IB/Group 4/Computer Science/Computer Organisation/The Information Layer

What is a base in a numeral system ?

A base, or a radix, is the amount of unique digits that is used to represent numbers in a numeral system. Our current system, the decimal system is base ten, as we have 0-9, or 10 unique digits.

For computer science, base 2, or the binary system, is particularly important in understanding computer processing as it is how boolean logic gates are represented numerically, with 0 being false and 1 being true. This means that at its core, a computer's code is only comprised of a long series of 1s and 0s. However, with a smaller number of digits, numbers need a greater quantity of digits to be represented. (For example, 6 in binary is 0110.) As a result, computers use alternate numerical systems to represent numbers with a smaller quantity of digits, such as octal(base 8) and hexidecimal, (base 16) with bases that are to the power of 2 as it is easier to convert into binary.

What is positional notation ?

Positional notation is a type of notation used to represent a positional system (aka a place value system), such as numbers in our modern day decimal system. Simply put, in a positional system, the value of a digit in a number is determined by its position. For example, in the number 343, the first number 3 represents 300, whereas the last 3 represents only 3. To express this in a more formulaic manner, every digit in a number is multiplied by 10ⁿ, where n represents the position of the number. This is method is demonstrated in the example of 943 below:

	$9$	$\times$	$10^{2}$	$=$	$9$	$\times$	$100$	$=$	$900$
$+$	$4$	$\times$	$10^{1}$	$=$	$4$	$\times$	$10$	$=$	$40$
$+$	$3$	$\times$	$10^{0}$	$=$	$3$	$\times$	$1$	$=$	$3$
									$943$

However, this method is only applicable to our base 10 number system. To express this idea formally, the number in the base-$R$ number system has $n$ digits is represented as follows, where $d_{i}$ represents the digit in the $i$th position in the number:

$d_{n}\times R^{n-1}+d_{n-1}\times R^{n-2}+\ldots +d_{2}\times R+d_{1}$

While this may seem complicated, it is easy to apply, such as for the number 63578 in base 10. Here $n$ is 5, as the number has 5 digits, and $R$ is 10.

$6\times 10^{4}+3\times 10^{3}+5\times 10^{2}+7\times 10^{1}+8$

This formula can also be applied to other a different base system, such a base 3 or base 12. For example, if we wished to write 943 in a base 13 system:

	$9$	$\times$	$13^{2}$	$=$	$9$	$\times$	$169$	$=$	$1521$
$+$	$4$	$\times$	$13^{1}$	$=$	$4$	$\times$	$13$	$=$	$52$
$+$	$3$	$\times$	$13^{0}$	$=$	$3$	$\times$	$1$	$=$	$3$
									$1576_{10}$

In this way, positional notation can help us represent and understand numbers in other base systems.

Binary

What is binary ?

The term binary refers to any encoding in which there are two possible values. In computer science they refer to the electronic values 0 and 1 (see figure below). Booleans values are binary. In computers we use binary instead of the decimal system as it offers a much simpler and efficient way to perform calculations through electronics as opposed to a decimal system. This can be explored in the following sections regarding binary gates.

Image: 150 pixels

Interpretation of voltages to binary by the computer. In a system where the max voltage is 5 Volts, then the input current in an electrical wire is interpreted as a logical 0 by a transistor if the voltage is between 0 volts and 0.8 volts. On the other hand, if the voltage is between 2 volts and 5 volts, then the transistor interprets the input signal as a logical 1 (right diagram). It is important to note that there is a non-usable area of voltage that does not correspond to either a logical 0 or 1. This non-usable area is particularly useful since voltage is always slightly fluctuating. Having this grey zone allows us to better identify electrical signals (0 and 1) from each other. Similar voltage conversions from volts to binary digits also exist for output signals from transistors.¹

The base of a number system specifies the number of digits used in the given system. The digits always begin with 0, and then continue through one less than the base. For example, there are 2 digits in base 2: 0 and 1. There are 8 digits in base 8: 0 through to 7. There are 10 digits in base 10: 0 through 9. The base also determines what the positions of digits mean. When one adds 1 to the last digit in the number system, one has to carry the digit position to the left.

