Ever since the first computer was developed in the early 1900’s the

computer has been using math to solve most of it’s problems. The Arithmetic

and Logical unit helps the computer solve some of these problems. All type

of math can be solved on computer’s which it uses.

Binary Arithmetic

A computer understands two states: on and off, high and low, and so

on. Complex instructions can be written as a combination of these two

states. To represent these two conditions mathematically, we can use the

digits 1 and 0. Some simple mathematical operations, such as addition and

subtraction, as well as the two’s complement subtraction procedure used by

most computer’s.

Evaluating an Algebraic Function

It is frequently necessary to evaluate an expression, such as the one

below, for several values of x.

y= 6×4+4×3-5×2+6x+4

First to start with developing the power’s of x to perform the

necessary multiplications by the coefficients, and finally produce the sum.

The following steps are the way the computer “thinks” when it is

calculating the equation.

1.Select x

2.Multiply x by x and store x2

3.Multiply x2 by x and store x3

4.Multiply x3 by x and store x4

5.Multiply x by 6 and store 6x

6.Multiply stored x2 by 5 and store 5×2

7.Multiply stored x3 by 4 and store 4×3

8.Multiply stored x4 by 6 and store 6×4

9.Add 6×4

10.Add 4×3

11.Subtract 5×3

12.Add 6x

13.Add 4

Binary Coded Decimal

One of the most convenient conversions of decimal to binary coded

decimal’s is used today in present day computer’s. BCD(Binary Coded

Decimal) is a combination of binary and decimal; that is each separate

decimal digit is represented in binary form. For example the chart below

represents the Binary and Decimal conversions.

DecimalBinary

0 0

1 1

2 10

3 11

4 100

5 101

6 110

7 111

8 1000

9 1001

10 1010

BCD uses one of the above binary representations for each decimal

digit of a given numeral. Each decimal digit is handled separately.

For example, the decimal 28 in binary is as follows:

(28)10 = (11100)2

The arrangement in BCD is as follows:

28

0010 1000

Each decimal digit is represented by a four-place binary

number.

Direct Binary Addition

In binary arithmetic if one adds 1 and 1 the answer is 10. The answer

is not the decimal 10. It is one zero. There are only two binary digits in

the binary system. Therefore when one adds 1 and 1, one gets the 0 and a

carry of 1 to give 10. Similarly, in the decimal system, 5 + 5 is equal to

zero and a carry of 1. Here is an example of binary addition:

column 4 3 2 1

0 1 1 1

+ 0 1 1 1

1 1 1 0

I n column 1, 1+1=0 and a carry of 1. Column 2 now contains 1+1+1.

This addition, 1+1=0 carry 1 and 0+1=1, is entered in the sum. Column 3 now

also contains 1+1+1, which gives a carry of 1 to column 4. The answer to

the next problem is found similarly.

1 0 0 1 1 0 1 1

+ 0 0 1 1 1 1 1 1

1 1 0 1 1 0 1 0

Direct Binary Subtraction

Although binary numbers may be subtracted directly from each other, it

is easier from a computer design standpoint to use another method of

subtraction called two’s complement subtraction. This will be illustrated

next. However direct binary subtraction will be discussed.

Direct Binary Subtraction is similar to decimal subtraction, except

that when a borrow occurs, it complements the value of the number. Also

that the value of the number of one depends on the column it is situated.

The values increase according to the power series of 2: that is 20, 21,23,

and so on, in columns 1, 2, 3 and so on. Hence, if you borrow from column 3

you are borrowing a decimal 4. ex column 3 2 1 1 1 0 – 1 0 1 0 0 1

In the example a borrow had to be made from column 2, which

changed its value to 0 while putting decimal 2 (or binary 11) in

column 1. Therefore, after the borrow the subtraction in column 1

involved 2-1=1; in column 2 we had 0-0=0; and in column 3 we had

1-1=0.

If the next column contains a 0 instead of a 1 , then we

must proceed to the next column until we find one with 1 from

which we can borrow.

ex

1 0 0 0

– 0 1 0 1

After the borrow from column 4,

0 1 1 (11)

– 0 1 0 1

0 0 1 1

Notice that a borrow from column 4 yields an 8(23). Changing

column 3 to a 1 uses a 4, and column 2 uses a 2, thus leaving 2

of the 8 we borrowed to put in column 1.

ex

0 1 1 0 0 0 1 0

– 0 0 0 1 0 1 1 1

After the first borrow:

0 1 1 0 0 0 0 (11)

– 0 0 0 1 0 1 1 1

After the second borrow (from column 6):

0 1 0 1 1 1 (11) (11)

– 0 0 0 1 0 1 11

0 1 0 0 1 0 11

These operations are stored in the computer’s memory then

performed in the computer’s Arithmetic/Logic Unit in the CPU.

Approximations

In computer’s, it is very important to consider the error

that may occur in the result of a calculation when numbers which

approximate other numbers are used. This is important to the use

of computer’s because of computers are usually very long and

involve long numbers.

Division

It is possible to divide one number from another by

successively subtracting the divisor from the dividend and

counting number of the subtractions necessary to reduce the

remainder to a number smaller than the divisor.

For example, to divide 24 by 6:

Number ofIs remainder smaller

subtractions than divisor?

24

– 61 No

18

– 62 No

12

– 63 No

6

– 64 Yes

0

This shows how the computer “thinks” when it is calculating a

problem using the division operation.

Here is another example when there is a remainder.

For example to divide 27 by 5:

Number of Is remainder smaller

Subtractionsthan divisor?

27

– 51No

22

– 52No

17

– 53No

12

– 54No

7

– 55Yes

2

Therefore 27 = 5, with a remainder of 2.

These two diagrams show the flow of thinking for the operation of

division in a calculation.

Evaluating Trigonometric Relations

For many problems in mathematics, the relationships between

the sides of a right triangle are important, and this, of course,

may suggest a general definition of trigonometry. hat is,if a

computer is available, how trigonometric functions can be done by

hand. It is interesting to consider some of the features of this

field from a computer-oriented point of view.

It is not necessary to consider the last three functions in

the same sense as the first three because, if any one of the

first three one can get, the last three one can get by the

reciprocal of the first three.

Reference to the triangle above shows that:

tan A = a

b

and that tan A is related to sin A and cos A by the following:

sin A = a/c = a = tan A

cos Ab/cb

Something similar is shown below using the Pythagorean

Theorem:

a2 + b2 = c2

and dividing by c2:

a2 + b2 = c2

c2 + c2 = c2.

Applications of Computer Math

Computer Math is used in various ways in the mathematics and

scientific field. Many scientists use the computer math to

calculate the equations and using formulas, there by making

calculating on computer much faster. For mathematicians computer

math can help mathematicians solve long and tedious problems,

quickly and efficiently.

The introduction of computer’s into the world’s technology

has drastically increased the amount of knowledge helped by the

computer’s. The different aspects of using computer math are

virtually limitless.