19. Weaving Code – Amandine David

13.
Amandine David (FR/BE)

Weaving Code

Texts by Amandine David et Raoul Sommeillier

A number base is the use of a specific number and its successive powers to form larger numbers.

A base 10 system includes the following: one (\(1=10^0\)), ten (\(10=10^1\)), one hundred (\(100=10^2\)), one thousand (\(1000=10^3\)), ten thousand (\(1000=10^4\)), etc.

The fact that the decimal system is used in most of the world, is most likely related to the fact that we use our ten fingers to count. We would probably use a base 8 system if we had 8 fingers!

Different base systems:

Base 60A sexagesimal system used in ancient Mesopotamia, from which we borrowed the number 60 for minutes and seconds. It has many divisors: 2, 3, 4, 5, 6, 10, 12, 15 and 30.
Base 20The vigesimal system was used by French Gauls and was common in the Middle Ages. Traces of this system are apparent in the names of some French numbers: “eighty” is quatre-vingt (“four-twenty”); “ninety” is quatre-vingt-dix, (“forty-twenty-ten”).
Base 12The duodecimal system is used to count months, hours and eggs by the dozen.
Base 10Our commonly used counting base.
Base 5The quinary number base is partially used in the roman numeral system. In addition, certain Japanese coded systems use a bi-quinary system.
Base 2Prominent in computer science, the base 2 system is inspired by the binary characteristic of electronic components. Like electrical contacts, they have two possible values or states: open or closed, blocked or unblocked, 0 or 1.
Bases
8 and 16
Octal and hexadecimal systems are more concise and easier for “us humans” to understand than the binary system with its endless streams of 0s and 1s.

Except for zero, any number could be used as a base, including fractions and irrational numbers. A pi-based number system is entirely possible.

There is an additional factor to take into consideration with regards to number bases: they can be positional or not.
Both decimal and binary systems are positional number systems.
Take number 1975, written in our familiar base 10 system. The value of each number depends on
– the number itself (the value when on its own)
– and its position

In 1975, number 5 is for 5 units = 5 x 1
– number 7 is for tens = 7 x 10
– number 9 is for hundreds = 9 x 100
– and number 1 is for thousands = 1 x 1000

As the example shows, positional notation is key when forming numbers.

The number farthest to the right is for units. The following numbers are for tens, then hundreds, etc. Numbers in a decimal system are read from right to left. As a result, units will always be in the first position (position 0) regardless of number size.

To determine the position weight of a number, use the following rule:

\[Weight\ of\ a\ number = base^{position}\]

The value of a number is the product of its weight and the value of the digit itself.
The weight of a number is the power of the number base and exponent y is the position of the number when starting with the units and counting from right to left.
Consider also that the position of a digit in a number is equal to the number of digits to its right.

The number 1975 in decimal base 10 can be described as follows:
– The weight of number 5 is 100, its value is 5 x 100 = 5 x 1 = 5
– The weight of number 7 is 101, its value is 7 x 101 = 7 x 10 = 70
– The weight of number 9 is 102, its value is 9 x 102 = 9 x 100 = 900
– The weight of number 1 is 103, its value is 1 x 103 = 1 x 1000 = 1000

The binary numeration system, also called binary code or binary numeration, is a base 2 number system.

The binary system includes only two numbers: 0 and 1.

Note that in a base 2 system, number 2 does not exist. Similarly, in our decimal system (base 10), number 10 does not exist.

The binary system is positional and follows the same rules as our decimal system. From right to left: first units; then twos, fours, eights, sixteens, thirty-twos and so forth.

Take binary number 10110.
As seen above, the weight value of a number depends on its position and the base system:

\[Weight = 2^{position}\]

Apply this rule to \(10110_{(2)}\).

Positions\(4\)\(3\)\(2\)\(1\)\(0\)
Binary numbers\(1\)\(0\)\(1\)\(1\)\(0\)
Weight\(2^4=16\)\(2^3=8\)\(2^2=4\)\(2^1=2\)\(2^1=1\)
Number values\(1*16=16\)\(0*8=0\)\(1*4=4\)\(1*2=2\)\(0*1=0\)

The total value is then \(16+4+2=22\).
There is a sixteen, a four, and a two: 16 + 4 + 2 = 22.

Binary codes are essential in computer science: the core element is a bit.

The word bit is the combination of “binary” and “digit”.

A series of eight bits strung together makes a byte. Common binary numbers or words are 8, 16, 32 and 64. For better readability, a space is left between each group of four bits: \(0100\ 0001\).

Note: “word” is the commonly used to express the value of a binary number: e.g.: a 32-bit word or a 64-bit word.

Pages: 1 2 3