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2.1.10

Outline the way in which data is represented in the computer.

 

Teaching Note:

To include strings, integers, characters and colours. This should include considering the space taken by data, for instance the relation between the hexadecimal representation of colours and the number of colours available.

TOK, INT Does binary represent an example of a lingua franca?

S/E, INT Comparing the number of characters needed in the Latin alphabet with those in Arabic and Asian languages to understand the need for Unicode.


 

Sample Question:

sdfsdfsf

JSR Notes:

This is actually a HUGE question to really understand it, but ultimately only one little assessment statement, so the way I'll approach it is with three "tiers":

In class, for instruction, we'll have to do these below tiers in the reverse order than they appear here, but for study (since there's so much here), if you can understand the top tier, and are able to write the same sort of answer in your own words, you're good to go; otherwise, keep on reviewing on down through the notes.

 

(Tier 1 & 2 ) The Summary of Likely IB Exam Questions & Their Answers

So these are the four most obvious questions that come verbatim from this assessment statement and teaching notes:

Q-A: "Outline the way in which data is represented in the computer" " include(ing) strings, integers, characters and colours.

Q-B: (Discuss) "the space taken by data, for instance the relation between the hexadecimal representation of colours and the number of colours available."

Q-C: "Compare the number of characters needed in the Latin alphabet with those in Arabic and Asian languages to understand the need for Unicode."

Q-D: "Does binary represent an example of a lingua franca?"

 

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Data Representation

Q-A: "Outline the way in which data is represented in the computer" " include(ing) strings, integers, characters and colours.

Answer - All data is represented on a computer in a binary way. In the RAM of a computer this binary representation is in the form of open circuits (the 1s of the binary code) and closed circuits (the 0s of the binary code). On hard drives, the binary representation is in the form of little regions that have one level of magnetism or another.

So all data, be they numeric data or color data or whatever, ultimately have to be able to be translated into a binary form.

Integers

Integers are able to be stored simply as the binary equivalent of their decimal value. So the decimal number 6, in an 8-bit integer context would be 0000 0110. But usually, integer data types are "signed", meaning that they can represent positive or negative numbers. This is achieved by having the Most Significant Bit (the MSB) being a negative value, which, the way that additional places in a number system work, will be equal to the total positive value represented by the rest of the bits.

For example, for an 8-bit signed integer 1000 0000 is -128, and 0111 1111 is +127.
or for a 16-bit signed integer, 1000 0000 0000 0000 is - 32 768, and 0111 1111 1111 1111 is +32 767.

Characters

Each specific character (for example, 'A', 'g', '!', '#') in a given character set has an equivalent binary code. In the ASCII character set, for example, 'A' is represented by the following 8-bit binary number: 0100 0001.

Note that since both characters and integers share certain binary equivalents, they can be casted (i.e. converted) into each other. So for example, ASCII character 'A' is the binary equivalent of decimal integer 65 (they are both represented by 0100 0001), so casting between them (in Java) can work as follows:

System.out.println( (char) 65) prints A.
And System.out.println( (int) 'A') prints 65.

Strings

Strings are groups of characters. You can think of them as being a "string" of characters, in the same way that your mother may have a string of pearls. So they are groups of 8-bit, or 16-bit characters strung together. This grouping is more properly referred to as an array. So, properly put, Strings are arrays of characters.

Colors

For colors, it depends on the color model, but using the common RGB model, one byte is used for each of the red (R), green (G) and blue (B) values. Red would then be 1111 1111 0000 0000 0000 0000. Since this model uses three bytes per pixel, it is referred to as 24-bit color.

 

If you were to answer the general question in one sentence:
Data is represented by binary values according to various universally agreed upon code sets, which match up particular letters/colors/sound with particular combinations of 0s and 1, and the number of things able to be represented by those sets increases exponentially with additional digits.

And if you were to draw a basic summary diagram for this assessment statement, particularly for Q1A, the following would suffice nicely:

data represented on a computer, summary

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Hex & Colors

Q-B: Discuss "the space taken by data, for instance the relation between the hexadecimal representation of colours and the number of colours available."

Tier 1 Answer: The number of bits used per character/number/color determines the number of characters/numbers/colors which can be represented. With all data, each extra bit (binary digit) which is used to represent individual characters/numbers/colors doubles the number of characters/numbers/colors available. But since computers work with groups of 8 bits usually, we tend to think of the influence of having an extra 8 bits used per thing. And so an extra 8 bits will double, and double, and double and so on the initial number of colors etc. 8 times, or increase it by a factor of 2^8 (256). So every additional 8 bits of memory used for representing something has not just double the number of possibilities, but 256 times. (For example, a 16 bit color set can represent 256 times more individual colors than 8 bit color set.)