What is a bit ?

A bit is the smallest unit of data that a computer can process and store. It can either store a 1 or a 0 and represents on/off, true/false, yes/no.

What is a byte ?

1 byte is a unit of data that consists of/stores 8 bits. 1 byte represents 2⁸ distinct values.

How do I convert a decimal number to a binary number ?

If we want to convert a decimal number to a binary number, we will also use our previous conversion table.

For example let's convert decimal number 56:

• First we look at the biggest binary power that can entirely fit in 56. This is 32 = 2⁵. It is not 64 because 56 does not fit entirely into 64.
• Then we take the remainder of 56 - 32 = 24. We check which is the biggest binary power that can entirely fit into 24. This is 16 = 2⁴.
• Then we take the remainder of 24 - 16 = 8. We check which is the biggest binary power that can entirely fit into 24. This is 8 = 2³.
• Finally 8 - 8 = 0 so we have reached the end.

Now we mark the powers of two that we used for our calculus as a binary 1 and the other powers as 0. We get 00111000

decimal to binary conversion table

	27	26	25	24	23	22	21	20
=	128	64	32	16	8	4	2	1
=	0	0	1	1	1	0	0	0

How do I convert a binary number to a decimal number ?

To convert a binary number into decimal, we just need to sum all powers of 2 that have a 1 in the binary digit.

First off, we must count the number of digits in the Binary Number and correlate them with the powers of 2, starting from $2^{0}$ up to $2^{8}$.

For each digit in binary we must take multiply the number by the corresponding exponent of 2. For example:

Take the binary number 1011. Following the previous rules concerning positional numbers, we achieve the following table:

	$1$	$\times$	$2^{3}$	$=$	$1$	$\times$	$8$	$=$	$8$
$+$	$0$	$\times$	$2^{2}$	$=$	$0$	$\times$	$4$	$=$	$0$
$+$	$1$	$\times$	$2^{1}$	$=$	$1$	$\times$	$2$	$=$	$2$
$+$	$1$	$\times$	$2^{0}$	$=$	$1$	$\times$	$1$	$=$	$1$
									$11$

Now following the same logic, but written differently, if we convert the following byte into decimal: 10011100

binary to decimal conversion table

	1	0	0	1	1	1	0	0
=	27	26	25	24	23	22	21	20
=	128	64	32	16	8	4	2	1
=	128			16	8	4

We only add the columns where the binary digit is equal to 1.

Finally, we add each number up. this is then the number that is converted from binary.

So 10011100 = 128 + 16 + 8 + 4 = 156

How many possible numbers can I store in a byte ?

A byte is 8 bits, which means that the amount of values that can be stored in a byte of data is 2⁸ = 256.

How many possible numbers can I store in a an n-bit number ?

Since there are n possible bits that each can take two possible values (1 or 0), we can store $2^{n}$ possible numbers in a n-bit number.

In a byte this means we can store 2⁸ = 256 different numbers.

What is the biggest number I can get in a an n-bit number ?

Since there are n possible bits that each can take two possible values (1 or 0), the biggest number that we can get is 2ⁿ -1. Indeed the smallest number that we can get is a 0 hence the -1 in the formula.

In a byte this means the biggest number we can get is 2⁸ -1 = 255, being 11111111, and the smallest number is 0 being 0000000.

