Why do we need to distinguish between rational and irrational numbers? What’s the importance of knowing where they fall between the two?
In: Mathematics
As for “Real world applications”, there really aren’t any. Any irrational or transcendental number can be sufficiently calculated to the point of removing any practical error.
For instance, NASA uses ~15 decimal places of Pi for most of their calculations. If you were to look at using Pi to 15 decimal places versus Pi to significantly more decimal places, and were looking at a task like putting an orbiter around Mars (about 55 million kilometers away at closest points), then Pi to 15 decimal places still gives you precision to within a millimeter.
1/4 is a rational number – 1/3 is an irrational number. Why?
Because 1/4=0,25 and thus the value is complete, while if you say 1/3= you get 0,333333 but that isnt the complete answer, cause you can always add more 3’s.
Most (almost all) of the real numbers are irrational. Now this doesn’t concern your average day if you don’t work in certain fields, but it is essential in math, and therefore in science and engineering. Rational based polynoms lead out of the rationals, even out of the reals, but you definitely need irrational numbers to have a complete number line.
Interestingly, despite their abundance, proving that a number is indeed irrational is usually not easy.
same reason you need to distinguish between sharks and piranhas.
it probably wont matter to you if you’re not doing anything in a related field, but when you do the differences are massive and it is important.
rational numbers are all number you can write as a fraction (you could write 0.153 as 153/1000, so this includes all ending decimal numbers)
while irrational are those like Pi where that isn’t possible.
Same as between non-integer rational numbers and integers. Or between even numbers and odd numbers. There are differences between them, and interesting patterns.
Most of maths involves looking for interesting patterns in things, or poking at things to see what happens.
Most people don’t need to know, any more than they need to know e, √-1, or the prime factors of some large number. But these things are fundamental to mathematics if you want to go further than balancing your cheque book or totting up your bills.
Because, you can develop a definite value for a rational number, it is exact. Irrational numbers are approximations of the value, the more precise you get with it the closer you are to the true value, but you will never quite reach it. You can take pi to 25 quintillion decimal places, but you still won’t have its exact value.
There is a story that goes with our discovery of numbers. It’s like evolving technology. That’s why new kinds of numbers are important. It relates to their applications.
Selling goods requires whole numbers. (Counting numbers)
Accounting requires zero and negative integers. (the integers)
Measuring and building stuff requires fractions (rational).
Trigonometry and circles requires radicals and pi (irrational numbers.). The fact that pi was not a fraction astounded Greek philosophers.
Understand complex fluid dynamics and advanced physics require complex numbers.
We haven’t found any new kinds of numbers beyond this, although I suppose you could argue quantum numbers used in quantum computers could be a new kind of number.
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