Appendix
A worked example
The following examples A and B are solutions to the same question.
Example A is a bare answer, merely showing working; whereas Example B is a full solution in good grammar with clear explanation of the processes used. Example A is fine as draft work but is unlikely to be of use for future revision reference without annotation.
Example A: Solve \(4\cosh(x)+3\sinh(x)=4\).
\(\frac{4}{2} (e^x + e^{-x}) + \frac{3}{2} (e^x-e^{-x}) = 4\)
\(\frac{7}{2}e^x + \frac{1}{2}e^{-x} = 4\)
\(7e^x+e^{-x} = 8\)
Put \(y=e^x\)
\(7y+\frac{1}{y} = 8\)
\(7y^2-8y+1=0\)
\((7y-1)(y-1)=0\)
\(y=\frac{1}{7} \mbox{ or } y=1\)
\(e^x=\frac{1}{7} \mbox{ or } e^x=1\)
\(x=\ln(\frac{1}{7}) \mbox{ or } x=\ln(1)\)
\(x=\ln(1)-\ln(7) \mbox{ or } x=0\)
\(x=-\ln(7) \mbox{ or } x=0.\)
Example B: Solve \(4\cosh(x)+3\sinh(x)=4\).
Using the definitions of \(\cosh(x)=\frac{1}{2} (e^x + e^{-x})\) and \(\sinh(x)=\frac{1}{2} (e^x - e^{-x})\) we try and find values of \(x\) so that \[\frac{4}{2} (e^x + e^{-x}) + \frac{3}{2} (e^x-e^{-x}) = 4.\] Collecting terms gives \(\frac{7}{2}e^x + \frac{1}{2}e^{-x} = 4\) which simplifies to \[7e^x+e^{-x} = 8.\] Put \(y=e^x\) (so that \(e^{-x}=\frac{1}{y}\)) then \[7y+\frac{1}{y} = 8.\] Multiplying by \(y\) and rearranging gives the quadratic \(7y^2-8y+1=0\) which factorises as \[(7y-1)(y-1)=0.\] The solutions are \(y=\frac{1}{7} \mbox{ or } y=1\). Remembering that \(y=e^x\) gives \(e^x=\frac{1}{7} \mbox{ or } e^x=1\), so that \(x=\ln(\frac{1}{7}) \mbox{ or } x=\ln(1)\). Using the rules of logarithms \(x=\ln(1)-\ln(7) \mbox{ or } x=0\) gives the final solutions \[x=-\ln(7) \mbox{ or } x=0.\]
Common mathematical symbols
The Greek Alphabet
The Greek alphabet is regularly used in mathematics.
| Lower case | Upper case | Greek Name |
|---|---|---|
| \(\alpha\) | \(A\) | alpha |
| \(\beta\) | \(B\) | beta |
| \(\gamma\) | \(\Gamma\) | gamma |
| \(\delta\) | \(\Delta\) | delta |
| \(\varepsilon\) | \(E\) | epsilon |
| \(\zeta\) | \(Z\) | zeta |
| \(\eta\) | \(H\) | eta |
| \(\theta\) | \(\Theta\) | theta |
| \(\iota\) | \(I\) | iota |
| \(\kappa\) | \(K\) | kappa |
| \(\lambda\) | \(\Lambda\) | lambda |
| \(\mu\) | \(M\) | mu |
| \(\nu\) | \(N\) | nu |
| \(\xi\) | \(\Xi\) | xi |
| \(o\) | \(O\) | omicron |
| \(\pi\) | \(\Pi\) | pi |
| \(\rho\) | \(P\) | rho |
| \(\sigma\) | \(\Sigma\) | sigma |
| \(\tau\) | \(T\) | tau |
| \(\upsilon\) | \(\Upsilon\) | upsilon |
| \(\phi\) | \(\Phi\) | phi |
| \(\chi\) | \(X\) | chi |
| \(\psi\) | \(\Psi\) | psi |
| \(\omega\) | \(\Omega\) | omega |
Contributors
Contributions to previous editions were made by the following people whilst at the listed institutions:
| Heather Cooke | The Open University |
| Chris Good | The University of Birmingham |
| Toby Hall | University of Liverpool |
| Keith Hurst | University of Southampton |
| Michael Howkins | De Montfort University |
| Laurence Nicholas | Heriot-Watt University |
| Neil Pitcher | University of Paisley |
| Chris Sangwin | The University of Birmingham |
| Brian Smith | University of Central England |
| Jane Williams | The Open University |
Books
Some helpful books.
Courant, R., Robbins, H. and Stewart, I., What is Mathematics? OUP, 1996.
Polya, G., How to Solve It, Penguin, 1990.
Kahn, P., Studying Mathematics and its Applications, Palgrave, 2001.
Creme, P. and Lea, M.R., Writing at University, Open University Press, 2008.
Hurst, E. and Gould, M., Bridging the Gap to University Mathematics, Springer, 2009.
Tracey, E., The Student’s Guide to Exam Success, Open University Press, 2002.
Alcock, L. How to study for a Mathematics Degree, Oxford UP, 2012.
Pappas, T., The Joy of Mathematics, Wide World Publishing, 1993.
Earl, R. and Nicholson, J., The Concise Oxford Dictionary of Mathematics, OUP 2021.
Houston, K., How to Think Like a Mathematician: A companion to Undergraduate mathematics, Cambridge University Press, 2013.
Alcock, L., Mathematics Rebooted: A Fresh Approach to Understanding, OUP, 2017.
Nelson, D., The Penguin Dictionary of Mathematics, Penguin, 1998.
It is interesting to find out where your maths came from! If you want to know more about the people behind the names of your Theorems and formulae, a good place to start is the University of St. Andrews history of mathematics pages: https://mathshistory.st-andrews.ac.uk/