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Do calculators even routinely work with infinite precision (like (sqrt(2) x sqrt(2) = 2)? I guess you could achieve some results with Mathematia or something li

by Tichy 13y ago

Do calculators even routinely work with infinite precision (like (sqrt(2) x sqrt(2) = 2)? I guess you could achieve some results with Mathematia or something like that, but I have my doubts about the average calculator. Sometimes it is still useful to do calculations by hand, also to understand what is going on. (Edit: just tested the OS X calculator, sqrt(2) x sqrt(2) works out by chance, but sqrt(200) x sqrt(200) fails).

mike_esspe 13y ago
It just means you should choose a better calculator From my calculator (Genius Mathematical Tool): genius> sqrt(200)*sqrt(200) = 200.0
- Tichy 13y ago
  Well that is also wrong, or at least the .0 shows that the calculator is not doing the equations, it is just calculating. It doesn't know know that sqrt(2) x sqrt(2) = sqrt(2 x 2) = sqrt(2^2) = 2 And that is the kind of thing you get to understand if you do the calculations by hand.
  - richardwhiuk 13y ago
    A better way to look at sqrt(2) * sqrt(2) is: sqrt(2) * sqrt(2) = 2^(0.5) * 2^(0.5) = 2^(0.5+0.5) = 2^1 = 2
    - Tichy 13y ago
      That's nice, too, although I think the concept of fractional powers was only introduced much later in my school.
ronaldx 13y ago
A current-generation Casio scientific handles this well - the display shows exact numbers such as 1+√(2) which suggest it's doing the precise calculation where reasonable.
- Tichy 13y ago
  The problem is that something like sqrt(2) can not be represented as a number in a calculator. So as long as the calculator only operates with numbers, instead of doing algebra, it will do it wrong. In the case of sqrt(2) x sqrt(2), most calculators will convert sqrt(2) to an approximation of sqrt(2), then multiply those approximations. If you are lucky the result will be rounded to 2, but you will not always be lucky.
  - ronaldx 13y ago
    No, I don't believe you are correct: a calculator can represent sqrt(2) as sqrt(2), just the same as we can on paper (perhaps you are calling this an algebraic representation). In fact, I'm confident that Casio - probably the market leader in scientific calculators - is doing exactly this, storing the exact number. Casio does not appear to approximate simple surds as floats until it becomes necessary, and stores the exact number sqrt(2), just as it stores exact representations of rational numbers wherever reasonable (and, it correctly distinguishes whether it knows a number exactly or only as a float - it refuses to convert inexact floats into fractions, but will convert exact numbers). Calculator phone apps don't compete at this level - which is surprising, I would have thought smartphones would have put calculators against the wall.
m_mueller 13y ago
Edit: Sorry, I just noticed that actually I was missing the point - I thought your comment was in response to vacri. Leaving my comment here, just in case. I think your comment misses the point. It's important to learn when to use a calculator and when to simplify the equation instead. Your example obviously should be simplified first on paper, resulting in 2. This can easily be enforced by requiring the intermediate steps, such that a calculator won't work. The point of parent is, that learning pen-and-paper calculation is of basically no value (except if you go into computer science, where remembering those algorithms can come in handy). Your example is not calculation, it's arithmetics, a necessary step towards algebra.
- Tichy 13y ago
  I agree that probably training too much manual addition and multiplication is a waste of time these days. However, a bit of it still seems to be useful. For example I am often able to calculate the bill in a restaurant (not always, though). And sometimes it is just faster than going to the calculator, depending on the situation. Also a lot of the process of addition and multiplication might also be good practice for arithmetics?
comex 13y ago
Well, I was talking about basic arithmetic, addition/subtraction/multiplication/division of integers. Doing algebra (which is usually where you'd find square roots) on paper is certainly still useful, but usually the numbers aren't very large, so it's essentially orthogonal to the type of work mentioned here. Algebra and more advanced math has multiple ways to go about dealing with an expression and requires understanding to proceed, but long multiplication or addition is just rote application of an algorithm. (I suppose the fraction in the problem in the article makes it a bit trickier, but I suggest that the dilemma of ordinary four-function calculators not being able to do exact fractions is not a common real world problem, or they would be able to do them.)

