Abstract: The relationship between the width of the interval and the addition, subtraction, multiplication, and division of the Interval

SiCp exercise 2.9 is like a mathematical problem. We need to prove that the sum and difference width of the range is a function of the width of the added and subtracted intervals, but it is not true for multiplication and division.

The so-called width in the book is half of the gap between the start and end of the interval. In my opinion, it is more like half of the width of the interval. If you think of an interval as a line segment, the so-called width should be the difference between the start point and the end point. If you must remember half of the width as the width, it doesn't matter.

It is easier to prove that the sum of the intervals is the function of the width of the added intervals. See the following proof steps:

If interval 1 is (A1 B1) and interval 2 is (A2 B2 ),

The width of range 1 is

W1 = (B1-A1)/2

The width of range 2 is:

W2 = (B2-A2)/2

The sum of the two zones is:

W3 = (b1 + b2)-(A1 + A2)/2

W3 = (b1 + B2-A1-A2)/2

W3 = (b1-a1)/2 + (b2-a2)/2

W3 = W1 + W2

Similarly, subtraction can be proved by a similar method.

For multiplication and division, because the operation to obtain the maximum and minimum values is involved, I did not expect any way to prove that they do not have the function relationship as in addition and subtraction.

However, the question does not require us to prove it. We just need to give an example to illustrate that this is not the case for multiplication and division. Think of it as a proof-of-law. Find an invalid example.

In this example, the range width of multiplication and division is the same, but the width of the result after multiplication and division is different.

If you look for an example, you can easily find one:

(1 2) * (5 6) = (5 12)

Here, the multiplier width is 0.5, and the result width is 3.5.

In the following calculation:

(1 2) (100 101) = (100 202)

The width of the multiplier is 0.5, but the width of the result is increased ....

Abstract: The relationship between the width of the interval and the addition, subtraction, multiplication, and division of the Interval