Lab Activity 2

[ICoy Aj]

Erick jon c. apa

HA3

1. Binary Number System

A set of eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were known in ancient China through the classic text I Ching. In the 11th century, scholar and philosopher Shao Yong developed a method for arranging the hexagrams which corresponds to the sequence 0 to 63, as represented in binary, with yin as 0, yang as 1 and the least significant bit on top. There is, however, no evidence that Shao understood binary computation. The ordering is also the lexicographical order on sextuples of elements chosen from a two-element set.^[7]

Similar sets of binary combinations have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been widely applied in sub-Saharan Africa.

Gottfried Leibniz

In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text.^[8] Importantly for the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature".^[8] (See Bacon's cipher.)

The modern binary number system was studied by Gottfried Leibniz in 1679. See his article:Explication de l'Arithmétique Binaire^[9](1703). Leibniz's system uses 0 and 1, like the modern binary numeral system. As a Sinophile, Leibniz was aware of the I Ching and noted with fascination how its hexagrams correspond to the binary numbers from 0 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.^[10]

In 1854, British mathematician George Boole published a landmark paper detailing an algebraic system of logic that would become known as Boolean algebra. His logical calculus was to become instrumental in the design of digital electronic circuitry.^[11]

In 1937, Claude Shannon produced his master's thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled A Symbolic Analysis of Relay and Switching Circuits, Shannon's thesis essentially founded practical digital circuit design.^[12]

In November 1937, George Stibitz, then working at Bell Labs, completed a relay-based computer he dubbed the "Model K" (for "Kitchen", where he had assembled it), which calculated using binary addition.^[13] Bell Labs thus authorized a full research programme in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed January 8, 1940, was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on September 11, 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were John Von Neumann, John Mauchly and Norbert Wiener, who wrote about it in his memoirs.^[14][15][16]

^{2. Decimal Number System}

The Decimal Number System uses base 10. It includes the digits from 0 through 9. The weighted values for each position is as follows:

3. Octal Number System

The octal, or base 8, number system is a common system used with computers. Because of its relationship with the binary system, it is useful in programming some types of computers.

Look closely at the comparison of binary and octal number systems in table 1-3. You can see that one octal digit is the equivalent value of three binary digits. The following examples of the conversion of octal 225₈ to binary and back again further illustrate this comparison:

4. Hexadecimal Number System

A big problem with the binary system is verbosity. To represent the value 202 requires eight binary digits.

The decimal version requires only three decimal digits and, thus, represents numbers much more compactly than does the binary numbering system. This fact was not lost on the engineers who designed binary computer systems.

When dealing with large values, binary numbers quickly become too unwieldy. The hexadecimal (base 16) numbering system solves these problems. Hexadecimal numbers offer the two features:

• hex numbers are very compact
• it is easy to convert from hex to binary and binary to hex.

Since we'll often need to enter hexadecimal numbers into the computer system, we'll need a different mechanism for representing hexadecimal numbers since you cannot enter a subscript to denote the radix of the associated value.

The Hexadecimal system is based on the binary system using a Nibble or 4-bit boundary. In Assembly Language programming, most assemblers require the first digit of a hexadecimal number to be 0, and we place an H at the end of the number to denote the number base.

The Hexadecimal Number System:

uses base 16

includes only the digits 0 through 9 and the letters A, B, C, D, E, and F

In the Hexadecimal number system, the hex values greater than 9 carry the following decimal value:

[princezjane23]

<sup>NAME: LOVELY JANE C. KAMDAN
SECTION: HA3

1.BINARY NUMBER SYSYEM</sup>
   A method of representing numbers that has 2 as its base and uses only the digits 0 and 1. Each successive digit represents a power of 2. For example, 10011 represents (1 × 2⁴) + (0 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰), or 16 + 0 + 0 + 2 + 1, or 19.
  A positional system of numeration that uses binary digits and a radix of two binary numeration system, binary system, pure binary numeration system

positional notation, positional representation system - a numeration system in which a real number is represented by an ordered set of characters where the value of a character depends on its position.

^{2.DECIMAL NUMBER SYSTEM}
  A numbering system that uses ten digits, from 0 to 9, arranged in a series of columns to represent all numerical quantities. Each column or place value has a weighted value of 1, 10, 100, 1000, and so on, ranging from right to left.
  The number system we use every day, based on 10 digits (0,1,2,3,4,5,6,7,8,9).
Position is important, with the first position being units, then next on the left being tens, then hundreds and so on.

^{3.OCTAL NUMBER SYSTEM}
   A numbering system that uses eight digits, 0 to 7, arranged in a series of columns to represent all numerical quantities. Each column or place value has a weighted value of 1, 8, 64, 512, and so on, ranging from right to left.
  A positional system of numeration that uses octal digits and a radix of eight octal numeration system positional notation, positional representation system - a numeration system in which a real number is represented by an ordered set of characters where the value of a character depends on its position.

^{4.HEXADECIMAL NUMBER SYSTEM}
   A number system having a base 16; the symbols for the numbers 0--9 are the same as those used in the decimal system, and the numbers 10--15 are usually represented by the letters A--F. The system is used as a convenient way of representing the internal binary code of a computer.
   hexadecimal - (Or "hex") Base 16. A number representation using the digits 0-9, with their usual meaning, plus the letters A-F (or a-f) to represent hexadecimal digits with values of (decimal) 10 to 15. The right-most digit counts ones, the next counts multiples of 16, then 16^2 = 256, etc.