Helping Ada Lovelace with her Homework: Classroom Exercises from a Victorian Calculus Course – Problem 3

Lovelace was no stranger to matters concerning complex and imaginary numbers. Nevertheless, the next problem showed that, despite her familiarity with them, her algebraic skills still needed improvement. While reading an article in the Penny Cyclopaedia by De Morgan on “Negative and Impossible Quantities”,⁴ she came across the following passage [De Morgan 1840, 134]:

It can be easily shown that any algebraical expression, however complicated, which is a function of \(\sqrt{(-1)}\) can be reduced to the form \(A+B\sqrt{(-1)}\), where \(A\) and \(B\) are possible quantities. For instance (\(k\) being \(\sqrt{(-1)}\) \[ (a+bk)^{m+nk}=e^A  \cos⁡ B+k e^A  \sin⁡ B \,\,\,\,\,\,\,\,   [1]  \]
where \(A\) and \(B\) are determined as follows. Let \[r=\sqrt{(a^2+b^2)},  \,\, \mbox{ [and] } \,\,  \tan\theta=\frac{b}{a}, \,\, \mbox{[then]} \] \[A=m \log ⁡r-n\theta, B=n \log r⁡   +m\theta.\]

In a letter to De Morgan, Lovelace reported [LB 170, 9 Sept. [1841], f. 123r] that she had ‘tried a little to demonstrate this Formula’ (i.e. [1]), but was having trouble. Based on the assumption that \(\tan⁡ \theta=\frac{b}{a}\)  gives \(\sin⁡ \theta=b\) and \(\cos⁡\theta=a\), she obtained

\[\begin{array}{rcl}
(a+bk)^{m+nk}&=& (\cos⁡ \theta+k \sin⁡\theta)^{m+nk}\\
&=&\left (e^{k\theta} \right)^{m+nk}\\
&=&e^{k(m\theta)}\times \{(e^{k(n\theta)} )\}^k \\
&=&(\cos ⁡m\theta+k \sin ⁡m\theta )\times (\cos ⁡n\theta+k \sin ⁡n\theta)^k.\end{array}\]

However, since she could get no further, she believed that the complete demonstration ‘must be a very complicated process’ [LB 170, 9 Sept. [1841], f. 123v]. In fact, she was closer than she realized but, despite a clear working knowledge of the algebra of exponents, and indeed Euler’s Formula, it was again her lack of algebraic and trigonometric experience (and consequent intuition) that ultimately prevented her from obtaining the desired result.

Can you spot Lovelace’s mistake and prove formula [1]? Click here to see the proof.

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Continue to Problem 4.

Notes

4. The phrase ‘impossible quantity’ was commonly used at this time to mean an imaginary or complex number.