Decimals and Fractions are two ways to represent numbers. Representation matters, and I have often wondered if one of these two methods is superior, or better aligned with the mathematical reality. In this article we argue that fractions are more aligned with reality, but decimals are easier to work with for humans. That may argue that humnas are not aligned with reality - well we never suspected we are.
First let's see how fractions outperform decimals.
The struggle of decimal starts very early. While
$\frac{1}{2}$ = $0.5$
$\frac{1}{10}$ = $0.1$,
we have
$\frac{1}{3} = 0.3333\dots$
$\frac{1}{7} = 0.142857142857142857\dots$.
So decimal representation has to resort to infinite number of digits to represent a number as simple as $\frac{1}{3}$. Now, infinity is a part of reality, so it is legit to use it - that's why decimal representation is also a correct representation, but it is better if we can do without it, since using infinity brings with itself notions of convergent series et al.
We call decimals like $0.333\dots$ as repeating decimals.
Whether a decimal number is repeating or not is dependent on base. $\frac{1}{3} = 0.333\dots$ in base 10, but it is simply $0.1$ in base 3. In general a fraction $\frac{1}{p}$ terminates in base N if prime factorization of $p$ only contains primes which divide $N$. For instance a number $\frac{1}{p}$ will be terminating in base 10 if $p$ contains only $2$ and $5$ as prime factors. That is why 1/2, 1/5, 1/10 and 1/50 terminate, but 1/3, 1/6 or 1/28 don't.
We could have lived happily with the fact that some fractions are repeating and some are terminating, just as we live with the fact that some numbers are prime and some are not, or that some numbers are multiples of 7 and some are not. However, whether a fraction is repeating or not depends on the base that we use, which means that this is not a property of the number, this an artifcat of the notation that we use.
Next we look at irrational numbers. The first instinct may be that decimals make a good attempt to approximate them: $$ \sqrt{2} = 1.4142135624\dots$$ $$e = 2.718281828459045\dots $$ $$\phi = 1.61803398875\dots $$ (golden ratio) $$\pi = 3.14156\dots$$ and we simply cannot write them as fractions, since these numbers are irrational.
But behold! Continued fractions come to our rescue! There is nothing stopping from numerator and denominator of a fraction itself to be a fraction, and once we use that we see that $$ \sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dots}}} $$ $$ e= 2 + \dfrac{1}{1 + \dfrac{1}{2 + \dfrac{1}{1+ \dfrac{1}{1+\dfrac{1}{4 + \dfrac{1}{1+\dfrac{1}{1 + \dfrac{1}{6+\dfrac{1}{1+\dfrac{1}{1 + \dfrac{1}{8+\dots}}}}}}}}}}} $$ and very elgantly $$ e = 2 + \dfrac{2}{2 + \dfrac{3}{3 + \dfrac{4}{4+\dfrac{5}{5+\dfrac{6}{6+\dots}}}}} $$ Similarly, $$\phi = 1 + \dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dots}}} $$ $$ \pi = \dfrac{4}{1 + \dfrac{1^2}{2 + \dfrac{3^2}{2 + \dfrac{5^2}{2+\dfrac{7^2}{2 + \dots}}}}} $$
While digits in decimal expansions of numbers are arbitrary, we find patterns in the continued fraction representation of numbers, which supports the assertion that fractions are real deal, while decimals are a hack for the real world.
Consider the four numbers we considered above, $\sqrt{2}, e, \phi, \pi$. Of these, $\sqrt{2}$ and $\phi$ are "algebraic" numbers since they are the roots of equations all whose coeffients are integers. Specifically, $\sqrt{2}$ is the solution for $x^2 = 2$ and $\phi$ is the solution for $x + 1 = x^2$. The other two are "transcendental" numbers.
We won't go into the detail, but only mention that if we write a "simple" continued fraction for a number (all numerators of simple continued fraction are 1), then a continued fraction which termintes or else there is periodicity in the partial quotients then that number is algebraic. If there is no period then the number is transcental. That is why $$ \sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dots}}} $$ is algberaic, but $$ e= 2 + \dfrac{1}{1 + \dfrac{1}{2 + \dfrac{1}{1+ \dfrac{1}{1+\dfrac{1}{4 + \dfrac{1}{1+\dfrac{1}{1 + \dfrac{1}{6+\dfrac{1}{1+\dfrac{1}{1 + \dfrac{1}{8+\dots}}}}}}}}}}} $$ is transcendntal.
Thus, while decimal representation of a number throws no light on it being algebraic or trancendntal, fractional form makes it obvious. [Edit: I later found out that above claim is not true. Only when equation has degree 2 is that that quotients have periodicity. So I am no longer sure if we can separate transcendtal and algebraic numbers by looking at their continued fractions.]
We have argued above that fractions are more superior representations of numbers. If that is the case, then why are they not used more than decimals? Everywhere, we find decimals: GDP of India is 3.6 trillion - why is not $3\frac{3}{5}$ trillion? Stock market yesterday rose by 1.1%, why not by a more precise $\frac{1123}{1012}$%? Even when physicists or computer scientists, who are fully familiar with both the representations, practice their trade, they talk in decimals: server response time is 45.34ms and training loss is 6.2343603.
The reason, I think is that our brains can comprehend decimals better. In particular, comparing two decimals is very easy for our brains compared to comparing two fractions. It is difficult to find what is bigger: $\dfrac{64}{71}$ or $\dfrac{13}{14}$, but once you know that $\dfrac{64}{71} \approx 0.9014$ and $\dfrac{13}{14} \approx 0.9285$, you instantly know what the answer is.
Another reason is that adding fractions involves multiplying numbers and multiplication is a heavier operation. Adding two decimals involves only adding digits, and so it is simpler.
It might also be that decimals got more in use because fractions are unwieldy to write, while decimals can fit in with other language text.
Our brains are wired so that decimals appeal more to them. However, what about computers? We know that computers have instructions for integer addition, multiplication and division (add, mul and div respectively), and also floating point addition, multiplication and addition (fadd, fmul, fdiv respectively). Even programming languages have a data type for "float" / "double". Could it be beneficial to have fraction as a native data type?
My guess is "no". Decimals have the advantage that they can represent very large and very small numbers (like $10^{100}$ and $10^{-100}$ respectively), a fraction representation would struggle to do that since then we will need to store represent large integers which is not supported by architecture - for 64 bit integers, $2^{64}$ is far less than $10^{100}$. You can indeed represent larger integers using software but then operations become slower.
Further, to do a fraction addition, you will need three integer mulitplications and an integer addition (since $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a * d + b * c}{b * d}$). And then worry about overflows and underflows. I am not how well we can work it out.
My conclusion is that reality fundamentally is described by fractions, but the way we perceive that complex reality is better served by decimals - decimals are good practical tools for us to solve our day to day problems.