Is Mathematics Really An Exact Science? A Paradoxical Equation

A frustrated woman, and all around are mathematical formulas.

Mathematics is often regarded as an exact science, built on rigorous axioms and logical consistency. However, there exist cases that challenge this perception. One such case involves the equation:

1^x = -1

At first glance, it seems clear that this equation has no real solutions, as for any real value of x, will always be equal to 1 and never -1. But is this truly the case? Let’s take a deeper look.

Understanding the Problem

Revisiting Complex Numbers

To explore this equation further, we introduce the function:

f (α) = cos α + i sin α

where is i = √(-1) the imaginary unit. Differentiating this function, we obtain:

f´(α) = -sin α + i cos α

Since i² = -1, we can rewrite it as:

f´(α) = i (cos α + i sin α)

or simply:

f´(α) = i f (α)

f´(α) / f (α) = i

Solving the Differential Equation

Rearranging and integrating both sides (taking as integration constant ln C):

∫ (f´(α) / f (α)) dα = i ∫dα + ln C

ln f (α) = i α + ln C

ln f (α) – ln C = i α

Using logarithmic properties:

ln (f (α) / C) = i α

Exponentiating both sides:

f (α) / C = e^iα

Thus, we obtain:

f (α) = C e^iα

The integration constant C is determined for α = 0. Then, based on the initial formula, it will be:

f (0) = cos 0 + i sin 0

f (0) = 1 + 0

f (0) = 1

Since f (α) = C e^iα, for α = 0, it follows that f (0) = C e^0 = C.

So, we have that f (0) = 1 and f (0) = C, which means that:

C = 1

Thus, we arrive at Euler’s formula:

e^iα = cos α + i sin α

Applying Euler’s Formula

If α = 2 k π or α = (2 k + 1) π, where k ∈ Z, we have:

cos 2 k π = 1

sin 2 k π = 0

cos (2 k + 1) π = -1

sin (2 k + 1) π = 0

so it is easy to conclude that it will be:

1 = e^i2kπ

1^x = (e^i2kπ)^x = e^i2kπx

-1 = e^i(2k+1)π

After substituting into the equation 1^x = -1, we get:

e^i2kπx = e^i(2k+1)π

Since the bases on both sides of the equals sign are equal, the exponents must also be equal, so:

i 2 k π x = i (2 k + 1) π

From here we get that the solution to the equation 1^x = -1 is:

x = (2 k + 1) / 2 k, k ∈ Z, k ≠ 0

Therefore, there are infinitely many solutions to the equation 1^x = -1 in the set of real numbers. The solution of this equation is any rational number where the numerator is odd and one greater than the denominator, where, of course, the denominator must be different from zero.

Verifying the Solutions

Substituting into the Original Equation

Let’s check.

If k = 1, then x = 3 / 2, so:

1^x = 1^(3/2) = √(1^3 ) = √1 = 1

If k = 2, then x = 5 / 4, so:

1^x = 1^(5/4) = ∜(1^5 ) = ∜1 = 1

So, something is wrong, because the check concludes that 1^x is always 1, not -1, regardless of the fact that the solution to the equation shows otherwise.

A Different Approach

On the other hand, if the original equation 1^x = -1 is taken as the x-th root, it will be:

1 = (-1)^(1/x)

If k = 1, then x = 3 / 2, so 1 / x = 2 / 3, which means:

(-1)^(1/x) = (-1)^(2/3) = ∛(-1)^2 = ∛1 = 1

If k = 2, then x = 5 / 4, so 1 / x = 4 / 5, which means:

(-1)^(1/x) = (-1)^(4/5) = 1

So, when the equation is viewed from this side, then the check proves that there are indeed infinitely many real solutions.

Conclusion: The Paradox of Mathematical Exactness

This example illustrates a deeper issue: the interpretation of mathematical rules can influence the existence of solutions. Depending on the approach taken, the equation either has no solution or infinitely many solutions.

This paradox challenges the belief that mathematics is an entirely exact science. Instead, it suggests that even within mathematics, the axioms and methods of analysis can shape our conclusions, much like in “non-exact” sciences.

What do you think? Is mathematics truly absolute, or does its truth depend on the way problems are formulated?