In the example of color, we usually work with hexadecimal values, and 8 binary digits is equivalent to two hex digits, both have a maximum decimal value of 255. So when a color model goes from 8 bit color (for example FF) to 16 bit color (for example 33FF), the number of colors that can be represented goes up by 256 times. And it's the same with going from a 16 bit color set to 24 bit color set (for example AA33FF); once again an increase of 256 fold.

So that is to say, generally, with every two addition hex digits, 16^2 or 256 times more things can be represented. So the number of colors available in 8 or 16 or 24 bit color models is thereby 256^1, 256^2, and 256^3, which equals 256, and then 65,536, and then over 16 million. It's 24 bit color that you usually are working with on a phone or laptop, so your phone or laptop has the ability to represent over 16 million unique colors.

If you had to, or chose to answer in one sentence:
The space in computer memory taken by data natuarlly increases as we increase the number of things we wish to represent, but this gain is an exponential factor of the extra memory required; for example each 8 bits - or two hex digits - used to represent colors increases by 256 times the representable color set.

 

Tier 2 - Tweaked out a bit:

Hexadecimal Representation of Color

To get this, first you need to get the straight-forward relationship between binary and “hex” (hexadecimal) values representing color.

Of course, at the memory level, all colors (like all data) are represented by 0s and 1s, somehow.
But we wouldn't want to be working in Photoshop, for example, with the the following color:
       “0000 0000  1111 1111  0000 0000” (green in a 24 bit RGB color model)

Instead, a nice, short number (in hex) is much easier to work with:
       “00 FF 00” (which is the same green in a 24 bit RGB model, written in hex)
And, it's more than the fact that a hex representation is short, we can also easily read right off of the hex color value, if we know what we are doing. It's easy to see, in the example above that there's no Red, maximum Green, and no Blue. But even with non-limit values, and we can often get a good idea. Take the color 22 D1 3F, for example; that's also going to be pretty much a green shade of color (mainly look at the first digit of each pair).


The relationship between binary and hex is this:
------ one hex bit is the same as four binary bits ------

This is because in binary, four places can represent 16 total different values: 
                0000 (decimal 0) all the way up to 1111 (decimal 15)

and hex it’s just one digit which can represent exactly 16 total different values:
                0 (decimal 0) all the way to F (decimal 15)

So, yes, every four binary digits is indeed equivalent to 1 hex digit.


Examples:  (using a 24 bit color model)


                        R           G           B

decimal green:        0          255          0
binary green:     0000 0000   1111 1111   0000 0000
hex green:          0    0      F    F      0    0

decimal red:         255          0           0
binary red:       1111 1111   0000 0000   0000 0000
hex red:            F    F      0    0      0    0

decimal sky blue:     0           51         204
binary sky blue:  0000 0000   0011 0011   1100 1100
hex sky blue:       0    0      3    3      C    C 

So you can see that using the hex values is just easier.  Green is 00FF00, Red is FF0000, Yellow is FFFF00, and Sky Blue is 0033CC, all fairly easy values to write and even to remember.


Finally then, in this "Tier 2" explanation, let's address, once again, the assessment statement teaching note point what is "the relationship between the hexadecimal representation of colours and the number of colours available"?

First of all remember that color models generally don’t go up by one hex digit, they go up by two. And that is because the standard for working with data is the byte and the byte is usually 8 bits, which equals 2 hex digits. And so...

For every two hex digits added to the color model, there are 256 times more colors available.

Here are the three most common color models:

8-bit color (i.e. two hex values color, for example F4) can represent 256 different colors.
16-bit color (i.e. four hex values color ,for example 55DD) can represent 65, 536 (which is 256 x 256)
24-bit color (i.e. six hex values color, for example CC99CC) can represent 16,777,216 (which is 65,536 x 256)


RGB and CMYK models demonstrated


Hex values of various "pure" RGB colors


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Languages & Unicode

Q-C: "Compare the number of characters needed in the Latin alphabet with those in Arabic and Asian languages to understand the need for Unicode."

Tier 1 Answer: The number of characters needed in the Latin alphabet (the one English, for example, is based on) is way smaller than those needed for Arabic and Asian languages. We only need the basic A-Z characters 0-1 numbers and common punctuation symbols to be able to store Latin characters as with the English language. Arabic and Asian languages and other world languages would need thousands and thousands of characters to be represented; think about how Chinese dialects have tens of thousands of pictograms in their language.

Remembering that computers are made to work primarily with groups of 8 bits (a byte), one group of 8 bits would be enough for Latin character languages such as English. The number of characters which could be represented by 8 bits is 2^8, or 256. ASCII, the original computer character set now uses exactly that, so it represents the most common 256 Latin characters and symbols. To go up into the thousands of available representations, it makes sense to stick with the 8 bit groupings model, and so with 16 bits, 2^16 (65, 536) different characters can be represented. And this is exactly what the standard UNICODE code set contains.


