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Actual rocket scientists practically never work with fractional approximations. They’re good ways to explain things to elementary school students who haven’t yet learned decimal notation. And they’re of historic value, since ratios were a crucial way of expressing concepts that come out of geometric construction.But any rocket scientist is going to be working with high-precision floating point numbers for any calculation, and wouldn’t use either fraction for anything more than a back-of-the-envelope SWAG that would never make it into production.The danger, such as it is, comes not from the inaccurate approximation, but from the way the engineering calculations have to be done. 22/7 is way, way, way too inaccurate to even be considered. The error in 355/113 is actually quite small, and might in and of itself be within tolerances for spacecraft design, but as part of just one of many steps in the calculation the errors accumulate.And it would be incredibly inconvenient, since none of the other measurements are going to be expressed as fractions. Even if you had an ultra-precise ruler that would measure 32768ths of a meter… what kind of denominator do you get from calculating the area of a circle with radius 673/32768 with pi at 355/113? The fractions just get stupid, fast.Any calculation is going to keep pi in the form of a numerical constant until as late as possible, and then they’ll use a variety of tools from mathematical methods to minimize the errors that creep in during actual computations. Elementary school fractions never come into it at any point.

Because we are reasonable people.Writing down the fraction [math]\frac{1}{113}[/math] takes five strokes: three [math]1[/math]’s, one [math]3[/math] (ok maybe count that as two connected strokes), and one fraction line.Six strokes overall, and that’s being generous. And look what you get: at a glance you can see that it’s a reciprocal of a natural number, in fact the reciprocal of a prime. If I asked you to double it, or halve it, you’d have no trouble at all: [math]\frac{2}{113}[/math] and [math]\frac{1}{226}[/math]. Piece of cake.Now let’s express the very same number in decimal notation.You need 112 random-looking digits. Can you remember them? Can you do anything with them? Can you double the number in that representation? Go ahead, try. Now halve it. It took five seconds before; tell me how long it took now.In higher math, we sometimes write down specific numbers, but more often we discuss properties of numbers in general. For example, how do you add two rational numbers? Here:[math]\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}[/math]Short and sweet. What’s the formula, or algorithm, for adding the decimal representations of two rational numbers? There is one, but I bet you can’t write it down easily, if at all.We’re not “attempting to dodge decimals”. We just don’t use them, because they are damn near useless. What’s the use of a representation of rational numbers if you can’t even reasonably use it to add or divide them?The decimal representation is useful for one thing: it makes it relatively easy to compare numbers. With the decimal representation, it’s a tiny bit easier to see that [math]\frac{8}{13}<\frac{5}{8}[/math], which is [math]0.615\ldots<0.625[/math]. It’s not that hard to cross multiply, but the decimal expansions make it quite obvious.Similarly, it’s easy to recognize the close proximity of the decimal expansion of a rational number and the initial part of the decimal expansion of a real number.For example, suppose you took the fraction we started with, [math]\frac{1}{113}[/math], and multiplied it by [math]355[/math][math][/math]. Easy:[math]\displaystyle \frac{355}{113}[/math]If you look at the decimal expansion of this number, you may see something familiar:[math]\displaystyle \frac{355}{113}=3.141592920\ldots[/math]Ring a bell? Yeah, that’s very close to [math]\pi[/math]. Since we are trained to recognize the decimal expansion of [math]\pi[/math], rather than its continued fraction, it’s easier for us to see its close proximity to that fraction in decimal form rather than in fraction form.That’s fine. Different representations have different uses. It’s just that in more advanced math, decimal expansions become, um… pointless?Yeah, nice pun.

We know that, by definition, irrational numbers cannot be expressed as fractions of the form a/b, where a, b≠0,We don’t know that, actually. Every real or complex number, rational or not, can be expressed as a ratio [math]a/b[/math]. What makes a number rational is that it can be expressed as a ratio [math]a/b[/math] of two integers [math]a[/math] and [math]b[/math], and what makes a number irrational is that it cannot be expressed as a ratio of two integers.The numbers [math]6[/math], [math]0[/math], [math]-2[/math], [math]\frac{3}{5}[/math], [math]-0.18[/math] and [math]\frac{355}{113}[/math] are rational. They are all ratios of two integers.The numbers [math]\sqrt{2}[/math], [math]\pi[/math], [math]e[/math] and [math]-\sqrt[3]{\pi+\sqrt{3}}[/math] are irrational. None of them is the ratio of two integers.The fact that [math]\pi[/math] is the ratio of a circle’s circumference to its diameter says nothing about the rational or irrational character of [math]\pi[/math], because there’s no reason for the circumference or the diameter to be integers. In fact, since we know that [math]\pi[/math] is irrational, they cannot both be integers.