∴ means "therefore" in mathematical proofs and arguments. It's a shorthand notation used to indicate a logical consequence of the preceding statement(s).
Origin of the Symbol
The symbol ∴ was first used in print by Johann Rahn in his 1659 book Teusche Algebra (also known as Teach Yourself Algebra).
How it's Used
When writing a mathematical proof or derivation, ∴ is placed before a statement that logically follows from the previous steps. For example:
- a = b + c
- b = d - e
- ∴ a = (d - e) + c
In this case, line 3 uses ∴ to indicate that the equation a = (d - e) + c is a direct consequence of substituting the value of 'b' from line 2 into the equation in line 1.
Comparison with Similar Symbols
While ∴ means "therefore", other related symbols are sometimes encountered:
- ∵ (inverted ∴) often means "because".
Examples in Mathematical Contexts
Here are a few more examples of how ∴ is used:
- In Geometry: If two angles are vertical angles, then they are congruent. ∠A and ∠B are vertical angles. ∴ ∠A ≅ ∠B.
- In Algebra: If x + 2 = 5, then subtracting 2 from both sides gives x = 3. ∴ x = 3.
Conclusion
The symbol ∴ serves as a convenient and widely understood abbreviation for "therefore" in mathematical reasoning, streamlining the presentation of logical arguments and proofs.
