowledge Required: Can add two numbers within 20

### Basics of Digital Recording

Basics of Digital Recording CONVERTING SOUND INTO NUMBERS In a digital recording system, sound is stored and manipulated as a stream of discrete numbers, each number representing the air pressure at a

### Numbering Systems. InThisAppendix...

G InThisAppendix... Introduction Binary Numbering System Hexadecimal Numbering System Octal Numbering System Binary Coded Decimal (BCD) Numbering System Real (Floating Point) Numbering System BCD/Binary/Decimal/Hex/Octal

### JEDMICS C4 COMPRESSED IMAGE FILE FORMAT TECHNICAL SPECIFICATION FOR THE JOINT ENGINEERING DATA MANAGEMENT INFORMATION AND CONTROL SYSTEM (JEDMICS)

JEDMICS C4 COMPRESSED IMAGE FILE FORMAT TECHNICAL SPECIFICATION FOR THE JOINT ENGINEERING DATA MANAGEMENT INFORMATION AND CONTROL SYSTEM (JEDMICS) APRIL 2002 PREPARED BY: NORTHROP GRUMMAN INFORMATION TECHNOLOGY

### ANALOG VS. DIGITAL. CSC 8610 & 5930 Multimedia Technology. Today in class (1/30) Lecture 2 Digital Image Representation

CSC 8610 & 5930 Multimedia Technology Lecture 2 Digital Image Representation Today in class (1/30) 6:15 Recap, Reminders 6:25 Lecture Digital Image Representation 7:30 Break 7:40 Workshop 1 discussion

### 4 Operations On Data

4 Operations On Data 4.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, students should be able to: List the three categories of operations performed on

### MEMORY STORAGE CALCULATIONS. Professor Jonathan Eckstein (adapted from a document due to M. Sklar and C. Iyigun)

1/29/2007 Calculations Page 1 MEMORY STORAGE CALCULATIONS Professor Jonathan Eckstein (adapted from a document due to M. Sklar and C. Iyigun) An important issue in the construction and maintenance of information

### CHAPTER 5 Round-off errors

CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any

### Monitors and Graphic Adapters

Monitors and Graphic Adapters To the process of displaying the information a graphic adapter and monitor are involved. Graphic adapter: an element between a processor (and its I/O bus) and a monitor. They

### Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 4 CRT Display Devices

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 4 CRT Display Devices Hello everybody, and welcome back to the lecture on

### Binary Numbering Systems

Binary Numbering Systems April 1997, ver. 1 Application Note 83 Introduction Binary numbering systems are used in virtually all digital systems, including digital signal processing (DSP), networking, and

### Chapter 1: Digital Systems and Binary Numbers

Chapter 1: Digital Systems and Binary Numbers Digital age and information age Digital computers general purposes many scientific, industrial and commercial applications Digital systems telephone switching

### Grade 6 Math Circles. Binary and Beyond

Faculty of Mathematics Waterloo, Ontario N2L 3G1 The Decimal System Grade 6 Math Circles October 15/16, 2013 Binary and Beyond The cool reality is that we learn to count in only one of many possible number

### Why Vedic Mathematics?

Why Vedic Mathematics? Many Indian Secondary School students consider Mathematics a very difficult subject. Some students encounter difficulty with basic arithmetical operations. Some students feel it

### 1. Staircase Sums. Here is an example of a kind of arrangement that we'll call a "staircase". It has 4 steps and the first step is of height 2.

Arithmetic Sequences and Series 1 You will need: Lab Gear and/or graph paper. One Step at a Time 1. Staircase Sums Here is an example of a kind of arrangement that we'll call a "staircase". It has 4 steps

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### Quick Start. Welcome To Your Little Ego

Welcome To Your Little Ego 1 2 1. USB Input (digital) 2. USB Mode Indicator 3. Sample Rate LEDs 4. Filter Select Switch 5. Headphone Output (analog) 4 3 SET 5 Quick Start If you have an Apple computer,

### Unit 6 Number and Operations in Base Ten: Decimals

Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,

### Linotype-Hell. Scanned File Size. Technical Information

Technical Information Scanned File Size Linotype-Hell Once you start scanning images and reproducing them as halftones, you find out very quickly that the file sizes can be enormous. It is therefore important

### The Hexadecimal Number System and Memory Addressing

APPENDIX C The Hexadecimal Number System and Memory Addressing U nderstanding the number system and the coding system that computers use to store data and communicate with each other is fundamental to

### Everything you wanted to know about using Hexadecimal and Octal Numbers in Visual Basic 6

Everything you wanted to know about using Hexadecimal and Octal Numbers in Visual Basic 6 Number Systems No course on programming would be complete without a discussion of the Hexadecimal (Hex) number

### DIGITAL-TO-ANALOGUE AND ANALOGUE-TO-DIGITAL CONVERSION

DIGITAL-TO-ANALOGUE AND ANALOGUE-TO-DIGITAL CONVERSION Introduction The outputs from sensors and communications receivers are analogue signals that have continuously varying amplitudes. In many systems

### Lecture 2. Binary and Hexadecimal Numbers

Lecture 2 Binary and Hexadecimal Numbers Purpose: Review binary and hexadecimal number representations Convert directly from one base to another base Review addition and subtraction in binary representations

### = Chapter 1. The Binary Number System. 1.1 Why Binary?

Chapter The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base-0 system. When you

### encoding compression encryption

encoding compression encryption ASCII utf-8 utf-16 zip mpeg jpeg AES RSA diffie-hellman Expressing characters... ASCII and Unicode, conventions of how characters are expressed in bits. ASCII (7 bits) -

### The counterpart to a DAC is the ADC, which is generally a more complicated circuit. One of the most popular ADC circuit is the successive

The counterpart to a DAC is the ADC, which is generally a more complicated circuit. One of the most popular ADC circuit is the successive approximation converter. 1 2 The idea of sampling is fully covered

### Technical Information. Digital Signals. 1 bit. Part 1 Fundamentals

Technical Information Digital Signals 1 1 bit Part 1 Fundamentals t Technical Information Part 1: Fundamentals Part 2: Self-operated Regulators Part 3: Control Valves Part 4: Communication Part 5: Building

### Decimals and other fractions

Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

### Numerical Matrix Analysis

Numerical Matrix Analysis Lecture Notes #10 Conditioning and / Peter Blomgren, blomgren.peter@gmail.com Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research

Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order

### Digital Versus Analog Lesson 2 of 2

Digital Versus Analog Lesson 2 of 2 HDTV Grade Level: 9-12 Subject(s): Science, Technology Prep Time: < 10 minutes Activity Duration: 50 minutes Materials Category: General classroom National Education

### MassArt Studio Foundation: Visual Language Digital Media Cookbook, Fall 2013

INPUT OUTPUT 08 / IMAGE QUALITY & VIEWING In this section we will cover common image file formats you are likely to come across and examine image quality in terms of resolution and bit depth. We will cover

### ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-470/570: Microprocessor-Based System Design Fall 2014.

REVIEW OF NUMBER SYSTEMS Notes Unit 2 BINARY NUMBER SYSTEM In the decimal system, a decimal digit can take values from to 9. For the binary system, the counterpart of the decimal digit is the binary digit,

### 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:

Exercises 1 - number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b) - 435 2. For each of

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite

ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,

### Today s topics. Digital Computers. More on binary. Binary Digits (Bits)

Today s topics! Binary Numbers! Brookshear.-.! Slides from Prof. Marti Hearst of UC Berkeley SIMS! Upcoming! Networks Interactive Introduction to Graph Theory http://www.utm.edu/cgi-bin/caldwell/tutor/departments/math/graph/intro

### Review of Scientific Notation and Significant Figures

II-1 Scientific Notation Review of Scientific Notation and Significant Figures Frequently numbers that occur in physics and other sciences are either very large or very small. For example, the speed of

### CS101 Lecture 11: Number Systems and Binary Numbers. Aaron Stevens 14 February 2011

CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 14 February 2011 1 2 1 3!!! MATH WARNING!!! TODAY S LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS

