The problem asks us to classify given numbers into different sets of numbers: - N: Natural numbers - Z: Integers - Q: Rational numbers - I: Irrational numbers - R: Real numbers

Number TheoryNumber SetsReal NumbersRational NumbersIrrational NumbersIntegersNatural Numbers
2025/3/30

1. Problem Description

The problem asks us to classify given numbers into different sets of numbers:
- N: Natural numbers
- Z: Integers
- Q: Rational numbers
- I: Irrational numbers
- R: Real numbers

2. Solution Steps

Let's analyze each number and determine the sets it belongs to:
-3: Negative integer.
* Z: Yes (Integers include negative and positive whole numbers and zero)
* Q: Yes (Any integer can be written as a fraction, e.g., -3/1)
* R: Yes (All rational numbers are real numbers)
124\frac{12}{4}: Simplify to

3. * N: Yes (Natural numbers start from 1: 1, 2, 3...)

* Z: Yes (3 is an integer)
* Q: Yes (3 can be written as 3/1)
* R: Yes (All rational numbers are real numbers)
π-\pi: Pi is an irrational number, so pi-pi is also irrational.
* I: Yes
* R: Yes (Irrational numbers are real numbers)
153-\frac{15}{3}: Simplify to -

5. * Z: Yes (Integers)

* Q: Yes (Any integer can be written as a fraction, e.g., -5/1)
* R: Yes (All rational numbers are real numbers)

1. 2: Decimal. $1.2 = \frac{12}{10} = \frac{6}{5}$

* Q: Yes (Can be expressed as a fraction)
* R: Yes (All rational numbers are real numbers)
32\frac{3}{2}: Fraction.
* Q: Yes (Fraction)
* R: Yes (All rational numbers are real numbers)
15-\frac{1}{5}: Fraction.
* Q: Yes (Fraction)
* R: Yes (All rational numbers are real numbers)
7\sqrt{7}: Square root of a non-perfect square.
* I: Yes (Cannot be expressed as a fraction of two integers)
* R: Yes (Irrational numbers are real numbers)

1. 136666...: Repeating decimal.

* Q: Yes (Repeating decimals can be expressed as fractions)
* R: Yes (All rational numbers are real numbers)
36-\sqrt{36}: Simplify to -

6. * Z: Yes (Integers)

* Q: Yes (Any integer can be written as a fraction, e.g., -6/1)
* R: Yes (All rational numbers are real numbers)
11-\sqrt{11}: Square root of a non-perfect square.
* I: Yes (Cannot be expressed as a fraction of two integers)
* R: Yes (Irrational numbers are real numbers)

3. Final Answer

Here's the classification of each number:
-3: Z, Q, R
12/4: N, Z, Q, R
-π: I, R
-15/3: Z, Q, R
1.2: Q, R
3/2: Q, R
-1/5: Q, R
√7: I, R
1.136666...: Q, R
-√36: Z, Q, R
-√11: I, R

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