Everyone’s heard about chatGPT, the latest and most sophisticated chatbot to date.We all know it can bs proficiently about ‘soft’ topics like English literature. I decided to quiz it about a hard topic, mathematics. As you probably know, I have a PhD in math, so I won’t go easy.

Let’s start with some basic high school maths.

What is the Euclidean distance between two points (a,b) and (x,y)?

In Euclidean geometry, the distance between two points (a,b) and (x,y) is given by the square root of the difference of the squares of the x-coordinates plus the difference of the squares of the y-coordinates. This can be expressed mathematically as:

√((x - a)^2 + (y - b)^2)

For example, the distance between the points (3,4) and (6,8) would be:

√((6 - 3)^2 + (8 - 4)^2) = √(3^2 + 4^2) = √25 = 5

It is important to note that this formula only applies in Euclidean geometry, where the notion of distance is based on the Pythagorean theorem. In other types of geometry, the concept of distance may be different, and the formula for calculating distance will vary accordingly.

Should be “squares of the differences.”. Otherwise OK. It even came up with an example.

Calculate an approximation to the square root of 10 using Newton’s method.

To find the square root of 10 using Newton’s method, we can follow the steps outlined above. Here is an example of how this might work:

1. Start with an initial guess for the square root of 10. Let’s say we guess that the square root of 10 is 3.
2. Use the initial guess to calculate a new approximation for the square root of 10. The formula for this is: x1 = (x0 + (n / x0)) / 2

Substituting our values into this formula, we get: x1 = (3 + (10 / 3)) / 2 = 5

It screwed up the arithmetic. Not 5, but 3.166.

3. Use the new approximation to calculate a better approximation for the square root of 10. The formula for this is the same as before: x2 = (x1 + (n / x1)) / 2

Substituting our values into this formula, we get: x2 = (5 + (10 / 5)) / 2 = 3.4

4. Repeat this process until the desired accuracy is achieved. In this case, we can stop after just two iterations, since our approximation is already accurate to within 0.1.

Therefore, our final approximation for the square root of 10 is 3.4. This is only an approximation, but it is a very good one, and it is much faster and easier to calculate than the exact square root of 10.

3.4 is not a very good approximation.

What is the integral of 1/x for x=1 to x=2?

The integral of 1/x for x=1 to x=2 is the area under the curve of the function 1/x from x=1 to x=2. This area can be found by using the definite integral, which is a mathematical tool used to calculate the area under a curve.

The definite integral of a function from a to b is defined as the limit of a sum of areas of rectangles under the curve of the function as the widths of the rectangles approach 0. In other words, it is the limit of the sum of the areas of many very thin rectangles that cover the area under the curve of the function from a to b.

In order to find the definite integral of 1/x from x=1 to x=2, we can use the formula for the definite integral:

∫f(x)dx = lim(Δx→0) ∑f(x)Δx

where f(x) is the function we are finding the integral of, x is the variable, and Δx is the width of the rectangles.

In this case, our function is 1/x, so we can plug this into the formula to get:

∫(1/x)dx = lim(Δx→0) ∑(1/x)Δx

We can then use this formula to calculate the definite integral of 1/x from x=1 to x=2 by summing the areas of many very thin rectangles under the curve of 1/x from x=1 to x=2. The result of this calculation is the exact value of the definite integral of 1/x from x=1 to x=2.

In general, the definite integral of 1/x from x=1 to x=2 is equal to the natural logarithm of 2, which is approximately 0.693. This means that the area under the curve of 1/x from x=1 to x=2 is approximately 0.693.

The explanation is irrelevant but the answer is correct.

What is the integral of 1/(1+x^2)?

The integral of 1/(1+x^2) is the inverse tangent function, denoted as atan(x) or tan^(-1)(x). To see why this is true, we can start by rewriting the fraction as the difference of two squares:

1/(1+x^2) = (1 – x^2)/(1+x^2)

(This is just wrong.)

Next, we can use the formula for the derivative of the inverse tangent function to simplify this further:

d/dx atan(x) = 1/(1+x^2)

Therefore, the integral of 1/(1+x^2) is the inverse tangent function.

This is correct.

What is a measure? (Not on the high school curriculum.)

In the context of mathematics, a measure is a way of assigning a numerical value to a set in order to capture its size or extent. For example, the measure of a line segment might be its length, the measure of a rectangle might be its area, and the measure of a set of points might be the number of points it contains. Measures are often used in integration and probability theory, among other branches of mathematics.

Totally correct.

Does every set have a measure?