### Digital Image Processing. Prof. P.K. Biswas. Department of Electronics & Electrical Communication Engineering

Digital Image Processing Prof. P.K. Biswas Department of Electronics & Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture - 27 Colour Image Processing II Hello, welcome

### Information, Entropy, and Coding

Chapter 8 Information, Entropy, and Coding 8. The Need for Data Compression To motivate the material in this chapter, we first consider various data sources and some estimates for the amount of data associated

### plc numbers - 13.1 Encoded values; BCD and ASCII Error detection; parity, gray code and checksums

plc numbers - 3. Topics: Number bases; binary, octal, decimal, hexadecimal Binary calculations; s compliments, addition, subtraction and Boolean operations Encoded values; BCD and ASCII Error detection;

### ANALOG VS DIGITAL. Copyright 1998, Professor John T.Gorgone

ANALOG VS DIGITAL 1 BASICS OF DATA COMMUNICATIONS Data Transport System Analog Data Digital Data The transport of data through a telecommunications network can be classified into two overall transport

### CDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012

CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Data Representation Binary Codes Why 6-3-1-1 and Excess-3? Data Representation (1/2) Each numbering

### OPERATIONS: x and. x 10 10

CONNECT: Decimals OPERATIONS: x and To be able to perform the usual operations (+,, x and ) using decimals, we need to remember what decimals are. To review this, please refer to CONNECT: Fractions Fractions

### Newton s First Law of Migration: The Gravity Model

ch04.qxd 6/1/06 3:24 PM Page 101 Activity 1: Predicting Migration with the Gravity Model 101 Name: Newton s First Law of Migration: The Gravity Model Instructor: ACTIVITY 1: PREDICTING MIGRATION WITH THE

### Signed Binary Arithmetic

Signed Binary Arithmetic In the real world of mathematics, computers must represent both positive and negative binary numbers. For example, even when dealing with positive arguments, mathematical operations

### Signaling is the way data is communicated. This type of signal used can be either analog or digital

3.1 Analog vs. Digital Signaling is the way data is communicated. This type of signal used can be either analog or digital 1 3.1 Analog vs. Digital 2 WCB/McGraw-Hill The McGraw-Hill Companies, Inc., 1998

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### CS321. Introduction to Numerical Methods

CS3 Introduction to Numerical Methods Lecture Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506-0633 August 7, 05 Number in

### V.D.U. / Monitor glossary pg. 153. Display Screen vs. Monitor. Types of Monitors. 1. Cathode Ray Tube (CRT)

V.D.U. / Monitor glossary pg. 153 A display device is an output device that conveys text, graphics, and video information to the user. Information on a display device is called a soft copy because it exists

### CSI 333 Lecture 1 Number Systems

CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...

### Systems I: Computer Organization and Architecture

Systems I: Computer Organization and Architecture Lecture 2: Number Systems and Arithmetic Number Systems - Base The number system that we use is base : 734 = + 7 + 3 + 4 = x + 7x + 3x + 4x = x 3 + 7x

### Comparison of different image compression formats. ECE 533 Project Report Paula Aguilera

Comparison of different image compression formats ECE 533 Project Report Paula Aguilera Introduction: Images are very important documents nowadays; to work with them in some applications they need to be

### Digital Design. Assoc. Prof. Dr. Berna Örs Yalçın

Digital Design Assoc. Prof. Dr. Berna Örs Yalçın Istanbul Technical University Faculty of Electrical and Electronics Engineering Office Number: 2318 E-mail: siddika.ors@itu.edu.tr Grading 1st Midterm -

### Audio Editing. Using Audacity Matthew P. Fritz, DMA Associate Professor of Music Elizabethtown College

Audio Editing Using Audacity Matthew P. Fritz, DMA Associate Professor of Music Elizabethtown College What is sound? Sounds are pressure waves of air Pressure pushes air molecules outwards in all directions

### Today. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic

Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1

### Activity 1: Bits and Bytes

ICS3U (Java): Introduction to Computer Science, Grade 11, University Preparation Activity 1: Bits and Bytes The Binary Number System Computers use electrical circuits that include many transistors and