Does matlab supports rational number types?

I see Matlab stores numbers in floating points [1], which causes round of errors!

For eg:

The above round off error is resulted, because 10^-5 is not a valid 64 bit floating point number in matlab.

However in *Mathematica*, numbers can be stored in rational format [2] (ie integer/integer format), which avoids any round off error for the same computations.

So my question is, will it be possible to avoid such round of errors in matlab?

[1]: http://folk.uib.no/fra001/radu/num2_bergen/float.pdf

[2]: https://reference.wolfram.com/language/tutorial/LinearAlgebraMatrixTypes.html

[Adam]

I think you need the Symbolic Maths toolbox to have handling for this.

[Stephen23]

All binary floating point numbers are rational numbers.

Also note that the proposed solution just shifts the problem: how should those two integers be stored? Presumably they would be finite integers stored on your computer... which means that there will always be rational numbers that cannot be represented (unless you happen to have infinite memory installed on your laptop).

Using (finite) rational numbers (in the sense the OP proposed) does not completely avoid errors: there are many more irrational numbers than rational numbers, some of which are commonly used. For example, there is no way to exactly represent e.g. sqrt(2) or pi or e as ratios of finite integers. Any attempt to store those values numerically will always include some error.

Answer 1

[John D'Errico]

MATLAB uses floating point numbers by default. This makes it more efficient to do things like work well on large scale linear algebra problems, but it introduces problems like this. There is no fractional integer class in MATLAB. (Well, there was one posted on the FEX that I recall, that worked with my own vpi.)

Yes, you can use the symbolic toolbox to do the kinds of things you want to do. But you then need to recognize that those computations will be quite a bit slower. You trade off speed for ultimate precision. And given the number of people who are never satisfied with the speed of their computers on any given problem, speed is important.

In the end, the real solution is to learn to use floating point arithmetic. That is a skill and sometimes an art. It is the gist of scores of numerical methods courses, teaching students how to do computations without falling under the wheels of a limited number of digits in their computations.

[Anver Hisham]

Yes, this clarifies my doubt. Thanks for the answer.

[Dana Massie]

rational arithmetic is useful in a number of corner cases where float arithmetic cannot produce a precise result. It is not possible to represent many fractional values precisely in a given number base (ie decimal or binary). Rational arithmetic can be used for some of those cases. In decimal, 1/3 is a repeating fraction, and cannot be represented exactly in a finite set of digits.

[Walter Roberson]

quite true, rationals can be useful. Unfortunately, there is no hardware support for rationals.

[Omar Lakkis]

The power 5 is a distraction, in fact the example works with squares in the following form

0.01-(0.1)^2

and the reason is that the rational number 1/10 has the infinite (periodic) binary expansion: 0.000110011... This means that using 0.1 in a calculation, except in the simplest situation, will necessarily introduce rounding errors.

A commong misconception is that the smaller the numbers the harder are they to represent as floats, but this is not true. It is instructive to compare the above with the following (luckily exact) operation

0.00390625-(0.0625)^2