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Most computer systems operate using binary logic. These two voltage levels represent exactly two different values and by convention, the values are zero and one. These two values coincidentally, correspond to the two digits used by the binary number system. Since there is a correspondence between the logic levels used by the computer and the two digits used in the binary numbering system, it should come as no surprise that computers employ the binary system.
The binary number system uses base 2 which includes only the digits 0 and 1. In the United States among other countries, every three decimal digits is separated with a comma to make larger numbers easier to read. For example, , is much easier to read than We adopt a similar convention for binary numbers. In order to make binary numbers more readable, we add a space every four digits starting from the least significant digit, on the left of the decimal point.
For example, the binary value will be written as We typically write binary numbers as a sequence of bits. There are defined boundaries for the different sizes of a bit:. In any number base, we may add as many leading zeroes without ever changing its value, but in the binary number system, adding leading zeroes adjusts the bit to a desired size. For example, we can represent the number 5 as a:. The boundary for a Word can be defined as either bits or the size of the data bus for the processor.
Therefore, a Word and Double Word is not a fixed size but varies from system to system - depending on the processor. For the and microprocessors, a word is a group of 16 bits. We number the bits in a word starting from bit zero b0 through fifteen b15 as follows:.
Remember, b0 is the LSB and b15 is the msb and when referencing the other bits in a word use their bit position number. Notice that a word contains exactly two bytes.
Naturally, a word may be further broken down into four nibbles. Nibble zero is the low order nibble b0-b4 and nibble three is the high order nibble bb15 of the word. The other two nibbles are "nibble one" and "nibble two". The three major uses for words are. A double word is exactly that, two words, so a double word quantity is 32 bits. Naturally, this double word can be divided into a high order word and a low order word, four bytes, or eight nibbles.
Now just like the decimal system, we multiply each digit by base 2 and its weighted position valuethen find the sum, like this:.
There are two methods, division by 2 and subtraction by the weighted position value. For this, we divide the decimal number by 2, if the remainder is 0 write down a 0. If the remainder is 1then write down a 1. This process is continued until the quotient is 0. The remainders will represent the binary equivalent of the decimal number. The remainder values are written from the least significant digit to the more significant digit.
So each new remainder value is written to the left of the previous. Start with a weighted position value greater than the number. If the number is less than the weighted value, write down a 0 and subtract 0. If the number is greater than the weighted value, write down a 1 but subtract the weighted value instead. This process is continued until the result is 0.
When performing the subtraction method, the digits will represent the binary equivalent of the decimal number from the most significant digit to the lesser significant digit. So each new digit is written to the right of the previous. The Octal system is based on the binary system with a 3-bit boundary. The Octal system uses base 8 which includes only the digits 0,1,2,3,4,5,6 and 7 any other digit would make the number an invalid octal number.
First, break the binary number into 3-bit sections from the LSB to the MSB then convert the 3-bit binary number to its octal equivalent. This yields the binary number or in our more readable format. To convert from Octal to Decimal, multiply the value in each position by its Octal weight and add each value. The typical method to convert from decimal to octal is repeated division by 8.
For this method, divide the decimal number by 8, and write the remainder on the side as the least significant digit. This process is continued by dividing the quotient by 8 and writing the remainder until the quotient is 0.
When performing the division, the remainders which will represent the octal equivalent of the decimal number are written beginning at the least significant digit right and each new digit is written to the next more significant digit the left of the previous digit.
Consider the number A big problem with the binary system is verbosity. For example, eight binary digits are required to represent the value while the decimal version requires only three decimal digits.
Especially with large values, binary numbers become too unwieldy. The hexadecimal numbering system solves these problems. Hexadecimal numbers are very compact and 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 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 0and we place an H at the end of the number to denote the number base. The hexadecimal number system uses base 16which includes only the digits 0 through 9 and the letters A, B, C, D, E, and F. This table provides all the information you'll ever need to convert from one number base into any other number base for the decimal values from 0 to For example, to convert 0ABCDH into a binary value, simply convert each hexadecimal digit according to the table above.
For example, given the binary numberthe first step would be to add two zeros in the MSB position so that it contains 12 bits.
Then separate the binary value into groups of four bits, so the revised binary value reads Then we look up the binary values in the table above and substitute the appropriate hexadecimal digit:.
To convert from Hex binare option bot uberprufunge Decimal, multiply the value in each position by its weighted value then add. Using the table above and the previous example, 0AFB2Hwe expect to obtain the decimal value To convert from decimal to hex is slightly more difficult.
The typical method is repeated division by While we may also use repeated subtraction by the weighted position valueit is more difficult for large decimal numbers. Divide the decimal number by 16, and write the remainder on the side as the least significant digit.
When performing the division, the remainders which will represent the hex equivalent of the decimal number are written beginning at the least significant digit right and each new digit is written to the next more significant digit the left of the previous digit. When you use hex numbers in an program, the Assembler usually requires the most significant hex digit to be 0 even if this number of digits exceed the size of the register.
This is an Assembler requirement and your value will be assembled correctly. If individual digits are A through F, we must convert to its decimal equivalent values from the table above then add. If the sum is greater than or equal to 16, we replace the sum with the sum minus 16 and carry over 1. Our decimal number system is known as a positional number system because the value of the number depends on the position of the digits.
For example, the number is different from even though the same digits are used. Other ancient number systems used additional symbols to represent larger values, but in a positional number system the value of each digit is determined by its place in the full number. The lowest place value is the rightmost position, and each successive position to the left has a higher place value.
The rightmost position represents the "ones" column, the next position represents the "tens" column, the next position represents "hundreds", etc. Therefore, the number represents 1 hundred and 2 tens and 3 ones, whereas the number represents 3 hundreds and 2 tens and 1 one.
The values of each position correspond to powers of the base of the number system. So for our decimal number system, which uses base 10, the place values correspond to powers of Other number systems use different bases. The binary number system uses base 2, so the place values of the digits of a binary number correspond to powers of 2.
For example, the value of the binary number is determined by computing the place value of each of the binary digits of the number:. The same principle applies to any number base. For example, the number base 5 corresponds to. In order to convert a decimal number into different number bases, we have to be able to express the number in terms of powers of the other base.
To convert the decimal number to base 4, we must figure out how to express as the sum of powers of 4for example:. Another method is to repeatedly divide the decimal number by the base in which it is to be converted until the quotient becomes zero. As the number is divided, the remainders - in reverse order - form the digits of the number in the other base. The answer is formed by taking the remainders in reverse order: Computer arithmetic is performed on data stored in fixed length memory locations, typically 8, 16, or 32 bits.
Special problems occur in manipulating fixed length numbers. The number base 10 is larger than the maximum 8-bit number; this results in a carry beyond 8 bits in the binary representation and a carry beyond two digits in the hexadecimal representation.
When doing arithmetic using fixed length numbers, these carries are potentially lost. Decimal examples Odometer numbers are useful for both addition and subtraction complements of signed numbers. Do subtraction as addition by using complements, i.