Problem 46 - Intermediate Algebra

Question

List the elements of $$ \left\\{-3,-\frac{8}{5}, 0, \frac{2}{3}, 1, \sqrt{3}, 2, \pi, 4.75,916 . \overline{6}\right\\} $$ that belong to the following sets. Odd prime numbers

Step-by-Step Solution

Answer

There are no odd prime numbers in the set.

Step 1: Identify the Set's Elements

First, copy down all the elements: \(-3, -\frac{8}{5}, 0, \frac{2}{3}, 1, \sqrt{3}, 2, \pi, 4.75, 916.\overline{6}\) and then determine if they are integers since only integers can be prime numbers.

Step 2: Determine Prime Numbers in General

A prime number is an integer greater than 1 that has no divisors other than 1 and itself. An odd prime number is a prime number that is not divisible evenly by 2. Note that 2 is the only even prime number.

Step 3: Filter Out Non-integers

Since prime numbers must be integers, discard any elements that aren't integers: \(-\frac{8}{5}, \frac{2}{3}, \sqrt{3}, \pi, 4.75, 916.\overline{6}\). This leaves us with integer candidates: \(-3, 0, 1, 2\).

Step 4: Evaluate Each Integer

Now, evaluate each remaining integer to see if it is an odd prime number. - \(-3\) is not a prime as prime numbers must be greater than 1.- \(0\) is not a prime.- \(1\) is not a prime as it does not have exactly two distinct positive divisors.- \(2\) is a prime number, but it is even.

Step 5: Conclude No Odd Primes

Since there are no odd integers greater than 2 in the list of integers, we conclude that there are no odd prime numbers present in the initial set.

Key Concepts

Integers

Integers are the set of whole numbers that can be either positive, negative, or zero. They do not include any fractions or decimal components. The integer set is represented by the symbol \( \mathbb{Z} \). Here’s a breakdown of what constitutes an integer:

• Positive integers: 1, 2, 3,...
• Negative integers: -1, -2, -3,...
• Zero: 0

In our original exercise, we must first identify which numbers are integers because prime numbers can only be whole numbers. This means discarding any fractions, decimals, or irrational numbers from our list.

Prime Numbers

Prime numbers are unique and hold a special place in mathematics. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime number has only two distinct positive divisors: 1 and the number itself.
To elaborate:

• A number like 7 is prime because its only divisors are 1 and 7.
• A number like 9 is not prime because it can be divided by 3 as well as 1 and 9.

In our exercise, the focus is on identifying if any element in the set is an odd prime number. It’s vital to remember here that 2 is the only even prime number. Thus, to be an odd prime number, the number must also be odd.

Odd Numbers

Odd numbers are integers that are not divisible evenly by 2. When you divide an odd number by 2, you will always have a remainder, whereas even numbers divide perfectly by 2 without any remainder.
Some properties of odd numbers include:

• The sequence of odd numbers: 1, 3, 5, 7, 9,...
• If you add or subtract two odd numbers, the result will be even.
• When you multiply two odd numbers, the result is always an odd number.

For an integer in our list to be an odd prime number, it must first be odd. This eliminates even numbers right from the beginning. Working from this characteristic significantly narrows down potential candidates for being an odd prime.

Number Sets

Number sets categorize numbers into groups based on their properties. Understanding which set each number belongs to helps in identifying characteristics like being an odd prime number.

Some key number sets include:

• Natural numbers (\( \mathbb{N} \)): 1, 2, 3,...
• Whole numbers: Similar to natural numbers but include zero: 0, 1, 2,...
• Integers (\( \mathbb{Z} \)): ..., -3, -2, -1, 0, 1, 2, 3,...
• Rational numbers (\( \mathbb{Q} \)): Numbers that can be expressed as fractions such as \( \frac{1}{2} \), \( -\frac{3}{4} \)
• Irrational numbers: Numbers that cannot be expressed as simple fractions, like \( \pi \) and \( \sqrt{3} \)

In solving the original exercise, it was crucial to sift through these numbers to sort them into their respective sets, recognize which are integers, and then further filter them based on the criteria for prime numbers.