CALCULATOR AND COMPUTER ARITHMETIC

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1 MathScope Handbook - Calc/Comp Arithmetic Page 1 of 7 Click here to return to the index 1 Introduction CALCULATOR AND COMPUTER ARITHMETIC Most of the computational work you will do as an Engineering Student will be performed either on a calculator or by means of a computer program. This section is intended to illustrate the ways in which calculators and computers store numbers and perform arithmetic operations. It is not intended as a guide to the use of such devices. Decimal Systems - Base 10 In everyday life information are represented in the decimal system 0,1,,,4,5,6,7,8,9 and the base 10 using the digits e.g = (x10 ) + (1x10 ) + (5x10 1 ) + (6x10 0 ) + (1x10-1 ) + (x10 - ) Notice how the position of a digit within the number indicates the power of 10 by which it is multiplied. Here is another example: = (x10 4 ) + (x10 ) + (6x10 ) + (9x10 1 ) + (6x10 0 ) + (4x10-1 ) + (8x10 - ) + (6x10 - ) Having understood this representation it is clear that we can use any positive integer as the base. As you know, when using a calculator you key in numbers in the usual decimal system and the answers are displayed in the decimal system. However, the machine will actually store the numbers using a different base. Depending on the type of machine the base will be either or 16. Binary System - Base The digits are 0, 1 E.g = (1x ) + (0x ) + (1x 1 ) + (1x 0 ) + (1x -1 ) + (0x - ) + (1x - ) The fact that only two digits are required was the attribute of the binary system, which was so useful in the development of machines for storing numbers and doing arithmetic, since switch off 0 and switch on 1 provided an obvious code. In decimal arithmetic the above number is: / + 0/4 + 1/8 =

2 MathScope Handbook - Calc/Comp Arithmetic Page of 7 4 Hexadecimal System - Base 16 Some modern machines use this system in which the digits are 0,1,,,4,5,6,7,8,9,A, B, C, D, E, F In decimal terms A=10, B=11,C=1,D=1,E=14,F=15 ( ) ( ) ( 1 ) ( 0 ) ( 1 ) ( ) 5AD 4C = A 16 + D C 16 which, as a decimal number is : 5x x56 + 1x /16 + 1/56 = As an example, the decimal number 0.15 can also be expressed as: - Binary: Hexadecimal: 14A. To check this = = = A. = x x16-1 = = Conversion Between Bases As the example above shows, to convert from binary or hexadecimal to decimal is a simple computation. The algorithm for performing the reverse operation is illustrated in the following example.

3 MathScope Handbook - Calc/Comp Arithmetic Page of 7 Decimal number 0.15 Conversion to binary: Integer part Divide by the new base repeatedly = 165, remainder R 1 = 0 = 8, remainder R = 1 = 41, remainder R = 0 = 0, remainder R 4 = 1 = 10, remainder R 5 = 0 = 5, remainder R 6 = 0 =, remainder R 7 = 1 = 1, remainder R 8 = 0 = 0, remainder R 9 = 1 The integer part is then R9 R8 R7 R1 Fraction Part = Multiply repeatedly by the new base until the fractional part becomes zero (It may not do in which case stop the process when enough accuracy is achieved) 0.15x = 0.5 = = I 1 +F 1 F 1 x = 0.5x = 0.5 = = I +F F x = 0.5x = 1.0 = 1+0 = I +F The fraction is then I 1, I, I = Thus

4 MathScope Handbook - Calc/Comp Arithmetic Page 4 of 7 Conversion to hexadecimal: Integer part Divide by 16 repeatedly 0/16 = 0 remainder R 1 = A 0/16 = 1 remainder R = 4 1/16 = 0 remainder R = 1 The integer part is: R R R 1 = 14A Fraction part Multiply repeatedly by 16 stopping as described above 0.15 x 16 =.0 = = I 1 + F 1 The fraction part is 0. and so = 14A. 6 Addition and Multiplication in Binary Arithmetic The basic rules are: 0+0 = 0, 1+0 = 0+1 = 1, 1+1 = 10 0x0 = 0, 1x0 = 0x1 = 0, 1x1 = 1 Addition: Multiplication: x = The rules for hexadecimal arithmetic are omitted. 4

5 MathScope Handbook - Calc/Comp Arithmetic Page 5 of 7 7 Normal Form and Floating Point Arithmetic Even though your calculator or computer will be storing numbers and operating on them in non-decimal form this is not visible to you as the user. You input data in decimal form and the machine gives you the answer in decimal form. In the following description of how computing machines store and operate on numbers we can without any loss of validity ignore the conversion process and we shall use the decimal system in all examples. A computing machine has only a finite space in which to store a number. If you compute the quantity /00 the exact answer is recurring. This is beyond the storage capacity of any machine. Your computer would actually store this as three pieces of information, which we call the normal form of the number. In this case (10 - ) (The actual number of sixes stored would be machine dependent, typically equivalent to about 1 decimal digits). The normal form of a number then consists of three parts:- (sign) (mantissa) (b n ) sign :- mantissa :- plus or minus a fraction with a fixed number of digits b :- the arithmetic base ( or 16 in a machine, 10 in our examples) n :- an integer, positive or negative with a maximum value which is machine dependent. The integer n is chosen so that 1 b mantissa < 1 With b=10 we have 0.1 mantissa < 1 so that in normal form is (10 ) but not (10 4 ) or (10 ) Computer arithmetic is called Floating Point Arithmetic. In essence it is as follows 5

6 MathScope Handbook - Calc/Comp Arithmetic Page 6 of 7 Addition/Subtraction: To add/subtract two numbers: Adjust the value of n (exponent) of the smaller number to be equal to the exponent of the larger number. Add/subtract the numbers. Adjust (round off) the mantissa to fit the available number of digits. Multiplication/Division: Multiply/divide the two mantissas. Combine the exponents. Adjust and round off the product mantissa. Addition (Five digit mantissa) (10 ) (10-1 ) = (10 ) (10 ) = (10 ) = (10 ) Multiplication (Five digit mantissa) [ (10 )] x [0.611 (10-1 )] = (10 1 ) = (10 1 ) As can be seen in the above examples, each of the final answers is subject to a small error due to the restriction on the number of digits in the mantissa. In some calculations the accumulated effect (accumulated rounding error) can be quite serious. Consider the simple computation The exact value is We will now perform the computation as a hypothetical calculator with a 5 digit mantissa would = 0 ( 10 4 ) = 0 ( 10 4 ) = An error of over 10% All calculators and computers have rather better than 5digit mantissa but the example nevertheless illustrates possible pitfalls. 6

7 MathScope Handbook - Calc/Comp Arithmetic Page 7 of 7 Click here to return to the main document 7

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