Question

Determine a definition for what it means to add two real numbers. Then, prove that the real numbers are closed under addition.

Answer 1

Defination of sum of two real number : Suppose $\small x$ and $\small y$ be two real number then addition of two real number  $\small x$ and $\small y$ is denoted by and is defined by ,

y

The number can be obtained by moving the length of the segment joining 0 and y from the point coressponding to x to the right if y is positive and to the left if y is negative .

y

Now for any two real number $\small x$ and $\small y$ their sum represents length of a ponit from the origin which is also an real number . Hence real numbers are closed under addition .