The circumference of a circle bears a constant ratio ∏ to its diameter
The surface area of a sphere bears a constant ratio ∏ to its square of its diameter.
I dont understand these statements. What they imply?
Revered members, please help

 

The circumference of a circle bears a constant ratio ∏ to its diameter
The surface area of a sphere bears a constant ratio ∏ to its square of its diameter.
I dont understand these statements. What they imply?
Revered members, please help

Those statements are just expressing some things we know about circles and spheres. That is, as follows:

1. The ratio of circumference (C) of a circle to diameter (D) is the same for every Euclidean circle (C/D=pi)
2. The ratio of surface area (A) of a sphere to the diameter (D) squared is the same for every Euclidean sphere (A/D^2=pi)
3. These constant ratios are equal to each other and turn out to be a transcendental number that we chose to call pi and represent by the Greek letter \(\pi\).

 

Thanks steveb.What is meant by the word transcendental number? General meaning of the word is " superior or surpassing all others". But i presume you used that word in mathematical context.

 

Googled "transcendental number " and here's the one from wikipedia - In mathematics, a transcendental number is a number (possibly a complex number) that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e.

 

Thanks steveb.What is meant by the word transcendental number? General meaning of the word is " superior or surpassing all others". But i presume you used that word in mathematical context.

Above you see there are official mathematically correct definitions of transcendental numbers. I don't really know them myself, but I trust wmodavis knows what he's talking about.

Before progressing to those definitions, it is helpful to keep in mind the simplest notion that we learn in elementary math classes. A key characteristic of both pi and e is that they are numbers that can't be represented as fractions of two integers. Another way to say this is that their decimal representations go on forever without ever repeating.

For example 1/3=0.3333333.... , with the 3 repeating forever, is a rational number and is not a transcendental number. However pi=3.141592653589793...... goes on forever without repeating any pattern of numbers. So, it is transcendental.

When we think of the real number line, we have to include both rational numbers (ratios of integers) and transcendental numbers of this type.