Exercises - Binary representation of numbers

1. What is the decimal representation of 1001 ?

2³ + 2⁰ = 8 + 1 = 9

2. What is the decimal representation of 01000101 ?

2⁶ + 2²+ 2⁰ = 64 + 4 + 1 = 69

3. What is the binary representation of 42 ?

42 - 32 = 10 with 32 = 2⁵

10 - 8 = 2 with 8 = 2³

2 - 2 = 0 with 2 = 2¹

Hence the binary representation: 00101010

4. What is the binary representation of 129 ?

129 - 128 = 1 with 128 = 2⁷

1 - 1 = 0 with 1 = 2⁰Hence the binary representation: 01000001

5. How many different integers can I represent in 2 bytes ?

2 bytes are 16 bits (2 x 8bits), hence 2¹⁶ = 65536 different numbers can be represented in 2 bytes

6. What is the biggest number I can represent in 2 bytes ?

2 bytes are 16 bits (2 x 8bits), hence 2¹⁶ -1 = 65535 is the biggest number that can be represented in 2 bytes.

Challenge: Construct a method to convert denary numbers to binary numbers. It is recommended to create an algorithm similar to how we convert the numbers, then make sure the program follows the algorithm.

Remember: you can add a character to the end of the string by simply adding them (string+char). You can also use the static method pow(a, b) in the Math class.

Follow my algorithm (a short clue for you) if you find it hard to make one out:

1. Find the max exponent of 2 depending on the input (a while loop can make your understanding clearer rather than recursion)
2. Go through all the exponents, fit in the value 1 or 0.

Sample Solution

Below is a sample Java solution to convert a denary number to its binary representation:

public class Main {
   public static void main(String[] args) {
       System.out.println(convert(23));
   }

   public static String convert(int numberToConvert) {
       String binaryResult = "";
       int remainingValue = numberToConvert;

       int maxExponent = 0;
       while (remainingValue >= Math.pow(2, maxExponent + 1)) {
           maxExponent++;
       }

       System.out.println(maxExponent);

       int currentPowerValue;

       while (maxExponent >= 0) {
           currentPowerValue = (int) Math.pow(2, maxExponent);
           if (remainingValue < currentPowerValue) {
               binaryResult += "0";
           }
           if (remainingValue >= currentPowerValue) {
               binaryResult += "1";
               remainingValue -= currentPowerValue;
           }
           maxExponent--;
       }

       return binaryResult;
   }
}

Hexadecimal

What is Hexadecimal ?

How are digits represented in bases higher than 10? In bases higher than 10, the digits above the integer 9 will be represented as symbols. All digits are written in the following conversion table.

decimal	hexadecimal	binary
0	0	0000
1	1	0001
2	2	0010
3	3	0011
4	4	0100
5	5	0101
6	6	0110
7	7	0111
8	8	1000
9	9	1001
10	A	1010
11	B	1011
12	C	1100
13	D	1101
14	E	1110
15	F	1111

Hexadecimal is a base 16 numbering system which uses 16 possible digits. Its primary attraction is its ability to represent very high numbers with fewer digits than in the binary or decimal system.

How do I convert a decimal number to a hexadecimal number ?

The method is as following:

• Apply floor division to the decimal number by 16. Write down the remainder starting from the left side.
• Continue applying floor division by16 to the previous result of the floor division until you reach 0. Keep writing down the remainders.
• Convert the remainders, from last to first, to hexadecimal using the table to the right.
• You have your hexadecimal number !

For example, if we want to convert the decimal number 2545 to hexadecimal:

• We apply floor division: 2545 // 16 = 159. The remainder of that division is 2545 % 16 = 1
• We apply floor division to our previous result: 159 // 16 = 9. The remainder of that division is 159 % 16 = 15
• We apply floor division to our previous result: 9 // 16 = 0. The remainder of that division is 9 % 16 = 9.
• We reached 0, so we stop dividing.
• We convert our remainders: 9 → 9, 15 → F, 1 → 1
• Therefore our hexadecimal number is: 9F1.

Exercises - Hexadecimal representation of denary

1. What is the denary representation of 21 ?

00100001 = 21(hex)

33

2. What is the denary representation of 217 ?

001000010111 = 217(hex)

512+16+4+2+1 = 535

3. What is the denary representation of 2126 ?

0010000100100110 = 2126 (hex)

2+4+32+256+8192 = 8486

How do I convert a hexadecimal number to a decimal number ?

This conversion is a bit simpler as we can use positional notation to convert our number. In fact, we only need to multiply each hexadecimal digit by its corresponding power of 16.

For example, if we want to convert the hexadecimal number ABC to a decimal number, we multiply $C$ by $16^{0}$ as there are 12 ones in the number, $B$ by $16^{1}$ as there are 11 sixteens, and $A$ by $16^{2}$ as there are 10 two-hundred-fifty-sixes.

	$A$	$\times$	$16^{2}$	$=$	$10$	$\times$	$256$	$=$	$2560$
$+$	$B$	$\times$	$16^{1}$	$=$	$11$	$\times$	$16$	$=$	$176$
$+$	$C$	$\times$	$16^{0}$	$=$	$12$	$\times$	$1$	$=$	$12$
									$2748$

How do I convert a binary number to a hexadecimal number?

To convert a binary number to a hexadecimal number, start by making sure the length of your binary number is a multiple of 4. If not, add 0s to the front of the number until it is a multiple of 4.

Next, split the binary number into groups of 4 digits.

Then, for each group of 4 digits, turn the binary number into a single number, by converting each digit in the 4-digit values to the corresponding power of 2:

(The thousands becomes 8, the hundreds becomes 4, the tens becomes 2, and the units becomes 1. However, if the binary value at a place is a 0, then it remains 0.)

For instance, the binary value 1111 becomes 8421, while 0101 becomes 0401.

Then, add up the digits. 8421 becomes 8+4+2+1 = 15, while 0401 becomes 4+1 = 5.

Finally, convert that value to hexadecimal. If the value is from 0 to 9, it stays that value in hexadecimal. If it is 10 or higher, it changes to the corresponding alphabet letter: 10 is A, 11 is B, et cetera.

For example if we want to convert the binary number: 1011111011

• Our number has only 10 bits, so we add two 0s to the left side: 001011111011
• We divide our number in groups of 4 bits: 0010 1111 1011
• We turn each group into a single number: 0+0+2+0 = 2, 8+4+2+1 = 15, 8+0+2+1 = 11
• We convert the single number to hexadecimal numbers: 2 → 2, 15 → F, 11 → B
• Our hexadecimal number is 2FB

Exercises - Hexadecimal representation of numbers

1. What is the decimal representation of 1A4 ?

	1	$\times$	$16^{2}$	$=$	1	$\times$	$256$	$=$	$256$
$+$	A	$\times$	$16^{1}$	$=$	10	$\times$	$16$	$=$	$160$
$+$	4	$\times$	$16^{0}$	$=$	4	$\times$	$1$	$=$	$4$
									$420$

2. What is the decimal representation of A8D ?

	A	$\times$	$16^{2}$	$=$	10	$\times$	$256$	$=$	$2560$
$+$	8	$\times$	$16^{1}$	$=$	8	$\times$	$16$	$=$	$128$
$+$	D	$\times$	$16^{0}$	$=$	13	$\times$	$1$	$=$	$13$
									$2701$

3. What is the hexadecimal representation of 2008 ?

2008 // 16 = 125, with remainder of 8.

125 // 16 = 7, with remainder of 13.

7 // 16 = 0, remainder of 7.

Convert each answer to hexadecimal values, 7 = 7, 13 = D, 8 = 8.

Hence the hexadecimal representation: 7D8

4. What is the hexadecimal representation of 47806 ?

47806 // 16 = 2987, with remainder of 14.

2987 // 16 = 186, with remainder of 11.

186 // 16 = 11, with remainder of 10.

11 // 16 = 0, remainder of 11

Convert each answer to hexadecimal values, 11 = B, 10 = A, 11 = B, 14 = E.

