Question

Determine which numbers in the set are (a) natural numbers, (b) integers, (c) rational numbers, and (d) irrational numbers.$$ \left\\{\frac{8}{2},-\frac{8}{3}, \sqrt{10},-4,9,14.2\right\\} $$

Short answer

(a) Natural numbers: \(\frac{8}{2}, 9\)\n (b) Integers: \(\frac{8}{2}, -4, 9\)\n (c) Rational numbers: \(\frac{8}{2}, -\frac{8}{3}, -4, 9, 14.2\)\n (d) Irrational Number: \( \sqrt{10} \)

Step-by-step solution

Classifying as Natural Numbers

Natural numbers are positive whole numbers. So, from the set, \(\frac{8}{2}\) which simplifies to 4, 9 are natural numbers.

Classifying as Integers

Integers are all whole numbers including negatives. So, the integers in the set are \(\frac{8}{2}\) (which is 4), -4, and 9.

Classifying as Rational Numbers

Rational numbers can be expressed as a fraction of two integers. From the set, all numbers except \( \sqrt{10} \) are rational numbers, because they can all be written as a fraction. For example, 14.2 can be written as \( \frac{142}{10} \).

Classifying as Irrational Numbers

Irrational numbers cannot be expressed as fractions. From the set, \( \sqrt{10} \) is an irrational number. These cannot be expressed accurately as fractions.

Key concepts

Understanding Natural Numbers

When we talk about natural numbers, we're referring to the set of positive whole numbers that we often begin counting with as children. These numbers are the building blocks of basic arithmetic and include the numbers 1, 2, 3, and so forth.

Now, looking at the set given in the exercise, we can identify which numbers qualify as natural numbers. For instance, the fraction \( \frac{8}{2} \) simplifies to the number 4, which is a natural number because it's a positive whole number. Similarly, the number 9 also fits into this category as it's another positive whole number.

In educational terms, it's important to note that natural numbers are denoted by the symbol \( \mathbb{N} \) and are fundamental when learning about higher-level mathematical concepts like sets and sequences.

Integers Explained

The realm of integers expands upon natural numbers by including zero and negative whole numbers. Symbolized by \( \mathbb{Z} \), integers are a critical part of number theory and real-world applications like accounting for debts or elevations below sea level.

In the given exercise, integers are identified by their lack of fractional or decimal parts. The number -4, despite being negative, is an integer. The simplified form of \( \frac{8}{2} \) which is 4, and 9 are also integers because they are whole numbers and can be positive or negative. Integers do not include fractions or decimals, making them distinct from other number classifications.

Rational Numbers Demystified

Rational numbers, indicated by the symbol \( \mathbb{Q} \), can be expressed as the quotient of two integers with the denominator not being zero. They encompass a wide range of figures, from whole numbers to fractions and even recurring decimals.

For example, from our set, \( -\frac{8}{3} \) and 14.2 (which can be converted to \( \frac{142}{10} \) as a fraction) are both rational. Even the negative and positive whole numbers identified before as integers are rational, since they can be written as a fraction with 1 as the denominator. The only exception in the set is the \( \sqrt{10} \) since it cannot be expressed as a fraction of two integers, thus not fitting the profile for rational numbers.

Irrational Numbers Unveiled

Irrational numbers are perhaps the most enigmatic of number classifications. They cannot be expressed as simple fractions of two integers, which means they are not rational numbers. Irrational numbers have non-repeating, non-terminating decimal expansions.

This might sound complex, but a classic example is \( \sqrt{10} \) from our set. Since there's no fraction consisting of integer numerator and non-zero integer denominator equal to \( \sqrt{10} \), it's classified as irrational. Other common examples include \( \pi \) and \( e \), which are cornerstone constants in mathematics. These numbers are crucial when dealing with continuous quantities and can arise in contexts varying from geometry to complex mathematical theories.