Identify all the sets to which 3.1214122144 … belongs

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To identify all the sets to which the number 3.1214122144… belongs, we need to analyze its properties. From the given number, it appears to be an irrational number with a repeating pattern.

1. Real Numbers: The number 3.1214122144… is a real number because it can be represented on the number line.

2. Irrational Numbers: The number 3.1214122144… is an irrational number because it has a non-repeating and non-terminating decimal expansion.

3. Algebraic Numbers: An algebraic number is a real or complex number that is a root of a non-zero polynomial equation with integer coefficients. Since the given number does not satisfy this condition and cannot be expressed as the solution of any polynomial equation, it is not an algebraic number.

4. Rational Numbers: A rational number is a number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. The given number 3.1214122144… cannot be expressed in this form, so it is not a rational number.

5. Natural Numbers, Whole Numbers, and Integers: The number 3.1214122144… does not belong to any of these sets because they consist of positive whole numbers, including zero, and the given number is not an integer.

Hence, the number 3.1214122144… belongs to the set of real numbers and irrational numbers.

To identify all the sets to which 3.1214122144... belongs, we first need to determine the pattern in the decimal representation.

Looking at the decimal, we can see that the digits 1214 repeat after the decimal point. Therefore, the number can be represented as 3.12(14), where the parentheses indicate the repeating pattern.

Now, let's identify the sets to which this number belongs:

1. Rational Numbers: The number 3.12(14) can be expressed as a fraction. To do this, we let x = 3.12(14) and multiply the equation by a power of 10 to eliminate the repeating part:
100x = 312.1414(14)
Now, we subtract the original equation from the above equation:
100x - x = 312.1414(14) - 3.12(14)
99x = 309
x = 309/99 = 103/33
Therefore, 3.12(14) is a rational number.

2. Real Numbers: The number 3.12(14) lies on the real number line, as it can be expressed as a terminating or repeating decimal.

3. Irrational Numbers: The number 3.12(14) cannot be expressed as a fraction of integers, hence it is not an irrational number.

Therefore, 3.12(14) belongs to the sets of rational numbers and real numbers.

If the ... indicates some set of repeating digits, the number is rational

If not repeating, then it is irrational

Either way, it is real