Math CentralQuandaries & Queries


Question from Jen, a student:

If you have (pi/2) and you want to add it to (12/13), how would you go about it?


I'm sorry to report it, but
$$\frac{\pi}2 + \frac{12}{13}$$
is as simple an expression as you are going to get. Not all real numbers have a symbol to represent them. To understand what this means you can compare your given numbers with counting numbers, such as 2 and 3. Every counting number has its own symbol, and the sum of two counting numbers is a counting number. So, for example, $2 + 3 = 5$, and it is easy to understand what addition means for counting numbers. Same for fractions, although some people find adding fractions complicated: the sum of two fractions is a fraction, as for example,

$$\frac12 + \frac{12}{13} = \frac{13+24}{2\cdot 13} = \frac{37}{26}.$$

The symbols for counting numbers and fractions are agreed upon all over the world, and addition for these numbers is easily understood. Some other important numbers have their own "nice" symbols such as $\sqrt 2$ and $\pi$; but others do not, such as $\sqrt 2 + 3$ and $\pi+3$ and $\sqrt 2+\pi$. There are simply more numbers than there are symbols for them!

Of course, you can use your calculator to find out approximately what these sums are by using decimal approximations. Thus, $\frac{\pi}2$ is approximately $1.571$, while $\frac{12}{13}$ is approximately $.923$. From this you know that
$\frac{\pi}2 + \frac{12}{13}$ is approximately $2.494$. But this just tells you that the sum is some number between $2.4935$ and $2.4945$; it does not tell you exactly what that sum is.


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