Some Infographics

One of the trends in 2025 was prompting Gemini to create interesting infographics. In this post, I'll display three interesting mathematical infographics that I prompted Gemini to created.

Prompt 1: Create an infographic comparing real, complex, quaternion and octonian numbers.

Prompt 2: Create an infographic highlighting the differences between deficient, perfect and abundant numbers with clear examples to illustrate each type of number.

Prompt 3: Create an infographic showing how to derive the Zeckendorff representation of a number using the Fibonacci base system.

These infographics are certainly concise and informative and I'll create more in the future. For the time being, I'll go back over my previous posts and insert these three into their the appropriate posts.

Complex Chicanery

I came across an interesting video on YouTube, uploaded today on a channel named Higher Mathematics that posed the following problem:$$ \text{Solve for } x:\\1^x=5$$Clearly the problem has no solution amongst the real numbers but there turns out to be an infinity of solutions once we introduce complex numbers. The solution (with \(k=1,2,3 \dots \) ) then unfolds:$$\begin{align} e^{i \theta} &= \cos{\theta}+i \sin{\theta}\\e^{i 2k \pi} &= \cos{2k \pi}+i \sin{2k \pi}  \\&=1 \\ \therefore (e^{i 2k \pi)})^x &= 5\\ e^{i 2k \pi x} &= 5 \\\ln {e^{i 2k \pi x} }&= \ln{5} \\i 2k \pi x &= \ln{5} \\x &= \frac{ \ln{5}}{i 2k \pi} \\\ &=\frac{-i \ln{5}}{2k \pi}\end{align} $$The complex numbers for values of x given by \(k=1, 2,3\) are shown in Figure 1. They all map to the point \(5+0i\).

Figure 1

Thus we have a function \( f(x) \) with its domain D being the countably infinite set of points:$$ \left( \Big( 0,\dfrac{-i \ln{5}} {2k \pi} \Big) \text{ where k=1, 2, 3, ...} \right) $$ that maps all these points to \( (5, 0i) \) so that:$$ f(x) \rightarrow (5, 0i) \text{ for all x in D}$$

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Let's look now at a slightly different problem:$$2^x=x$$Right from the start, it can be said once again that there are no real solutions to this equality. This is clear once we graph the line \(y=2^x\) and \(y=x\) where we see that there are no points of intersection. See Figure 2.

Figure 2

This problem was posed on the same YouTube channel mentioned earlier. Here is a link to the video. So how do we find if any complex numbers satisfy this equality? Firstly, we take the natural logarithms of both sides and proceed from there:$$ \begin{align} 2^x &= x \\ \ln{2^x} &= \ln{x} \\ x \ln{2} &= \ln{x} \\ \frac{\ln{x}}{x} &= \ln{2} \\ \frac{\ln{x}}{e^{\, \ln{x}} }&= \ln{2} \\ \ln{x} \, {e^{\, -\ln{x}} } &= \ln{2} \\ -\ln{x} \, {e^{\, -\ln{x}} } &= -\ln{2} \\ W(-\ln{x} \, {e^{\, -\ln{x}} } ) &= W(-\ln{2}) \\ -\ln{x} &= W(-\ln{2} ) \\ \ln{x} &= -W(-\ln{2}) \\ x &= e^{-W(-\ln{2})} \\ x &= \frac{1}{e^{W(-\ln{2})}} \\ x &\approx -0.37927 - 0.72087 i\end{align}$$where W is the Lambert W function that I've written about in previous posts. See The Omega Constant and the Lambert W Function (June 24th 2020) and More on the Lambert W Function (February 16th 2021). To evaluate \(x\) using Wolfram Alpha the command ProductLog[-ln(2)] needs to be used.