Comparison with Competitors
Since we offer a comprehensive mathematics application, our competitors include not only math apps like Photomath, Mathway, Microsoft Math, and others but also large language models (LLMs). The advantages of our approach over LLMs are summarized in the section on word problems. Here, we focus on comparing the purely mathematical capabilities of our app, specifically comparing our solution with the most well-known app, Photomath.✱ Date of comparison: February 2025
Elegant Solution #1
- We use well-established methods such as the zero-point method
- This method is applied not only to absolute value equations and inequalities but also to more complex equations/inequalities in product or quotient form.
- The final solution is much clearer and more comprehensible.
- We include all steps (unlike other apps, which often omit some).
- No need to click through multiple layers—only a single optional click on a blue box for a more detailed explanation.
- We solve different parts of the equation in parallel, just like a human would. This means we don’t first simplify one complex part, then the other, and so on.
- Left: Photomath, Right: Calculio
Elegant Solution #2
- We prioritize smart, elegant solutions over brute force.
- Instead of expanding all brackets, we first rearrange and combine factors, reducing the problem to a simple quadratic equation.
- If you wonder why 6y is rewritten as -984y + 990y, it’s because our competitor artificially creates a quartic equation, which they then factorize numerically — without explanation — making it impossible for a student to replicate the process.
- Left: Photomath, Right: Calculio
Elegant Solution #3
- We avoid unnecessary operations unless they are truly needed.
- For example, rewriting a sine power using an identity is an unnecessary step that complicates and prolongs the process.
- Left: Photomath, Right: Calculio
Smart Substitution
- We implement multiple methods of substitution.
- The final solution is typically clearer and more concise.
- Students can often solve problems using basic quadratic equations instead of dealing with cubic or quartic equations, which usually require numerical methods.
- Left: Photomath, Right: Calculio
Systems of Equations #1
- Thanks to sophisticated methods, we handle even complex systems of equations that competitors struggle with.
- In this case, substitution is used to express one variable in terms of the other, which is then substituted into the second equation.
- Left: Photomath, Right: Calculio
Systems of Equations #2
- We implement common tips and tricks for solving systems of equations.
- Every step is thoroughly explained so students can apply the same approach to similar problems.
- Left: Photomath, Right: Calculio
Systems of Equations #3
- Sometimes, seemingly complex systems can be solved with elementary methods like the addition method.
- This quickly reduces the system to a much simpler form.
- Left: Photomath, Right: Calculio
Equations and Inequalities with Parameters
- Calculio is the only app in the world capable of handling this topic.
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Other apps typically:
- Provide no solution at all
- Only display the final result (often incomplete or incorrect) without any steps
- Only express the unknown variable without analyzing boundary cases or applying necessary conditions
- We also include a structured summary at the end of the problem
- Left: Photomath, Right: Calculio
Trigonometric Expressions
- Students often struggle with simplifying trigonometric expressions due to the vast number of formulas and relationships.
- We clearly explain how to simplify trigonometric expressions without unnecessary or convoluted steps.
- Left: Photomath, Right: Calculio
Trigonometric Equations
- We provide graphical representations of the unit circle to make the process clearer.
- Our solutions resemble those found in textbooks but are usually more detailed for better understanding.
- Left: Photomath, Right: Calculio
Trigonometric Inequalities
- Most competing apps do not support solving trigonometric inequalities at all.
- We handle this complex topic in a clear and understandable way.
- We also use graphical representations of the unit circle.
- Left: Photomath, Right: Calculio
Exponential and Logarithmic Equations #1
- We present solutions in a clear and understandable manner.
- We handle even complex equations
- Left: Photomath, Right: Calculio
Exponential and Logarithmic Equations #2
- We demonstrate commonly used tricks for solving challenging textbook problems.
- Left: Photomath, Right: Calculio
Exponential and Logarithmic Equations #3
- We can handle complex combinations, such as exponential equations involving trigonometric expressions.
- Left: Photomath, Right: Calculio
Reciprocal Equations
- A commonly discussed topic in high school mathematics.
- We present solutions clearly and understandably
- Left: Photomath, Right: Calculio
Expression Simplification #1
- We simplify even relatively complex expressions.
- Left: Photomath, Right: Calculio
Expression Simplification #2
- We elegantly simplify numeric trigonometric expressions
- Left: Photomath, Right: Calculio
Deduction #1
- We apply various deduction techniques to simplify calculations — our arsenal includes dozens of them.
- We explain every step in detail, including what, how, and most importantly, WHY we did it.
- Example: The sum of non-negative addends is zero only if each addend is zero
- Left: Photomath, Right: Calculio
Deduction #2
- Demonstration of deduction based on function value ranges.
- Left: Photomath, Right: Calculio