Hence the hexadecimal representation: BABE

Octal

Note

Not needed for the exam

What is Octal ?

Not needed for the exams

Octal is a numbering system in base 8, meaning only digits from 0 to 7 exist in this notation.

For example, one can calculate the decimal equivalent of 754 in octal (base 8) − simply put, finding $754_{8}=k_{10}$. As before, it is the case of expanding the number in its polynomial form and adding up the numbers:

	$7$	$\times$	$8^{2}$	$=$	$7$	$\times$	$64$	$=$	$448$
$+$	$5$	$\times$	$8^{1}$	$=$	$5$	$\times$	$8$	$=$	$40$
$+$	$4$	$\times$	$8^{0}$	$=$	$4$	$\times$	$1$	$=$	$4$
									$492$

Representing Text

What is ASCII?

The original 7-bit ASCII table (English version)

ASCII stands for American Standard Code Information Interchange. It was a 7-bit code, later changed to 8-bit, that had the purpose of setting a number to a letter and other foreign symbols which would allow for the interchanging of information between computers in a single country or using the same language. The total number of letters and symbols stored in ASCII was 128. Countries would input their own letters and symbols instead of English.

However, there was one major drawback, the fact that ASCII only held symbols with English letters, computers from around the world could not successfully interchange information and data with each other.

This drawback led to the creation of the larger database of letters and symbols, Unicode which allows all languages to store their letters in Unicode.

What is Unicode ?

Though ASCII was the industry standard for a very long time, it simply did not have enough characters and symbols to cover the fast internationalisation of the world. It could contain a total of 256 characters however only 128 were pre-programmed into ASCII, the other 128 characters were free to be personalized by each country. This initial solution was expected to work because this way each country could have it’s own organisation system. Unfortunately however, if a person in Japan were to send a email with Japanese characters to someone in the UK it would be an incomprehensible mess due the fact that their personalized characters were completely different.

The solution to this was Unicode. Unicode, now built on a 16-32 bit system instead of ASCII's 8 bit system and is able to cover 149,813 different characters, which is enough to cover every single character in every single language that exists, including emojis and symbols. The need to create any personalized characters was no more, further facilitating communications and translation processes between countries.

As of 2024, a total of 161 scripts are included in the latest version of Unicode (covering alphabets, abugidas and syllabaries), although there are still scripts that are not yet encoded, particularly those mainly used in historical, liturgical, and academic contexts. Further additions of characters to the already encoded scripts, as well as symbols, in particular for mathematics and music (in the form of notes and rhythmic symbols), also occur.

What is interoperability?

Interoperability refers to the ability of different computer systems to exchange and use information effectively, even if they were developed by different manufacturers, with the aim of fostering internationalization. This can be achieved through standardized protocols, data formats, and APIs that facilitate cross-platform communication and integration. Unicode, for instance, promotes interoperability through consistent character representation across different systems and applications. It also enables multilingual support, with the capability of displaying special characters from diverse languages (such as Arabic and Chinese).

What is UTF-8 ?

UTF-8 is a way of translating between Unicode characters and binary text. This is done by storing each Unicode character in the form of up to four one-byte binary strings. Each binary string is created by converting the code point of the Unicode character (letters and numbers that form an index) into a set of binary strings, where each string represents one character in the code point.

For example, the character "A", is represented as `"U+0041" (where "41" is the code point), which is encoded to "01000001" (which is 41 in hexadecimal).

How can I represent characters from different languages in a text?

The best way to represent characters from different languages is to use Unicode. ASCII is inconvenient for different languages as it was designed specifically for English, and does not contain any non-English characters. Meanwhile, Unicode has Chinese, Japanese, and Korean characters, the Cyrillic and Arabic alphabets, as well as any other symbol from a wide variety of languages.

Representing Floating point numbers

Note

Not needed for the exam

How are floats represented/stored ?

