Mathematical Discovery: Brazilian Marcelo Girard Turns Traditional Addition into a Playful Method with the Girard Theorem – THE GIRARD THEOREM
Marcelo Girard
A discovery made during adolescence and fueled by scientific curiosity is now gaining international attention. Brazilian researcher Marcelo Girard has published the Girard Addition Formula on Zenodo — a platform maintained by the European Organization for Nuclear Research (CERN). This innovative and alternative method for performing mathematical addition holds significant educational and exploratory potential.
Developed at age 14, the formula introduces a nontraditional approach to addition. Through positional decomposition (hundreds, tens, and units), substitution of digits with letters (such as A, B, C, D), and multiplications using correction factors (notably the number 9), the technique offers a playful and visual path to arrive at the same result as standard addition.
“My goal was never to replace traditional math, but to show that it can be presented in a creative and enchanting way,” says Girard.
The Essence of the Formula
The so-called Girard Addition Formula — or Girard Positional Decomposition Theorem (GPDT) — turns the operation of addition into a logical sequence that makes digit positions explicit and reveals hidden patterns in the decimal system. For instance, when adding two-digit numbers like 34 and 27, the formula reaches the same total (61) through a process involving multiplication by 9 on the tens digits.
The general expression for two-digit numbers represented as 10A + B and 10C + D is:
GirardSum = (A + D) + (B + C) + 9A + 9C
This model can be expanded to numbers with more digits by using powers of 10 minus 1 as correction factors — an elegant property of the decimal base system.
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How the Girard Method Works: Step-by-Step Addition
Let’s look at a practical example with the numbers 34 and 27:
Decomposition into letters:
- 34 → A = 3 (tens), B = 4 (units)
- 27 → C = 2 (tens), D = 7 (units)
Steps in Girard’s formula:
- Add A + D: 3 + 7 = 10
- Add B + C: 4 + 2 = 6
- Multiply A × 9: 3 × 9 = 27
- Multiply C × 9: 2 × 9 = 18
- Add all the results:
10 (A+D) + 6 (B+C) + 27 (A×9) + 18 (C×9) = 61
The final value? 61, which is exactly 34 + 27.
The Girard Formula: The Mathematical Rule
Written in mathematical terms, the Girard Method becomes:
S = (A + D) + (B + C) + 9A + 9C
This is the Girard Addition Formula — a compact way to represent the method. It shows that no matter which numbers you choose, the result will always match the standard addition.
Why Does It Work?
Despite its apparently complex look, the formula follows a solid mathematical structure. By decomposing numbers into tens and units and combining sums with strategic multiplications, the method simply reorganizes the operation in a creative way — while preserving its essence.
According to Girard, the idea came from observing repetitive patterns during simple additions. When he noticed that certain combinations consistently led to the same result, he decided to formalize the approach.
“I realized that by multiplying the tens by 9 and summing specific pairs of digits, I could reach the same result as traditional addition — but through a different, almost playful path. I created this formula when I was 14, in 1988, and now, in 2025, I finally decided to share it with the world,” explains Marcelo Girard.
ADVANCED EXAMPLE WITH THREE-DIGIT NUMBERS
Girard also extended his formula to three-digit numbers. Here’s an example:
237 + 245
Decomposition:
- 237 → H₁ = 2, T₁ = 3, U₁ = 7
- 245 → H₂ = 2, T₂ = 4, U₂ = 5
Formula applied:
Sum = (H₁ + U₂) + (T₁ + T₂) + (U₁ + H₂) + 99 × (H₁ + H₂) + 9 × (T₁ + T₂)
Substitute the values:
(2+5) + (3+4) + (7+2) + 99×(2+2) + 9×(3+4)
= 7 + 7 + 9 + 396 + 63 = 482
Guess what? 237 + 245 = 482, confirming the formula’s validity.
The Girard Theorem: Mathematical Proof
A theorem is a statement that can be proven mathematically. In the case of the Girard Positional Decomposition Theorem (GPDT), the proof shows that the formula works consistently, regardless of the numbers used.
Why is it a theorem and not just a “formula”?
Because it generalizes a rule for any number of digits. It has been algebraically demonstrated with rigorous mathematical proofs. It also has potential applications beyond addition, such as error checking and algebra teaching.
Educational Potential and Classroom Use
Although more elaborate than the traditional method — requiring about 20% more operational steps — the GPDT stands out as a powerful educational tool. It:
- Clarifies positional value, making the decimal system more intuitive
- Encourages early algebraic reasoning by working with variables
- Can be adapted for games and educational apps, boosting student engagement
- Provides built-in result verification using 9-multiplier checksums
Educators are already studying how to integrate the method into teaching environments to broaden foundational math instruction, making learning more visual, interactive, and accessible.
A Creative View of Mathematics
Marcelo Girard summarizes his mathematical philosophy with an unusual metaphor:
“Numbers are alien letters. I’m just an astronaut.”
His approach represents more than just an arithmetic technique — it’s an invitation to imagination. The Girard Addition Formula shows that even math, often seen as rigid and immutable, can be a rich ground for invention, beauty, and storytelling.
International Recognition and Free Access
Girard’s full work is available for free in the Zenodo repository, with an international DOI and both Portuguese and English versions. The publication marks a major step in the democratization of science and the recognition of original ideas from beyond the major academic hubs.
Zenodo is a scientific repository developed and maintained by CERN (European Organization for Nuclear Research), and funded by the European Union.
🔗 Access the full study: https://doi.org/10.5281/zenodo.15232943
Tip for Teachers:
Use the formula with your students as a challenge or fun activity! Take two two-digit numbers, represent them as A, B, C, and D, and explore how the patterns of the decimal system are surprisingly revealed.
Future Possibilities Include:
- Spreadsheet and educational app implementations
- Redundant checks in encryption and math games
- Extensions to modular arithmetic, such as error detection in digit transpositions
Author Contact:
Marcelo Brito Girard – celogirard@gmail.com
