TEJ3M

Number Systems

Introduction

Welcome to TEJ3M - Computer Engineering Technology. Computer Engineering is a discipline that integrates several fields of electrical engineering and computer science required to develop computer systems. Computer engineers usually have training in electronic engineering, software design, and hardware-software integration instead of only software engineering or electronic engineering. Computer engineers are involved in many hardware and software aspects of computing, from the design of individual microprocessors, personal computers, and supercomputers, to circuit design. This field of engineering not only focuses on how computer systems themselves work, but also how they integrate into the larger picture.

Source: Wikipedia

Computer Lab Rules

You are responsible for knowing these rules. The over-riding principle is that computers are to be used strictly for educational purposes only.

You are not allowed to eat or drink (water may be left at desk) near the computers
Do not share your passwords with others. Do not let anyone else use your account!
You are not allowed to play games, apart from games that are part of your course work. No commercially prepared games can be brought into the lab or installed please.
You are not allowed to have streaming video. (too much bandwidth).
Do not bring bags and coats into the lab.
Do not alter things like the desktop/system files without permission

Assignment #1 - Tech Website

To get used to the computers and our computer network, you are to complete simple assignment whereby you will create a Google Site that you will use to submit some assignments in the course (you can use yours from last year and just add a new division of pages for grade 11 if you want). You will create the main Site that will be about you (include some details about yourself, some pictures and/or embedded content). Off of that main page include several sub-pages that will be titled with each unit of the course (number systems, computer systems, networking, electronics, robotics). On the number systems page I want you to create an entry called Assignment #1. Answer the following questions in that section:

What are elements of computer technology you feel are important
What don't you like about computer technology as a whole
What is the thing you like most about computer technology

The goal of this assignment is mostly to ensure you have remembered the Appropriate use of Technology Policy of the OCDSB, as well as the discussion about our school network and submission protocol. You will submit the published link to the Google Site in the Classroom Assignment. If you're looking for resources beyond what's described in class check out Google's information page(s).

The content for the assignment is:

a) Introduce yourself and your battlestation (your computer workstation - what type of computer is it? What are its specs?)

b) Why are you taking this course? (Is there something specific that you want to learn?)

c) What is one app or a piece of software or hardware that you used this summer and why did you like it?

d) What do you think is the most significant date in computer history? What was developed and why do you think it's important?

History of Computers

Number Systems

A numeral system (or system of numeration) is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. (source)

At first, it would seem like using any number system other than decimal is complicated and unnecessary. However, since the job of electrical and software engineers is to work with digital circuits, engineers require number systems that can best transfer information between the human world and the digital circuit world and those forms are usually binary, octal, or hexadecimal. It turns out that the way in which a number is represented can make it easier for the engineer to perceive the meaning of the number as it applies to a digital circuit. In other words, the appropriate number system can actually make things less complicated.

Number Conversions

While useful in their own right, often numbers are more meaningful in another number system. For example, an electrical engineer who is working on a hexadecimal "programming layer", may want to translate something into a binary "machine layer", as such, they need to be able to convert quickly between the base 16 number system (HEX) to the base 2 number system (BIN, or 0b).

The three systems of interest for us are decimal (dec), binary (bin) and hexadecimal (hex).

-Computers use what number system? Binary (base 2)

- 0 OFF (false)

- 1 ON (true)

- 1 binary digit - 1 bit

- 8 binary digits - 1 byte

- 4 binary digits - 1 nybble

-Hexadecimal number system is for humans and computers (eg. RGB colour codes)

Converting between Decimal to Binary

To convert between Decimal and Binary you can either arrange the columns by their respective placeholders, or use the remainder method of conversion

Consider the number 294. The steps below show how to convert this number to binary using repeated division by 2. The 'R' stands for the remainder of the division

Decimal to Binary Divide-by-2 method:

An easy method of converting decimal to binary number equivalents is to write down the decimal number and to continually divide-by-2 (two) to give a result and a remainder of either a “1” or a “0” until the final result equals zero. If there's a non-perfect division (e.g. the remainder is 0.5) then the remainder becomes 1 and the number rounds down.

NOTE: THE FIRST DIVISION IS ON THE RIGHT, THE LAST DIVISION IS ON THE LEFT

- e.g. the number 294
  - divided by 2
- is evenly 147 with a remainder of 0
  - divided by 2
- is 73.5->73 (there's a decimal so we round down) and the remainder is 1
  - divided by 2
- is 36.5->36 with a remainder of 1 (rounded down and the # becomes 36 with the R =1)
  - divided by 2
- is evenly18 with a remainder of 0
  - divided by 2
- is evenly 9 with a remainder of 0
  - divided by 2
- is 4.5->4 with a remainder of 1
  - divided by 2
- is evenly 2 with a remainder of 0
  - divided by 2
- is evenly 1 with a remainder of 0
  - divided by 2
- is 0.5->0 with a remainder of 1

So the number's : 1 0010 0110

Subtraction method

You could also just do raw-figuring using the subtraction method. Let's take the number 114

114 isn't bigger than 128 so there are no 128's
114 IS bigger than 64 so it contributes 1 and we subtract 64 from 114 leaving 50
50 is bigger than 32 so we take 32 away from 50 leaving 18
18 is bigger than 16 so we take away 16 from 32 leaving 2
2 isn't bigger than 8 so it contributes 0
2 isn't bigger than 4 so it contributes 0
2 is subtracted at the 2's column leaving 0
there's nothing left to subtract in the 1's column so it's 0

So 114 is: 01110010 in 0b (but we generally don't show leading 0's so 1110010)

Converting Binary to Decimal

To convert binary to decimal is relatively easy. All you have to do is list out the columns, then add up representative placeholders. Let's say for example we want to convert 101101(0b) to dec:

32 + 8 + 4 + 1 = 45

Converting Decimal to Hex

You can also use the dividing approach as with binary, but with alterations. Here are the steps:

Step 1: Divide the number by 16. Store the whole number for the next step. Take the decimal remainder (if there is one) and multiply by 16. Store it.