Not every set has a measure. In order for a set to have a measure, it must satisfy certain properties, such as being countably infinite or having a well-defined notion of “size” or “length.” For example, the set of real numbers has a measure (namely, its Lebesgue measure), but the set of all subsets of the real numbers does not have a measure. In general, only certain types of sets can be assigned a measure in a meaningful way.

Completely correct.

What is the measure of the union of two sets?

The measure of the union of two sets is the sum of the measures of the individual sets. This is true because the union of two sets can be thought of as the sum of the elements in the sets, where each element is counted only once, even if it appears in both sets. For example, if the measure of set A is 3 and the measure of set B is 5, then the measure of the union of A and B is 8.

This is wrong.

Only if they are disjoint

The statement “the measure of the union of two sets is the sum of the measures of the individual sets” is only true if the two sets are disjoint. If the two sets have one or more elements in common, then the measure of their union will be the sum of the measures of the individual sets minus the measure of the intersection of the sets. This is because the intersection of the sets will be counted twice in the union, so we need to subtract it out in order to avoid overcounting. For example, if the measure of set A is 3, the measure of set B is 5, and the measure of the intersection of A and B is 2, then the measure of the union of A and B is 6 (the sum of the measures of the individual sets minus the measure of the intersection).

GPT corrects itself. This is right.

What is the measure of the set of real numbers between 0 and 1?

The measure of the set of real numbers between 0 and 1 is 1. This is because the set of real numbers between 0 and 1 is a continuous interval on the number line, and the measure of any continuous interval is equal to the difference between its upper and lower bounds. In this case, the upper bound is 1 and the lower bound is 0, so the measure of the set of real numbers between 0 and 1 is 1.

Absolutely right.

What is the measure of the set of irrational numbers between 0 and 1?

The measure of the set of irrational numbers between 0 and 1 is also 1. This is because the set of irrational numbers between 0 and 1 is a subset of the set of real numbers between 0 and 1, and the measure of a subset is always less than or equal to the measure of the set it is a subset of. In this case, the measure of the set of irrational numbers between 0 and 1 is equal to the measure of the set of real numbers between 0 and 1, which is 1.

Right, although the reasoning is incorrect.

What is the measure of the set of rational numbers between 0 and 1?

The measure of the set of rational numbers between 0 and 1 is 0. This is because the set of rational numbers between 0 and 1 is a countable set, which means that it can be put into one-to-one correspondence with the set of natural numbers (i.e. the set of whole numbers including 0: {0, 1, 2, 3, …}). Because the set of natural numbers has measure 0 (it is a countable set and the measure of any countable set is 0), the measure of the set of rational numbers between 0 and 1 must also be 0.

Right again.

What is the measure of the set of algebraic numbers between 0 and 1

The measure of the set of algebraic numbers between 0 and 1 is 0. This is because the set of algebraic numbers is a countable set, which means that it can be put into one-to-one correspondence with the set of natural numbers (i.e. the set of whole numbers including 0: {0, 1, 2, 3, …}). Because the set of natural numbers has measure 0 (it is a countable set and the measure of any countable set is 0), the measure of the set of algebraic numbers between 0 and 1 must also be 0.

Darn! It knows about the algebraic numbers! (A number is algebraic if it’s a root of a polynomial with rational coefficients.)

What is the measure of the intersection of two sets of measure 0?

The measure of the intersection of two sets of measure 0 is 0. This is because the measure of the intersection of two sets is always less than or equal to the measure of each individual set. Since the measure of each individual set is 0, the measure of their intersection must also be 0

Perfect reasoning!

What is the measure of the intersection of two sets each of measure 1/2?

The measure of the intersection of two sets is the size of the overlap between the two sets. The measure of a set is a number that indicates the size or magnitude of the set. In the case of two sets each with a measure of 1/2, the measure of their intersection would be 1/2, since the measure of a set is a property of the set itself and is not affected by the presence of other sets.

Wrongity wrong wrong wrong. If the sets are disjoint, the measure of the intersection is 0. If they overlap at all, the measure could be anything between 0 and 1/2.

Well, I’m going to quit while I’m ahead. Chat GPT did much better than I expected, although it’s obviously not infallible. I was impressed by two things: the way it manipulated the variables in the distance example, and several instances of apparently using reasoning. For example, it seems to have reasoned as follow: the algebraic numbers are countable; any countable set has measure 0; therefore the algebraic numbers have measure 0.

Based on my teaching experience, I’d say chatGPT scored like a second year math major. Except for the arithmetic errors! Two cheers for chatGPT!