Can the product of two irrational numbers be rational?
This question is, of course, a quibble of mathematicians (or teachers). However, with some knowledge of rational and irrational numbers, one can solve the problem of the product.
Rational and irrational numbers - you should know that
Hand on heart: what rational and irrational Counting is "somehow" hidden from most of them during their school days - but actually quite simple.
- Mathematicians differentiate between various number ranges. The simplest are the natural numbers, just as one counts.
- The next larger number range are the whole numbers. In addition to the natural numbers, there are also zero and negative numbers. After all, you also want to show debts or minus degrees in temperature.
- The rational numbers are again the next larger number range; By the way, rational means "reasonable". In addition, there are all numbers that can be written as a fraction or formulated differently: all finite and periodic decimal fractions. This subheading includes 1/3, for example, but also -2.5. Breaks occurred historically when the sharing of goods did not work - the Egyptians already knew such breaks.
- Irrational (i.e. unreasonable) numbers include all infinite decimal fractions. Well-known examples of such numbers are the root (2) (a proof that billions of students had to endure), the circle number Pi and Euler's number e. Irrational numbers cannot be represented as a fraction.
- Incidentally, rational numbers and irrational numbers together form the number range of real numbers, which is often casually referred to as "all numbers".
Unnatural numbers - instructions on how such a thing can exist
If there are natural numbers, then there must be unnatural ones too. With this …
The product of irrational numbers - anything is possible
But what happens when you calculate with irrational numbers? This is the question mathematicians (and sometimes teachers ask their students).
- Added or If you subtract two irrational numbers, the result is again irrational (or zero if the numbers are the same).
- But what happens when you multiply two infinitely long ones Decimal numbers? Which number range does the product belong to? The problem can be approached using examples. You don't need much more than the above.
- If you multiply the circle number Pi by Euler's number e, both of which have an infinite number of digits after the decimal point, the result will again be an irrational number.
- However, if you multiply the root (2) by the root (2), the result is the number "2", not just a rational number, but even a natural one.
- And even more: Root (2) x Root (18) = Root (36) = 6.
So the product of two irrational numbers may well be a rational number, but in general it is not.
