[Keith Lim]

NAME: keith Ryan lim

Section: HN3

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.

2. Decimal Number System

a positional system of numeration that uses decimal digits and a base of ten.

3. Octal Number System

a positional system of numeration that uses octal digits and a radix of eight.

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

[Joey Andrew Atsoca]

Name:Joey Andrew A. Acosta

Section:NH3

 

1. Binary Number System

-binary system, numeration system based on powers of 2, in contrast to the familiar decimal system, which is based on powers of 10. In the binary system, only the digits 0 and 1 are used. Thus, the first ten numbers in binary notation, corresponding to the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 in decimal notation, are 0, 1, 10, 11, 100, 101, 110, 111, 1000, and 1001. Since each position indicates a specific power of 2, just as the number 342 means (3 × 10²) + (4 × 10¹) + (2 × 10⁰), the decimal equivalent of a binary number can be calculated by adding together each digit multiplied by its power of 2; for example, the binary number 1011010 corresponds to (1 × 2⁶) + (0 × 2⁵) + (1 × 2⁴) + (1 × 2³) + (0 × 2²) + (1 × 2¹) + (0 × 2⁰) = 64 + 0 + 16 + 8 + 0 + 2 + 0 = 90 in the decimal system. Binary numbers are sometimes written with a subscript "b" to distinguish them from decimal numbers having the same digits. As with the decimal system, fractions can be represented by digits to the right of the binary point (analogous to the decimal point). A binary number is generally much longer than the decimal equivalent; e.g., the number above, 1011010_b, contains seven digits while its decimal counterpart, 90, contains only two. This is a disadvantage for most ordinary applications but is offset by the greater simplicity of the binary system in computer applications. Since only two digits are used, any binary digit, or bit, can be transmitted and recorded electronically simply by the presence or absence of an electrical pulse or current. The great speed of such devices more than compensates for the fact that a given number may contain a large number of digits.

2. Decimal Number System

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Probably the biggest stumbling block most beginning programmers encounter when attempting to learn assembly language is the common use of the binary and hexadecimal numbering systems. Understanding these numbering systems is important because their use simplifies other complex topics including boolean algebra and logic design, signed numeric representation, character codes, and packed data.

This section discusses several important concepts including the binary, decimal, and hexadecimal numbering systems, binary data organization (into bits, nibbles, bytes, words, and double words), signed and unsigned number systems, arithmetic, logical, shift, and rotate operations on binary values, bit fields and packed BCD (Binary Coded Decimal) data, and the ASCII (American Standard Code for Information Interchange) character set. This is basic material and the remainder of this tutorial depends upon your understanding of these concepts. If you are already familiar with these terms from other courses or study, you should at least skim this material before proceeding to the next chapter. If you are unfamiliar with this material, or only vaguely familiar with it, you should study it carefully before proceeding. All of the material in this chapter is important! Do not skip over any material.

Most modern computer systems do not represent numeric values using the decimal system. Instead, they typically use a binary or two's complement numbering system. To understand the limitations of computer arithmetic, you must understand how computers represent numbers.

Remember how mathematical operations are entered into a computer:

+ is used for addition

- is used for subtraction

* is used for multiplication

/ is used for division

^ is used to raise to a power

There are four number bases commonly used in programming. These are:

3. Octal Number System

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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

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The hexadecimal (base 16) number system operates the same way as the decimal (base 10) number system, except it is based on sixteen instead of ten. The operation of the decimal system is familiar.

The 4-digit base-10 number 5826 appears below, indicating how the value of the number is derived from the values of its 4 digits.

Everybody knows that 5826 means five-thousand eight-hundred twenty-six. But only because they have been taught that 1, 10, 100, and 1000 are part of the calculation even though they are never written. All that is actually written is a total of twenty-one (5 and 8 and 2 and 6). Only the interpretation that people supply, which is purely mental and unwritten, informs what is written with its intended value.

Hexadecimal operates the same way. Each digit is "weighted" by a "multiplier," with the results all added together. The multipliers in both systems are the powers of the system base (10 or 16). The powers of 10 are 1, 10, 100, 1000, etc. while those of 16 are 1, 16, 256, 4096, etc. So the same digits "5826" used in base 16 represent a value calculated as follows:

Though the digits are the same (5826) the values come out quite different, because the base and its "multiplier values" are different.

You will see hexadecimal numbers some of whose digits are letters instead of numbers. That's because the number of digits needed by any number system is the number's base. So base base 2 needs 2 digits, base 10 needs 10,  and base 16 needs 16. Base 2 has 0 and 1. Base 10 has 0 through 9. Base 16 borrows 0 though 9 but needs another 6. For those, we could invent some symbols. However, for convenience we employ the first 6 letters of the alphabet (A through F) instead. When we run out of digits at 9, we use A as the next digit. So A represents the value 10. B comes next, and represents 11. The hexadecimal digits and thevalue they stand for are:

Hexadecimal digit	Value
0	0
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9
A	10
B	11
C	12
D	13
E	14
F	15

 

Here's the derivation of the value of another 4-digit hexadecimal number, but this one uses some of the high-order digits A-F:

The highest you can count with a given number of digits (in any number system) is the number in which every digit contains the maximum value in the number system (1 in base 2, or 9 in base 10, or F in base 16). So the largest number you can represent with 4 digits in base 16 is:

Counting any higher than that would require that you utilize more digits. The very next number is 10000 in hexadecimal, or 65536 in decimal. It is a well-known value in computer science and is called "64K."

[keithaclc1305]

[Janet Batitang]

janet batitang

HN3

1.BINARY NUMBER SYSYEM
   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.