Use the interval method in math

Intervals in math - what is it?

• The term "interval" occurs not only in musicology, but also in mathematics. There it is a precisely limited, coherent subset of another set, usually a range of numbers.
• Intervals are given in square brackets. The specification [0,1] means the set of all numbers between zero and one. This interval also includes, for example, the numbers 0.5 and 0.99. The two limits 0 and 1 also belong to this interval - it is referred to as closed. Open intervals to which the marginal numbers do not belong are marked with round brackets.
• The interval method is about finding a number (for example a periodic fraction or a root) as precisely as you want by continuously reducing an interval.
• For example, the periodic fraction 1/3 lies in the interval [0.3, 0.4]. A more precise limitation, however, is provided by the intervals [0.33, 0.34], [0.333, 0.334] and so on.

Extracting roots with the interval method - this is how it works

As a student, you will probably encounter the interval method for the first time when you remove the square root of a given number calculator, so "on foot" should only be determined by arithmetic. As an example of the procedure, the square root of 7 should be calculated with an accuracy of two places behind the decimal point:

1. Assuming some basic knowledge in square numbers, the following applies: 2
2. Now restrict the found interval a little to the left and right in order to get a more precise result for the root value. For example, 2.5
3. In the next step of the interval procedure, 2.6
4. The sample gives 6.76 <7 <7.29. Now you know that √7 is between 2.6 and 2.7. The first decimal place is therefore a 6.
5. Since the accuracy should be two decimal places, you must now select an interval between 2.6 and 2.7 as a further restriction. For example, you could start with 2.65
6. The left interval limit 2.65 was chosen too large. A clever choice at this point is 2.64
7. Squaring the sample confirms your consideration, because the following applies: 6.97 <7 <7.02. So √7 lies in the interval [2.64, 2.65] and you have found √7 = 2.64 to two decimal places.
8. Check the result with the calculator! You will be amazed how accurate the result is.

By the way: The interval method can be continued in order to calculate the root even more precisely, i.e. with even more decimal places. However, you will have to struggle with this Counting to square in writing for the sample, because strictly speaking no pocket calculator is allowed here either. Fortunately there is in the mathematics More options, root "on foot" to pull.