Abstract
This chapter deals with the use of numerical variables inside math formulas. What does it mean “to apply a formula?” Why and how indices are used in math formulas? These issues are addressed first by explaining the use of summations (“sigma notation”) and products together with their properties and typical applications. Then we consider numerical functions and their compositions.
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Notes
- 1.
See the discussion in Sect. 5.1 about the use of the word “any” in mathematics.
- 2.
There is a famous anecdote about the formula (7.1.1), involving the great mathematician C. F. Gauss (1777–1855). It is told that, when Gauss was a child, one day his school teacher ordered to his pupils, as a punishment, to compute the sum of the numbers from 1 to 100. While his classmates were still doing heavy computations, the little Carl Friedrich wrote the solution on a chalkboard. Actually, at the age of nine, he had already discovered (7.1.1).
- 3.
Whenever a formula “does not convince us,” it is useful to check it on some numerical example:
$$\displaystyle{ \sum _{k=2}^{5}c =\mathop{ c}\limits_{\text{corresponds to }k = 2}+\mathop{c}\limits_{\text{corresponds to }k = 3}+\mathop{c}\limits_{\text{corresponds to }k = 4}+\mathop{c}\limits_{\text{corresponds to }k = 5} = 4c\;. }$$ - 4.
Why? After all we know that 00 makes no sense. The point is that we see ∑k=0 nqk as a function of the real variable q (for every n) and we have q0 = 1 for every q ≠ 0. Hence it is natural to let qk = 0 for q = 0 and k = 0.
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Bramanti, M., Travaglini, G. (2018). Formulae and Indices (Level A). In: Studying Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-91355-1_7
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DOI: https://doi.org/10.1007/978-3-319-91355-1_7
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