Problem 1 - Algebra 1: Concepts and Skills

Question

Define rational number. Which of the following are rational numbers? $$5, \frac{2}{3}, \frac{-17}{2}, \sqrt{3}, 1.45,0, \pi$$

Step-by-Step Solution

Answer

The rational numbers from the list are 5, \( \frac{2}{3} \), \( \frac{-17}{2} \), 1.45, and 0.

Step 1: Definition of rational numbers

A rational number is any number expressed as the quotient (or division) of two integers, with the condition that the denominator (the number below the division bar) is not zero.

Step 2: Identify rational numbers

Apply this definition to the numbers given: 5, \( \frac{2}{3} \), \( \frac{-17}{2} \), \( \sqrt{3} \), 1.45, 0 and \( \pi \). If it can be expressed as a quotient of two integers where the denominator is not zero, then it's a rational number.

Step 3: Check each number

5 can be expressed as \( \frac{5}{1} \). \( \frac{2}{3} \) and \( \frac{-17}{2} \) are already in the form of rational numbers. \( \sqrt{3} \) is an irrational number because it can't be expressed as a fraction. Similarly, \( \pi \) is an irrational number because its decimal representation never ends nor repeats. 1.45 can be expressed as \( \frac{145}{100} \). Lastly, 0 is a rational number because it can be expressed as \( \frac{0}{1} \).

Key Concepts

Definition of Rational Numbers

Rational numbers play a big role in our understanding of mathematics. A rational number, in simple terms, is any number that can be written as a ratio or a fraction of two integers. The word "integer" means whole numbers, which include both positive and negative numbers, as well as zero. For a number to be considered rational, there is also an important rule: the denominator cannot be zero. This is because division by zero is undefined in mathematics.

• The general form of a rational number is \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \).
• Examples include numbers like \( \frac{1}{2} \), \( 3 \), and even \( -4 \).

You might wonder what happens with decimal numbers. Many decimals can also be expressed as fractions, which makes them rational numbers too. A few examples are 1.5 (which is \( \frac{3}{2} \)) and 0.75 (which is \( \frac{3}{4} \)). As long as a number can be simplified to a fraction with integers as numerator and denominator, it's rational.

Identifying Rational Numbers

Now that we know what rational numbers are, let's learn how to identify them. It's quite simple! We need to check if the number can be written as a fraction of two integers with a non-zero denominator.
Consider the following:

• Whole numbers like 5 are rational because they can be expressed as \( \frac{5}{1} \).
• Fractions such as \( \frac{2}{3} \) or \( \frac{-17}{2} \) are explicitly in the form of rational numbers.
• Decimals that either terminate or repeat can often be turned into fractions, making them rational too, like 1.45, which equals \( \frac{145}{100} \).

To practice, take a number and see if you can write it as \( \frac{a}{b} \), with both \( a \) and \( b \) as integers (and \( b eq 0 \)). If you can, it's a rational number. If not, it's not rational. Understanding this helps in many areas of math, from basic operations to more complex algebraic equations.

Examples of Irrational Numbers

Unlike rational numbers, irrational numbers cannot be expressed as a simple fraction of two integers. This means their decimal forms do not terminate or repeat. Let's explore a few familiar examples to understand this concept:

• The number \( \sqrt{3} \), or the square root of 3, cannot be expressed as a fraction. Its decimal representation goes on forever without repeating.
• Perhaps the most famous irrational number is \( \pi \). This is used in many mathematical calculations involving circles. \( \pi \) is approximately 3.14159, but much like \( \sqrt{3} \), its decimal form is non-repeating and infinite.

Recognizing irrational numbers is key because they're everywhere in higher math and science. They help scientists and mathematicians model and calculate phenomena that can't be neatly summarized with whole numbers or simple fractions, showing the beauty and complexity of numbers beyond the everyday counting we know.