Unicode logo.jpg, Public Domain

If you had to, or chose to answer in one sentence:
The number of characters needed in the Latin alphabet which we use in English is limited, and fits well within the 256 characters permitted by ASCII, but Arabic, and Asian langauges together require much more than 256 characters, so doubling the space taken in memory to by each character to 16 bits allows 256 x 256 characters to be represented with UNICODE.

 

Tier 2 - Tweaked out a bit:

Latin vs. Other Languages & UNICODE

Basic ASCII (The American Standard Code for Information Interchange) initially used a 7 bit set of characters to represent letters, and in various extended ASCII sets, a full 8-bit byte is used. That means that for the basic ASCII, 2^7 characters - i.e. 128 - can be represented, and in the extended ASCII sets, 2 ^ 8, or 256 characters can be represented.

(The reason, by the way, that 7, not 8 bits were used initially is that in the original ASCII the 8th bit of the standard 8 bit byte was used for error checking during data transmission - get me to explain how the error checking worked in class sometime; there was a "check-bit".)

The Latin alphabet that we use in English has 26 lower case letters, and 26 upper case letters. So those, along with the 10 numeric digit characters (0 - 9), and 20 or so common symbols of the keyboard (!@#$%^&*( ) ;':"[]{},.< >/?\|) can all be represented within those 128 combinations of 0s and 1s of basic ASCII. Below is a picture of the first 128 characters of ASCII.

ascii table

But for "Arabic and Asian languages", there's not enough room in a 128 set, or even a full 256 full ASCII extended set to fit all the characters. There are tens of thousands of Chinese characters, for example.

And there you have it, in terms of the teaching note "comparing the number of characters needed..." "Arabic and Asian", and indeed a whole bunch of other kinds of languages together demand more than 256 characters to be represented. So we now have computers that work with UNICODE, which can represent thousands of characters. UNICODE uses 16 bits per character. And in so doing, computers can still work with the basic 8-bit byte. It's just that working in UNICODE, two bytes (16 bits) are read for each character. And in using 16 bits, the calculation for the number of different combinations of 0s and 1s is 2^16, or 65,536.

Try to go on-line and find a UNICODE explorer kind of website or application which allows you to view different categories and characters of UNICODE.

http://unicode.mayastudios.com/ - a good one for work with Java code

The fundamental reason why so many more characters can be represented is the same as when going from 8-bit color to 16-bit color.

With every extra bit you add, you double the number of possible combinations of 0s and 1s, and therefore the number of possible things you can represent, whether those things be colors, or in this case letters).

The math of this is: 2^numberOfBits.

2^8 = 256

2^16 = 65,536

So in extended ASCII, the 0s and 1s combinations go from

0000 0000
0000 0001
0000 0010
0000 0011
.
.
.
to
1111 1100
1111 1101
1111 1110
1111 1111

256 combinations in all.

 

And in UNICODE, the 0s and 1s combinations go from

0000 0000 0000 0000
0000 0000 0000 0001
0000 0000 0000 0010
0000 0000 0000 0011
.
.
.
to
1111 1111 1111 1100
1111 1111 1111 1101
1111 1111 1111 1110
1111 1111 1111 1111

65,536 combinations in all.

 

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Binary a Lingua Franca?

Q-D: "Does binary represent an example of a lingua franca?"

Answer: A lingua franca is a language which most people around the world can understand. So, yes, binary, in a way does represent a lingua fraca, in so far as the computers used by most people around the world are able to interpret the character codes sets ASCII and UNICODE using those 0s and 1.

Though, this being a TOK point, how about taking it further, or rather one level up, and claim that it is UNICODE which is the lingua franca. With UNICODE, most of the people of the world can encode their visual way of communicating.

 

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(Tier 3 for all of the above) The Actual Theory Behind All of This

- some of this will help you understand the answers above - look through to find what applies.

THIS IS NOT NECESSARY, BUT IS PLACED HERE AS BOTH A POSSIBLE AIDE, AND FOR ENRICHMENT

------------ OPTIONAL ------------

And for a review of all of this, you can't beat the appropriate Crash Course video.

 

(Jaime: Really good video from CMS years of the South Asian professor talking about the String pool on YouTube.... the point being that this is still one layer of abstraction at least higher than the way strings actually work, as arrays of chars, but each char is in the shared String pool.)

One last point that naturally quite often comes up is: "how does the computer know to interpret the 0s and 1s as characters, or numbers or colors etc.?"

The answer is simply that in the header of each file, the context of those 0s and 1s is stated. Something along the lines of "what is to follow is text from the ANSI extended UNICODE character set". You can see this easily with a TextEdit document if you check the box "Ignor Rich Text Commands" when you open it. The simplest text document with "hello world" as the only words looks like this, and it is the first line which states that what is following is to be interpreted as text:


{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
{\colortbl;\red255\green255\blue255;}
\paperw11900\paperh16840\margl1440\margr1440\vieww18180\viewh15500\viewkind1
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural

\f0\fs24 \cf0 hello world}