7 thoughts on “Is Mathematics Really An Exact Science? A Paradoxical Equation”

  1. Wow your article just blew my mind! I didn’t know that the current state of mathematics as an exact science could be questioned ever. When I was younger, I was even ashamed because I didn’t have the best grades in mathematics and I was even considering myself as an idiot for that. However, I’ve always selected the notions I wanted to explore in mathematics especially when it was about the calculation of interests and money. But what I truly found interesting was the fact that you could use different theories on the same equation and arrive to different solutions…I’ve always thought that you’d always reach the same solution…Ah, mathematics!!!!

    Reply
  2. I didn’t know that Mathematics wasn’t an exact science until reading this. However I find the idea that there can be multiple solutions to the one problem in mathematics to be a comforting notion. I find it comparable to life’s problems in this sense. One individual may find a solution in life that fits them that doesn’t work for another person having a problem with the same set of events but a different personality (way of working). Someone else’s truth does not have to be my own and yet I can still be respectful of their viewpoint and way of thinking.

    Reply
  3. This article presents a fascinating paradox that challenges the perception of mathematics as an exact and absolute science. The equation 1^x = -1 seems impossible at first glance, but through complex number analysis and Euler’s formula, the exploration reveals an unexpected outcome: infinitely many real solutions. However, the verification process contradicts these findings, leading to an intriguing contradiction.

    This paradox highlights a crucial point—mathematical truths are often shaped by the frameworks and conventions we apply. While mathematics relies on rigorous logical structures, different interpretations and methodologies can lead to different conclusions. This is especially evident when transitioning between real and complex number systems or applying different exponentiation rules.

    Ultimately, this discussion raises an important question: is mathematics truly an objective and fixed system, or does its precision depend on the context and definitions we choose? While mathematics may strive for exactness, cases like this remind us that the way we define and interpret mathematical principles plays a significant role in determining what is “true.” This paradox is a great example of how even in mathematics, seemingly absolute truths can be questioned.

    Reply
  4. This deep dive into the equation 1^x = -1 really opened my eyes to how complex mathematics can get, especially when we step outside the realm of real numbers. I found it fascinating how Euler’s formula and the introduction of complex numbers revealed infinitely many solutions, something I wouldn’t have expected for an equation that seems so straightforward at first glance. It’s interesting to see how different approaches, like using roots or exponents, can completely change the interpretation of the problem. This makes me wonder if there are other seemingly simple equations that hold hidden complexities like this one! It really shows that in math, context and perspective are everything.

    Reply
  5. Hey, Slavisa,

    I’ve always seen math as the ultimate truth—until I stumbled into paradoxes like this. The idea that 1^x = -1 could have infinitely many real solutions feels like a trick, but it really highlights how the tools we use (like exponentiation and logarithms) shape our conclusions.

    It reminds me of the first time I encountered Euler’s formula—it felt like magic, yet it worked every time. This problem is a perfect example of how math isn’t just about rigid rules; it’s about perspective, interpretation, and sometimes, a little bit of creative thinking.

    Let me know what you think. I love your articles and would like to come back for more.

    John

    Reply
  6. -Honestly, I just think it’s absolute truth. 

    -There’s SUPPOSED to be infinitely many solutions this time; the equation itself SAYS so. 

    -It also really depends on WHAT you’re trying to make the equation do, exactly; either come up with the proof(s) OR show that none such exist; but that does not seem possible this time. 

    -Maybe I’m wrong, but that is what I think about Mathematics now, up to this point. 

    ALEJANDRO G.

    Reply
  7. To summarize:

    Always, when the term 1^x appears, it is taken without thinking that it is 1, for any x.

    It is fascinating, however, that using Euler’s formula (which I proved here) shows that the equation 1^x = -1 has infinitely many real solutions, even though almost everyone thinks that this equation has no solutions.

    The solution of this equation is any rational number, where the numerator is odd and one greater than the denominator, where, of course, the denominator must be different from zero.

    What is even more strange is that when directly checking the solution of this equation, it turns out that 1^x equals 1, not -1.

    However, by classically rearranging the equation 1^x = -1 and checking afterwards, it is confirmed that this equation has infinitely many solutions.

    That’s why I gave the title of the article, because on the one hand, the given equation has no solution, and on the other – there are infinitely many solutions.

    It is disappointing that through this example, mathematics shows the characteristic of “non-exact” sciences, where something can be both true and false at the same time, depending on the way of looking at the problem.

    Reply

Leave a Comment