5, -41, 3/2, pie, e, O, square root of 5, 6 squared, -34/2 a) List all natural numbers, integers, rational numbers, List all irrational numbers:
The Correct Answer and Explanation is :
Let’s break down and classify the numbers you provided into natural numbers, integers, rational numbers, and irrational numbers. I’ll also explain the reasoning behind the classification of each.
Numbers Given:
- 5
- -41
- 3/2
- π (pi)
- e (Euler’s number)
- O (this might be a typo or something unclear, so I’ll skip it for now)
- √5 (Square root of 5)
- 6 squared (36)
- -34/2
Natural Numbers:
Natural numbers are positive whole numbers starting from 1 and extending to infinity. They do not include negative numbers, fractions, or decimals.
Natural numbers in the list:
- 5 (Positive whole number)
Integers:
Integers include all whole numbers, both positive and negative, and zero. They do not include fractions or decimals.
Integers in the list:
- 5 (positive whole number)
- -41 (negative whole number)
- 6 squared (36) (positive whole number)
- -34/2 (which simplifies to -17) (negative whole number)
Rational Numbers:
Rational numbers are numbers that can be expressed as a fraction or ratio of two integers, where the denominator is not zero.
Rational numbers in the list:
- 3/2 (It is a fraction and hence rational)
- -34/2 (Simplifies to -17, which is an integer, and all integers are rational)
- 6 squared (36) (This is an integer, so it’s rational)
Irrational Numbers:
Irrational numbers cannot be expressed as a simple fraction of two integers, and their decimal expansion is non-repeating and non-terminating.
Irrational numbers in the list:
- π (pi) (It cannot be expressed as a fraction, and its decimal expansion is non-terminating and non-repeating)
- e (Euler’s number) (It also cannot be expressed as a fraction, and its decimal expansion is non-terminating and non-repeating)
- √5 (The square root of 5 is irrational because it cannot be expressed as a fraction)
Summary of the Classification:
- Natural Numbers: 5
- Integers: 5, -41, 36, -17 (from -34/2)
- Rational Numbers: 3/2, -17 (from -34/2), 36
- Irrational Numbers: π, e, √5
Explanation:
- Natural numbers are the counting numbers, starting from 1 and going up to infinity. Here, only 5 qualifies.
- Integers include both positive and negative whole numbers and zero. Numbers like -41 and 36 are integers, as are the simplified -17.
- Rational numbers are those that can be written as a fraction of two integers. Rational numbers are always either integers or fractions. So, 3/2 and -34/2 fit the definition of rational numbers.
- Irrational numbers cannot be expressed as simple fractions and have non-repeating, non-terminating decimal expansions. This includes π, e, and √5.
This classification helps us organize numbers based on their properties and provides insight into their behavior in mathematical contexts.