What is e + π? What could an answer to this question even look like?
One possible solution is to give a decimal approximation to the result. The decimal expansions for e and π are infinite, so there's no way of reporting all of the digits of e + π. [?] But we could report some of those digits and that would tell us some important things about the number, such as how it compared to other numbers.
How could we compute that sum? We could take the first [some number] of digits of e and the first [some number] of digits of π and add them, then report the answer.
| e = | |
| π = | |
| e + π = |
Pay careful attention to what occasionally happens with the digits of the sum (including the first time you add a digit). How awful! Not only can we not report all of the digits, but we're sometimes not even reporting the right digits!
How could you tell if a digit of the sum above is safe to report, that is, how do you know it's actually a digit of the actual sum?
Real world measured values behave similarly to e and π: we can't have all of the digits at once, but we can have as many digits as we want. [?]
Shown below is the coordinate plane with a random point highlighted. Zoom in using the scroll wheel or buttons below to try to figure out as many digits of its x and y coordinates as possible, and enter them in the text boxes below. [?]
x: y:Saying that the decimal digits of π starts with 3.1415 is the same as saying that π is between what two numbers?
Similar to talking about real numbers in terms of their decimal digits, we can also talk about real numbers as falling into intervals of values with the endpoints of those intervals being some particularly nice numbers, such as those with only finitely many decimal digits. For instance, we can talk about π as being somewhere in the interval [3.1415,3.1416], that is, π is somewhere between 3.1415 and 3.1416.
Interval arithmetic then allows us to use these intervals to figure out what intervals other numbers fall into.
Just knowing that π falls between 3.1415 and 3.1416, and that e falls between 2.7182 and 2.7183, what is the largest and smallest possible value e+π could be?
It might be easier to answer this sort of question if we used variables instead of constants that have fixed values:
Just knowing that a falls between 1 and 2, and that b falls between 3 and 4, what is the largest and smallest possible value a+b could be?
The interactive below will allow you to explore this problem on the x-axis: The x-coordinate of Point 1 is restricted to being between 1 and 2 and the x-coordinate of Point 2 is restricted to being between 3 and 4. Click and drag the points to change their x-coordinates and the point marked "Sum" will move so that its x-coordinate is the sum of the x-coordinates of Points 1 and 2. What is the largest value you can get the x-coordinate of the Sum to be? What is the smallest?
At the same time, a similar problem is happening in the y direction, and the midpoint and distance between the two points is also shown. You can ask similar questions about the midpoint and distance:
Once you know a range that a value falls into, you can figure out what you know about its decimal digits. For instance, once we know that e+π falls between 5.8597 and 5.8599, we are certain that its decimal expansion starts with 5.859, and that the ten thousandths digit is either 7 or 8.
One may get the impression of interval arithmetic as being really quite easy from the interactive above. To figure out the largest possible value of a + b, simply add the largest possible value of a and the largest possible value of b. To figure out the smallest possible value of a + b, simply add the smallest possible values of a and b. The range of possible values of a + b is then the interval from the smallest value of a + b to the largest.
But some situations require a more subtle approach:
Knowing that a is between -1 and 2, what are the possible values of a2?
The interactive below will allow you to explore these sorts of questions. You can adjust the left and right end of the bar below the function by clicking and dragging the bars at the ends. This will change the interval of input values, and you can watch as the interval of corresponding output values changes.
|
Enter a function: | |