Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form. Why is this? 20 Unit1

Answer:An irrational number can never be represented precisely in decimal form. Step-by-step explanation:A number can be precisely represented in decimal form if you can give a rule for the construction of its decimal part, for example: 2.246973973973973... (an infinite tail of 973's repeated over and over) 7.35 (a tail of zeroes) If this the case, then the number is a RATIONAL NUMBER, i.e, the QUOTIENT OF TWO INTEGERS. Let's show this for the first example and then a way to show the general situation will arise naturally. Suppose N = 2.246973973973... You can always multiply by a suitable power of 10 until you get a number with only the repeated chain in the tail for example: 1) [tex]N.10^3=2246.973973973...[/tex] but also 2) [tex]N.10^6=2246973.973973...[/tex] Subtracting 1) from 2) we get [tex]N.10^6-N.10^3=2246973.973973973... - 2246.973973973...[/tex] Now, the infinite tail disappears [tex]N.10^6-N.10^3=2246973 - 2246=226747[/tex] But   [tex]N.10^6-N.10^3=N.(10^6-10^3)=N.(1000000-1000)=999000.N[/tex] We have then 999000.N=226747 and [tex]N=\frac{226747}{999000}[/tex] We do not need to simplify this fraction, because we only wanted to show that N is a quotient of two integers. We arrive then to the following conclusion: If an irrational number could be represented precisely in decimal form, then it would have to be the quotient  of two integers, which is a contradiction. So, an irrational number can never be represented precisely in decimal form.