anonymous
  • anonymous
Identify the number that does not belong with the other three. Explain your reasoning. 50.1 repeating 1, negative 50 over 2, negative 50.1, square root 50
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
Let's quickly go over rational and irrational numbers A rational number is a number which can be expressed in the form of \[\frac{p}{q}\] where \[q \neq 0\] and \[p,q \in \mathbb{Z}\] that is p and q are integers irrational numbers r just opposite Let's look at first number, \[50.111...\] Let \[S=50.111....\] therefore \[10S=501.111...\] Multiplying by 10 will take 1 out from the decimal, but still an infinitely amount of 1's in subtracting 2nd equation by first we get \[10S-S=(501.111...-50.111...)=(501-50)+(0.111...-0.111...)\] We have simply taken the decimal parts apart it's the same as writing \[2.34=2+0.34\] Now we have \[9S=451\]\[S=\frac{451}{9}\] Which is of the form \[\frac{p}{q}, p,q \in \mathbb{Z}, q \neq 0\] If you do the division of 451 by 9 in a calculator you will indeed get your number back again Thus 50.111..... is a RATIONAL number, in general all numbers with terminating decimals(decimals that end after some numbers) eg. \[2.45\]\[55.2\] and numbers with non terminating and repeating decimals(decimals that keep on repeating to infinity) \[2.474747... \] \[5.77777...\] are RATIONAL But if a number has non terminating and NON repeating decimals, it is IRRATIONAL eg. infamous pi \[\pi = 3.14159...\] You will never find a pattern that is repeating in the decimal expansion of pi Note that 22/7 is an approximation of pi, pi does not equal to 22/7 \[\pi \approx \frac{22}{7}\] another eg. \[\sqrt{2}=1.4142....\] In general square root of a number which is not a prefect square is IRRATIONAL By perfect square I mean a number who has a simple and integer square root \[\sqrt{4}=2\] So 4 is a perfect square, it has an integer square root but root 2 cannot be written as an integer so it is an irrational number Your 2nd number \[-\frac{50}{2}=-25\] It is a rational number where the denominator is 1(an integer not equal to 0) and numerator is also integer(-25) similarly your 3rd number has a terminating decimal, so it's a rational number however what about \[\sqrt{50}\] rational or irrational?? The notation \[p,q \in \mathbb{Z}, q \neq 0\] Means p and q are integers and q is not equal to zero

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