Problem 25 - Beginning and Intermediate Algebra

Question

Give three numbers between -6 and 6 that satisfy each given condition. Real numbers but not whole numbers

Step-by-Step Solution

Answer

Three examples are 1.5, -3.2, and 0.75.

Step 1: Understand the Range

The numbers need to be between -6 and 6, which means they must be greater than -6 and less than 6.

Step 2: Identify Real Numbers

A real number is any number that can be found on the number line. This includes rational and irrational numbers, but not imaginary numbers.

Step 3: Exclude Whole Numbers

Whole numbers are 0, 1, 2, 3, 4, 5, and 6, along with their negative counterparts. To satisfy the condition, avoid these numbers.

Step 4: Select Non-Whole Numbers

Choose numbers that are not whole numbers. For example, 1.5, -3.2, and 0.75.

Key Concepts

Number Line

The number line is a visual representation of all real numbers in a straight line. The number line extends infinitely in both the positive and negative directions. Each point on the line corresponds to a real number. For example, \(3\) and \(-3\) are points on the number line, as well as fractions like \(\frac{1}{2}\) and \(\frac{-1}{2}\). To find any number, simply locate its position based on its value.

The number line can help you understand the relationship between numbers. For example:

• Numbers to the right of \(0\) are positive, such as \(1\), \(2\), and \(3\).
• Numbers to the left of \(0\) are negative, such as \(-1\), \(-2\), and \(-3\).

The distance from \(0\) on the number line is called the absolute value of the number.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the numerator and the denominator are integers and the denominator is not zero. For example, \(\frac{1}{2}\), \(3\), and \(-4\) are all rational numbers. Rational numbers can also be represented as decimals, which either terminate or repeat.

Examples of rational numbers:

• \(\frac{3}{4}\) (which is \(0.75\) in decimal form)
• \(0.333...\) (repeating as \(0.\bar{3}\))
• \(\frac{-7}{5}\) (which is \(-1.4\) in decimal form)

These numbers fit perfectly on the number line.

Irrational Numbers

Irrational numbers, on the other hand, cannot be expressed as a simple fraction. They are non-repeating and non-terminating decimals. Examples of irrational numbers include \(\frac{\root{2}}\), \(\frac{\root{3}}\), and \(\frac{\root{5}}\). These numbers go on forever without repeating patterns.

Examples of irrational numbers:

• \frac{\root{2}} (approximately \(1.414...\))
• \(\frac{\root{3}} (approximately \)1.732...\()
• \)\frac{\root{5}} (approximately \(2.236...\))

Even though these numbers seem 'messy', they still fit somewhere on the number line, just like rational numbers.

Whole Numbers

Whole numbers are the set of non-negative integers including zero. This means \(0\), \(1\), \(2\), \(3\), \(4\), etc. Whole numbers do not include negative numbers or fractions.

Examples of whole numbers include:

• \(0\)
• \(1\)
• \(2\)
• \(3\)

Whole numbers can be easily found on the number line. They are useful in counting objects, representing quantities, and performing basic arithmetic operations.