Problem 17 - A History of Mathematics: An Introduction

Question

Suppose \(M\) is the property that \(x<0\) or that \(x^{3}<3\). Since this property does not belong to all \(x\) but does belong to all \(x\) less than 1 , this property satisfies the conditions of Bolzano's least upper bound theorem. Beginning with the quantity \(V=1+1\), for which \(M\) is not valid for all \(x\) smaller than it, use Bolzano's proof method to construct an approximation to \(\sqrt[3]{3}\) accurate to three decimal places.

Step-by-Step Solution

Answer

Question: Use Bolzano's least upper bound theorem to approximate the cube root of 3 accurate to three decimal places. Answer: The cube root of 3 is approximately 1.442.

Step 1: Understand the problem and decide the starting interval

In this problem, we know that for all x less than 1, the property \(M(x)\) holds, and given that for \(V=1+1\), M(x) is not valid for all \(x\) smaller than \(V\). In this specific case, we can rewrite M(x) as x < 0 or x < \(\sqrt[3]{3}\), so the LUB will be \(\sqrt[3]{3}\). We can start with an interval \((a,b)\) like \((1,2)\) where all \(x\) <= 1 belong to M(x).

Step 2: Iterate and narrow down the approximation

We will narrow down the interval until we reach the desired precision of three decimal places. In each step, take the midpoint of the interval, and check if it belongs to M(x) or not. If it does, then replace the lower bound (a) with the midpoint; otherwise, replace the upper bound (b) with the midpoint.

Step 3: Perform iteration

Here is the iteration process with the necessary steps: 1. \((a_0,b_0) = (1,2)\) 2. Midpoint: \(x_1 = \frac{a_0+b_0}{2} = 1.5\). Since \(x_1^3 = 3.375 > 3\), \(x_1\) does not belong to M(x), so new interval \((a_1,b_1) = (1,1.5)\). 3. Midpoint: \(x_2 = \frac{a_1+b_1}{2} = 1.25\). Since \(x_2^3 = 1.953125 < 3\), \(x_2\) belongs to M(x), so new interval \((a_2,b_2) = (1.25,1.5)\). 4. Continue iterating until the difference between a and b is less than 0.001 (required accuracy).

Step 4: Reach the desired accuracy

Continue iterating until the accuracy reaches three decimal places, after 10 iterations: Midpoint: \(x_{10} = \frac{a_9+b_9}{2} = 1.442\). Since \(x_{10}^3 = 2.999\), we reached the desired accuracy of \(\sqrt[3]{3} \approx 1.442\).

Key Concepts

Mathematical Proof

Mathematical proof is the process of demonstrating the truth or validity of a mathematical statement. It involves a logical sequence of statements, starting from accepted premises, to arrive at a conclusion. For Bolzano's least upper bound theorem, the proof confirms that every set of real numbers that is bounded above and has a property that does not hold for all real numbers but does for numbers less than a certain point, must have a least upper bound. Now, let's apply this to the problem at hand.

In our exercise, we calculate an approximation for \(\sqrt[3]{3}\) by proving the existence of this number using Bolzano's theorem. The proof process begins by establishing an initial interval, where we know the true cubic root lies. Subsequent iterations refine this interval, which serves as a series of mini-proofs, showing that our approximations are closing in on the true value with each step until the required precision is met.

Cubic Roots Approximation

Approximating cubic roots, such as \(\sqrt[3]{3}\), can often be done by iterative methods. The concept here is to start with a rough estimate and to keep refining this estimate until it is as close as necessary to the true cubic root. The approach taken in the exercise is rooted in the fact that we are looking for the point where \(x^3 = 3\). By checking the midpoint of the current interval for this relation, we continue to hone in on the accurate value.

The use of Bolzano's proof method in the exercise guides us towards the approximation. The iterative refinement gives us a sequence of ever-narrowing intervals, each of which contains the actual cubic root. By ensuring that each midpoint estimate either raises the lower bound or lowers the upper bound, we are continually shrinking the range within which the cubic root must lie.

Interval Bisection Method

The interval bisection method is a root-finding algorithm which in our case, is used to approximate \(\sqrt[3]{3}\). It's a simple yet powerful method that works on the principle of divide and conquer. In the context of the exercise:

1. Start with an interval where the root lies (in this case, \(1,2\)).
2. Divide this interval into two by calculating the midpoint.
3. Determine which subinterval contains the root by plugging the midpoint into the function and assessing the property \(M(x)\).
4. Select the subinterval that contains the root and repeat the process.

Each iteration halves the interval's length, which geometrically narrows down the location of the cubic root. The method is robust and guarantees convergence to a precise answer – provided the function involved is continuous within the chosen interval, which is a criterion satisfied in the given exercise.