Which number is an irrational number? (1 point)

Mathematics

1 answer:

NemiM [27]3 years ago

8 0

Answer: The answer is (B). 1.01257486...

Step-by-step explanation: We are given four real numbers out of which we are to select the one which is irrational.

Option (A) is 0.12, in which the digits after the decimal are terminating. So, the number is rational.

Option (B) is 1.01257486..., the digits after the decimal are non-terminating and non-recurring. So, the number is irrational.

Option (C) is 0.1212121212..., in which the digits after the decimal are non-terminating and recurring. So, the number is rational.

Option (D) is 0.11111111..., in which the digits after the decimal are non-terminating and repeating. So, the number is rational.

Thus, (B) is the correct option.

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Our goal here is to somehow "surgically remove" the repeating part of the number, so let's start by putting the original value in a variable and messing around with it a bit.

We'll let $x=7.\overline{6}$ . We want to cut the $0.\overline{6}$ bit off completely, so let's create the scalpel that'll let us do that. If $x=7.\overline{6}$ , then we can also say that $10x=76.\overline{6}$ . Maybe I was lying a bit: the $x$ is our real scalpel here, and $10x$ is where we'll be making the cut. Mathematically, a "cut" is almost always shorthand for subtraction, so let's see what our operation (cutting $x$ off of $10x$ ) leaves us with:

$10x-x=76.\overline{6}-7.\overline{6}\\9x=69$

The operation was a success! We can now simply divide either side by 9 to find $x = 69/9$ , which, when reduced by dividing the numerator and denominator by 3, gives us $x=23/3$