Section to go more in-depth on the content.

Floats are typically represented in 4 bytes. Out of these 32 bits: 1 bit is used for the sign of the number, 8 bits are used for the exponent and 23 bits are used for the floating point of the number. It is a way of storing numbers in scientific notation.

Putting the values into mathematical terms, it is calculated by ±A*$10^{-B}$. The sign is represented by the blue area, A is given by the red area and B is given by the green area.

It is interesting to note that if we want to represent negative and positive numbers (whole numbers or decimal) we need to use one bit for storing the sign. If we only store positive numbers we don't need to keep that sign bit.

Representing sound

Note

Not needed for the exam

How is sound represented/stored ?

After sound has been recorded with a microphone, which converts sound waves into a digital signal, computers use a technique called sampling. The computer samples the sound by taking measurements of the amplitude of the sound signal at regular intervals, often 44.1 kHz (or 44,100 times per second), which are then saved as numbers in binary format.

Representing images

How are colors represented/stored ?

Colors we see on the computer screens today are represented by binary numbers of 24 bits.

As we know, all colors that we can perceive originate from the 3 primary colors of light red, blue, and green. Depending on how much of the 3 colors we mix, we are able to achieve many different colors. Therefore, the computer only analyses the 3 primary colors and how much they are used when being mixed. This system is called an RGB representation.

In the total of 24 bits, we divide it into 3 sets of bits for the 3 different primary numbers: 8 bits are for blue, 8 bits are for green, and 8 bits are for red.

Let's first take an example of the color red. In the color red, the corresponding RGB value is: R - 255 (most amount of color we can use), G - 0 (using none of this color), B - 0 (using none of this color).

Therefore, in the 8 bits, the number 255 in base 10 can be transformed into base 2 in 8 bits, which is in this case 11111111. Hence the color red in binary of 24 bits would be 111111110000000000000000. This case applies for both green (000000001111111100000000) and blue (000000000000000011111111).

To summarize, here are the steps:

- divide the 24 bits into 3 sets of 8 bits to represent the colors red, green, and blue

- find the RGB representation of the color in base 10

- transform each the RGB values that are in base 10 to base 2

- combine the 3 sets of 8 bits and here we have the final binary representation of a color

Generally, 24 bits are sufficient to depict realistic colors (photo-realistic). However, even more bits can be used such as 32 bits or 48 bits to illustrate even more realistic colors.

What is a pixel ?

A pixel is a small square of a screen which can be used to display a color. Screens are able to display an element using a multitude of pixels. A pixel is usually composed of 3-4 colors RGB(Red, Green, Blue and Cyan) by changing their intensities.(Source: Wikipedia)

How does the .png file format store and represent images ?

PNG(Portable Network Graphics) supports palette-based images (with palettes of 24-bit RGB or 32-bit RGBA colors), grayscale images (with or without an alpha channel for transparency), and full-color non-palette-based RGB or RGBA images. The PNG working group designed the format for transferring images on the Internet, not for professional-quality print graphics; therefore, non-RGB color spaces such as CMYK are not supported. A PNG file contains a single image in an extensible structure of chunks, encoding the basic pixels and other information such as textual comments and integrity checks documented in RFC 2083.(source: Wikipedia)

The PNG enables lossless data: You can reduce the size of the file wihtout losing data.

How does the .jpeg file format store and represent images ?

JPEG(Joint Photographic Experts Group) is a norm for image compression for a fixed image. It only fixes algorithms and the format of decoding. It enables the companies and firms to choose their own way of coding images.(Source: Wikipedia)

Compressing a JPEG file loses data.

Exercises

Write the hexadecimal number F12 in denary and binary.

Correction

Denary: 3858

Binary: 111100010010

Represent the denary number 123 in hexadecimal and binary.

Correction

Hexadecimal: 7B

Binary: 1111011

Write the hexadecimal number F0A in decimal.

Correction

3850