Test your mathematical communication skills.

The following exercise is based on “Notes on Writing Mathematics” by Haynes Miller, Department of Mathematics, Massachusetts Institute of Technology. Each sentence or phrase has at least one flaw. How would you edit each one to improve its style or syntax?

1. The Euler’s constant is an irrational number.

2. Since *x* > 0, let *y* = √2.

3. If we draw a line between *P* and *Q*, it is easy to show that . . .

4. There is nothing special about 10, this analysis can be generalized by replacing *a* for 10 as the base.

5. Hence, *x* = 4.

6. The trick is to create matrices *X* and *Y* such that they form . . .

7. We can slowly increase the spring constant such that the system’s natural frequency slowly increases.

8. √2 = 1.414

9. For some finitely small [epsilon] . . .

10. **Definition**. The condition number of a matrix is a measure of how sensitive it is to numerical operations.

11. The induced cycles of a graph are a basis of the graph cycle space.

12. For the equation *X*^{3} + *Y*^{3} = *A*, with *X*, *Y*, and *A* positive integers, there exist infinite numbers *A* such that there are at least two solutions {*X*, *Y*} that satisfy this equation.

For answers and explanations, see “Notes on Writing Mathematics” (Section C: A miscellany of common errors), Haynes Miller, Massachusetts Institute of Technology.