by Tichy 13y ago

mike_esspe 13y ago

It just means you should choose a better calculator From my calculator (Genius Mathematical Tool): genius> sqrt(200)*sqrt(200) = 200.0

Tichy 13y ago
Well that is also wrong, or at least the .0 shows that the calculator is not doing the equations, it is just calculating. It doesn't know know that sqrt(2) x sqrt(2) = sqrt(2 x 2) = sqrt(2^2) = 2 And that is the kind of thing you get to understand if you do the calculations by hand.
- richardwhiuk 13y ago
  A better way to look at sqrt(2) * sqrt(2) is: sqrt(2) * sqrt(2) = 2^(0.5) * 2^(0.5) = 2^(0.5+0.5) = 2^1 = 2
  - Tichy 13y ago
    That's nice, too, although I think the concept of fractional powers was only introduced much later in my school.

ronaldx 13y ago

A current-generation Casio scientific handles this well - the display shows exact numbers such as 1+√(2) which suggest it's doing the precise calculation where reasonable.

Tichy 13y ago
The problem is that something like sqrt(2) can not be represented as a number in a calculator. So as long as the calculator only operates with numbers, instead of doing algebra, it will do it wrong. In the case of sqrt(2) x sqrt(2), most calculators will convert sqrt(2) to an approximation of sqrt(2), then multiply those approximations. If you are lucky the result will be rounded to 2, but you will not always be lucky.
- ronaldx 13y ago
  No, I don't believe you are correct: a calculator can represent sqrt(2) as sqrt(2), just the same as we can on paper (perhaps you are calling this an algebraic representation). In fact, I'm confident that Casio - probably the market leader in scientific calculators - is doing exactly this, storing the exact number. Casio does not appear to approximate simple surds as floats until it becomes necessary, and stores the exact number sqrt(2), just as it stores exact representations of rational numbers wherever reasonable (and, it correctly distinguishes whether it knows a number exactly or only as a float - it refuses to convert inexact floats into fractions, but will convert exact numbers). Calculator phone apps don't compete at this level - which is surprising, I would have thought smartphones would have put calculators against the wall.

m_mueller 13y ago

Edit: Sorry, I just noticed that actually I was missing the point - I thought your comment was in response to vacri. Leaving my comment here, just in case. I think your comment misses the point. It's important to learn when to use a calculator and when to simplify the equation instead. Your example obviously should be simplified first on paper, resulting in 2. This can easily be enforced by requiring the intermediate steps, such that a calculator won't work. The point of parent is, that learning pen-and-paper calculation is of basically no value (except if you go into computer science, where remembering those algorithms can come in handy). Your example is not calculation, it's arithmetics, a necessary step towards algebra.

Tichy 13y ago
I agree that probably training too much manual addition and multiplication is a waste of time these days. However, a bit of it still seems to be useful. For example I am often able to calculate the bill in a restaurant (not always, though). And sometimes it is just faster than going to the calculator, depending on the situation. Also a lot of the process of addition and multiplication might also be good practice for arithmetics?

comex 13y ago

Well, I was talking about basic arithmetic, addition/subtraction/multiplication/division of integers. Doing algebra (which is usually where you'd find square roots) on paper is certainly still useful, but usually the numbers aren't very large, so it's essentially orthogonal to the type of work mentioned here. Algebra and more advanced math has multiple ways to go about dealing with an expression and requires understanding to proceed, but long multiplication or addition is just rote application of an algorithm. (I suppose the fraction in the problem in the article makes it a bit trickier, but I suggest that the dilemma of ordinary four-function calculators not being able to do exact fractions is not a common real world problem, or they would be able to do them.)