one to one function

[musicgenes]

I am reading a book called calculus made easy. I have always wanted to learn the subject and now that I am retired and decided to do so. I did well in pre calculus some 30 plus years ago and reviewed the subject through a video I purchased from the Teaching Company.

I am studying what is a function and have a question the book discusses as follows:

A slightly more complicated example of a one to one function is the relation of a square's side to its diagonal. So I know the following:

a squares diagonal is the hypotenuse of an isoceles right triangle. Isoceles meaning two sides of a triangle are the same as are two of the angles.

Pythagorean therorem means that the square of the hypotenuse equals the sum of the other two squares sides.

Now I can visualize the triangle and each square drawn in relation to each side of the triangle. I know that a^2 + b^2 = c^2.

I am trying to figure out if a^2 + b^2 = c^2 are the areas or just the sides as in Area = length x length?

I want to figure out how to express the diagonal as a function of the squares side.

Sorry for the confusion. This might take a few questions before I catch on.

 

[mmm4444bot]

I know that a^2 + b^2 = c^2

I am trying to figure out if a^2 + b^2 = c^2 are the areas or just the sides

Technically speaking, a^2 + b^2 = c^2 is an equation. It does not represent areas or sides.

The equation expresses a relationship between the three sides a, b, and c.

The numbers a^2, b^2, and c^2 are areas of squares having the same respective side lengths a, b, and c.

So, you may think of the equation as modeling a relationship between the three triangle sides as well as between the three related-squares' areas.

If you're interested, check out this graphical explanation of the Pythagorean Theorem (2:21 YouTube video). Cheers :cool:

 

[musicgenes]

Thank You!

Thank you both for taking the time to reply to my question. Both your answers moved me closer to understanding this relationship. I will watch the YouTube video and hopefully gain a better understanding. I think I now see that the pythagorean theory of the right triangle alone is respective side lengths a, b, and c as stated by mmm4444bot and that a^2 + b^2 = c^2 represents the areas of the physical squares attached to each of line a, b and c. I will now try to relate this to JeffM's answer concerning my question about the relationship of the diagonal of a square to one of its sides. I am sure I will have a zillion other questions before I understand the subject of Calculus. I hope to gain enough understanding to take a college calculus class next year.

JeffM how are you able to write a^2 + b^2 = c^2 in the more readable format?

This may be skipping too many steps. If so, just say so.

\(\displaystyle a^2 + b^2 = c^2.\) True of any right triangle.

But the relationship of the diagonal of a square to one of its sides is LESS general.

\(\displaystyle a = b \implies c^2 = a^2 + b^2 = a^2 + a^2 = 2a^2 \implies c = a\sqrt{2}.\)

So for any a = the length of a side of a square, there is a UNIQUE length for the diagonal.

Did this help? It is a unique relationship between lengths given the length of any side.

Technically speaking, a^2 + b^2 = c^2 is an equation. It does not represent areas or sides.

The equation expresses a relationship between the three sides a, b, and c.

The numbers a^2, b^2, and c^2 are areas of squares having the same respective side lengths a, b, and c.

So, you may think of the equation as modeling a relationship between the three triangle sides as well as between the three related-squares' areas.

If you're interested, check out this graphical explanation of the Pythagorean Theorem (2:21 YouTube video). Cheers :cool: