Computer Science – How Do Calculators Represent Floating Points Perfectly?

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If you ask a programming language to calculate 0.6 + 0.7, you’ll get something like 1.2999998, and that’s because of how floating point numbers are represented in computers. But if you ask a calculator, or even the calculator app on your phone, to do the same thing, it will give you the correct answer: 1.3. Why? Do calculators not represent floats the same way that other computers do?

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The short version is, they're cheating. Calculators or calculator apps are, if anything, less capable than computers and programming languages in general. That said, they're also specialized for doing exactly that sort of thing, so they're more inclined to take 12.9999998 and round it for you to 13. Programming languages will do the same thing, but depending on which language and how precise the number, you might have to ask them to.

Take any calculator and try something like adding $1+10^{-100}-1$, and in my experience, you'll get $0$, not $0.0000\dots 001$.

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[Math] a good language to develop in for simple, yet customizable math programs

I'd suggest to look into python because it's a modern language with an easy syntax.

Also for mathematical application you could look at sage. That's basically an extension of python which provides an interface to many mathematical programs (among others maxima, octave, gap, R, etc..).

This is also convenient because you have a "built-in" representation of mathematical objects. For example, if you want to manipulate polynomials, you don't need to define a class to represent polynomials. You do something like that:

def gcd(P,Q):
    F, G, R = P, Q, P % Q

    while (R != 0):
        F, G, R = G, R, F % G

    return G

print gcd(5,6)    # Compute gcd of integers

R.<x> = PolynomialRing(QQ)
P, Q = x^7-1, x^14-1

print gcd(P,Q)    # Compute gcd of polynomials

[Math] Do calculators have floating point error

Calculators are computers, too; they're just smaller. Surely if we knew how to represent arbitrary real numbers inside calculators, we could do the same thing with desktop computers.

That said, it's possible—both on a calculator and on a computer—to represent some real numbers exactly. No computer I know of would represent $\frac12$ inexactly, since its binary expansion (0.1) is short enough to put inside a floating point register. More interestingly, you can also represent numbers like $\pi$ exactly, simply by storing them in symbolic form. In a nutshell, instead of trying to represent $\pi$ as a decimal (or binary) expansion, you just write down the symbol "$\pi$" (or, rather, whatever symbol the computer program uses for $\pi$).

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[Math] a good language to develop in for simple, yet customizable math programs

[Math] Do calculators have floating point error

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