Step 2: Take the whole number from the last step and divide by 16 as per the last step. Repeat these 2 steps until the last whole number is 0. This will be the last step

Step 3: The final hexadecimal value will be the reverse order you stored (like in dec to bin conversions)

EG Decimal number = 450(10)

Step 1: 450/16 gives 28.125, Store 28 for the next step. 16x0.125=2, so the remainder is = 2

Step 2: 28/16 gives 1.75; Store 1 for the next step. Multiply 0.75x16, the remainder of 12 = C

Step 3: 1/16 gives 0; 16x0 and remainder = 1

Step 4: 0/16 gives 0 and remainder = 0

Therefore, the hexadecimal value is 01C2(16)

EG Decimal number = 118(10)

Step 1: 118/16 gives 7.375, Store 7 for the next step. Multiply 16x0.375 so the remainder = 6

Step 2: 7/16 gives 0.4375; 0.4375 x 16 = 7 Store 0 for the next step. remainder of 7

Step 3: 0/16 gives 0 and remainder = 0

Therefore, the hexadecimal value is 076(16)

173/16 = 10.8125 and we'd take the 0.8125x16 = 13

Notice with 10 I SHOULD have that go to the next step which is

10/16 = 0.625, but since we'd just multiply it back by 16 to get the remainder (10) it's simpler to remember: If the whole number of the decimal that's left is <16, then that is the remainder for the next step.
e.g. in step 1 above:
173/16 = 10.8125 and we'd take the 0.8125x16 = 13

since 10 is <16 it's the next remainder. so the remainder order is:
13 = D
10 = A
So it's AD(16)

Converting Hex to Decimal

Is exactly like converting binary to decimal, simply list out the columns and add the results.

Let's take the number 1D16

16's column | 1's column

(1 x 16) + (13 x 1) = 2910

How about this one A6316

256's column | 16's column | 1's column

(10x256) + (6x16) + (3x1) = 265910

Try this:

C0216 to base 10 (equals 3074 )

Converting Binary to Hex and Hex to Binary

This is the easiest type of conversion because we break the columns in hex down into 'octets' (also known as a byte).

HEX to BIN

To convert we simply break the Hex into two separate column, then assign the binary values to the octet. To convert hexadecimal F8 to binary, write down the binary for F first, then the binary for 8.

So, the answer is 1111 1000

This seems too easy, and it is. Use a calculator to convince yourself. Convert hex number 1A to binary.

Answer: 0001 1010

BIN to HEX

Is the exact same as HEX to BIN, but in reverse. To convert, we break the binary value into octects, then convert each to hex separately. Let's convert 10110 to HEX. First break it into octects and include leading zeroes on the octet to the left

As you can see it's 16

Step 1:
0001 0110

Step 2 (convert each octet to hex)

1 6

What about the binary number 1101101

0110 1101

6 13 (which is D)

So the HEX value is 6D. Pretty easy!

Assignment #2 - Number Conversions

Go to the following website, or watch the youtube video in Classroom or refer to our notes. Answer the questions from the google doc worksheet and when you're finished post it on your Google Site as Assignment #2 on the Number Systems webpage. Double check your answers using the programmer-menu on windows calculator.

1. Convert these binary numbers to decimal:

a. 11110000

b. 10010010

2. Convert the following decimal values to binary:

a. 38

b. 89

3. Convert the following hex values to decimal:

a. 10

b. AC

4. Convert the following decimal values to hex:

a. 121

b. 1F3

5. Convert the following hex values to binary:

a. 10

b. AF

6. Convert the following binary values to hex:

a. 10010010

b. 01101101

7. Convert the following numbers with the indicated bases into decimal:

a. 1105

b. 1313

8. Why do we use lowercase suffixes to denote what number system we're working in?

CONVERSION QUIZ REVIEW

<A5>

Be able to convert from

ASCII

ASCII (American Standard Code for Information Interchange) is the most common character encoding format for text data in computers and on the internet. In standard ASCII-encoded data, there are unique values for 128 alphabetic, numeric or special additional characters and control codes.

Source

ASCII encoding is based on character encoding used for telegraph data.

Characters in ASCII encoding include upper- and lowercase letters A through Z, numerals 0 through 9 and basic punctuation symbols. It also uses some non-printing control characters that were originally intended for use with teletype printing terminals when the system was developed in 1963

As far as we're concerned, ASCII characters may be represented in the following ways. Let's take the letter Z (capital):

as pairs of hexadecimal digits (eg 5A)

as decimal numbers from 0 to 127; (eg 90) or

as binary (101 1010)
In a computer the word "cat" is 0110 0011(2) , 0110 0001(2) , and 0111 0100(2) .

Activity

Open NOTEPAD. Hold down the ALT key and enter the ascii decimal code. E.g. ALT+54

What character is brought up?

try using this chart to try a few more. This is an extended table for even MORE.
If the War of the Worlds is roughly 58,000 words, how many binary entries is that? What do we have to assume?

This website can provide more details about ASCII

Google Sites

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