31–40 of 64 posts
Certainly, people don't need Wolfram Alpha as much.
On another point, it sucks to know what this means for Algebraic Geometry (the computational variant), which you could partly motivate, until now, for its use in constructing CASes.
Earlier quoted context omitted.
How is going from two functions with one variable to three functions with a variable and a constant a simplification?
If you can't recognize how much simpler the simplified version is, I'm not sure exactly what to tell you. But let's think about it in terms of assembly steps: 1. Multiply the input by itself 2. Add 1 3. Take the square root. There is often a fast square root function available. The above is a fairly simply sequence of SIMD instructions. You can even do it without SIMD if you want. Compare this to sinh being (e^x - e^…
I've only an A-Level in Further Maths from 1997, but understand complex numbers and have come across complex inverse trig functions before. My takeaway for other people like me from this is "computer is correct" because the proof shows that we can't define arccosh using a single proof across the entire complex plane (specifically imaginary, including infinity). The representation of this means we have both complex fu…
For a programmer, it's clear that we have lost the sign information but not the magnitude.
Simple. Makes most sign and solution reasoning explicit instead of implicit when solving quadratics or otherwise working with square roots.
This is a general pattern in CAS. For a more basic case, it’s not obvious sqrt(square(x)) will simplify to x without any further assumptions on x.
More generally it's not at all clear what 'simplify' means. Is x*x simpler than x^2? Probably? Is sqrt(5)^3 simpler than 5^(3/2)? I don't know. It entirely depends on what you're going to be doing with the expression later.
Does anyone else think that the latest LLMs - some of which can be used locally for free - combined with proof-verifying software like Coq or Lean for mistake-detection, might make many uses of Computer Algebra Systems like Mathematica obsolete? Certainly, people don't need Wolfram Alpha as much. On another point, it sucks to know what this means for Algebraic Geometry (the computational variant), which you could par…
Coq/Lean target very different use cases.
I've only an A-Level in Further Maths from 1997, but understand complex numbers and have come across complex inverse trig functions before. My takeaway for other people like me from this is "computer is correct" because the proof shows that we can't define arccosh using a single proof across the entire complex plane (specifically imaginary, including infinity). The representation of this means we have both complex fu…
Simple rule to keep in mind that even math savvy people seem to forget about is that: sqrt(x²) = |x| with bars for absolute value. For a programmer, it's clear that we have lost the sign information but not the magnitude. Simple. Makes most sign and solution reasoning explicit instead of implicit when solving quadratics or otherwise working with square roots.
i would disagree with that (pun intended).
Earlier quoted context omitted.
For x = -i, square(x) = -1, sqrt(square(x)) = i. Meanwhile, abs(x) = 1. You're right that it simplifies to abs(x) for real x, but that no longer holds for arbitrary complex values.
for arbitrary complex values sqrt() gives 2 answers with +- signs so sqrt(square(-i)) = +-i, one of which is x