What is QED?
QED stands for “Quod Erat Demonstrandum,” which is a Latin phrase that translates to “which was to be demonstrated.” It is commonly used in mathematics and philosophy to signify that a proof has been completed or a statement has been proven beyond doubt.
Origin of QED
The use of QED dates back to ancient Greek mathematicians and philosophers, who would end their proofs with the phrase “ὅπερ ἔδει δεῖξαι” (oper edi dei deiknai), which has a similar meaning to QED. It has since become a standard way to conclude a proof in various academic disciplines.
Application in Mathematics
In mathematics, QED is used at the end of a proof to indicate that the argument has been successfully completed and that the proposition has been demonstrated. For example, after proving a theorem, a mathematician might write “QED” to signal the end of the proof.
Example:
Proof: The sum of two even numbers is even.
Assume a and b are even numbers.
Since both a and b are even, they can be expressed as 2n and 2m respectively.
a + b = 2n + 2m = 2(n + m)
Since n + m is an integer, a + b is divisible by 2.
Therefore, the sum of two even numbers is even.
QED
Philosophical Significance
In philosophy, QED is used to conclude an argument or demonstrate the validity of a statement. It serves as a clear and concise way to signify that the point being made has been adequately supported and proven.
Case Studies
One famous historical example of the use of QED is in the proof of Fermat’s Last Theorem. Mathematician Andrew Wiles famously ended his proof of this centuries-old problem with the phrase “I think I’ll stop here.”
Modern Usage
While QED is traditionally associated with academic writing, it has also entered popular culture as a way to signify the end of an argument or discussion. In informal settings, someone might playfully say “QED” after making a compelling point or proving someone wrong.
Conclusion
In conclusion, QED is a powerful phrase that signifies the completion of a proof or argument. Whether used in mathematics, philosophy, or casual conversation, it carries the weight of centuries of intellectual tradition and rigor.
