Real numbers are the foundation of our number system, encompassing all the numbers you can think of on a number line. They include both rational and irrational numbers, giving us a complete set to work with in algebra and beyond.
Note
Real numbers include all positive and negative numbers, as well as zero. They can be represented as points on a continuous number line.
Rational numbers are numbers that can be expressed as a fraction of two integers (where the denominator is not zero). These include:
Example
The number 3/4 is rational because it can be expressed as a fraction of two integers: 3 (numerator) and 4 (denominator).
Irrational numbers are numbers that cannot be expressed as a simple fraction. They have decimal representations that neither terminate nor repeat. Some famous irrational numbers include:
Common Mistake
Many students mistakenly believe that all decimals are rational. However, non-repeating, non-terminating decimals like 0.101001000100001... are irrational.
Understanding the properties of real numbers is crucial for solving algebraic problems. Let's explore some key properties:
For any two real numbers $a$ and $b$:
Tip
The closure property ensures that when we perform basic operations on real numbers, we always get another real number as the result.
For any real numbers $a$ and $b$:
Example
$5 + 3 = 3 + 5 = 8$ $2 \times 4 = 4 \times 2 = 8$
For any real numbers $a$, $b$, and $c$:
Note
The associative property allows us to group numbers differently without changing the result.
For any real numbers $a$, $b$, and $c$:
This property is particularly useful in algebra for expanding expressions.
Example
$3(2 + 5) = 3(2) + 3(5) = 6 + 15 = 21$
Real numbers can be ordered on a number line, which is a fundamental concept in algebra. Here are some key points:
<, >
, ≤, ≥) to compare real numbers.
Tip
When comparing irrational numbers, it can be helpful to approximate them as decimals to determine their order on the number line.
An interesting property of real numbers is their density. Between any two real numbers, there are infinitely many real numbers.
Example
Between 0 and 1, we have: 0.1, 0.01, 0.001, 0.0001, and so on... As well as: 0.5, 0.25, 0.125, 0.0625, and so on... And countless others!
This property is crucial in advanced mathematics and helps us understand the continuous nature of the real number line.
Understanding the real number system is fundamental to success in Algebra I and beyond. By mastering these concepts, you'll have a solid foundation for tackling more complex mathematical ideas in the future.