A famous one: 10:3=3.3333~ /a:b=c / why is c*b not a?
It’s because strictly speaking c*b is NOT defined.
Why? Multiplication is Not defined for a number with infinite decimals, ie, c = 3.3333…
This is an old trick I used before.
I’ll prove to you that
1 = 0.999… (the 9’s goes on forever ) you can try it.
Let y = 0.999 … (1)
Now multiply both sides by 10 to get,
10y = 9.999 … (2)
Now (2) – (1), gives us,
9y = 9 or
y = 1
but in the beginning we assumed that y = 0.999…
and thus we have proved that 1 = 0.999…
The answer is we cannot multiply or divide numbers with infinity of digits by our usual way of definition.
Nevertheless, we use the “techniques” above a lot in approximation for example, the sum to infinity of a geometric series with common ratio less than 1.
Certain rational numbers allow their decimal representation repeat in a pattern. Like above, after the decimal point, the pattern is “3”. Rational numbers are numbers of the form p/q, where both p and q are integers.
Other example is 56/11 = 5.090909… where the repeating pattern after the decimal point is “09”.
Working backwards, one can show that the number
m = 5.090909…(1) is actually a rational number.
Consider, 100m = 100* 5.090909…
(2) – (1), gives us,
100m – m = 509.090909… – 5.090909…
99 m = 509 – 5 = 504
and thus, m = 504/99 = 56/11 done.
A careful observer may ask, the two “methods” look the same! How come in proving the former ( 1=0.999… ) this “method” is considered wrong while in proving the latter ( 56/11 = 5.090909… ) the method is viewed acceptable? I cant answer that with absolute certainty. I don’t think anyone can. That’s where the mystery is. And I am a mathematics student.
After some thought, I conclude that one possible reason is because of the idea of writing 56/11 as 5.090909… is “created” by us. The idea of “rewriting a fraction in decimal form”. And the weirdness is due to the varying effects of the concept of “infinity”. In some cases, “infinity” can mean this when a mathematical operation is done on it, while on others it may mean that…
Well, all this a product of human thought.
It shows the frailty of the human defined concept.
Note that “infinity” is not a “number” but rather is a “concept”!