Page 1.2: Real Numbers
Although some of the terminology might be new, the ideas presented in this section should be somewhat familiar to us.
For us, our entire study of calculus will deal with real numbers . In order to rigorously define what we mean by a real number, a lot of machinery that is beyond the scope of our calculus courses are required and so we will rely on our intuition to understand real numbers. Allow us to review some types of numbers.
Definition: The natural numbers are defined to be the set containing the numbers \(1, 2, 3, 4, \ldots\). We denote the set of natural numbers by \( \mathbb{N} \). So \( \mathbb{N} = \{1, 2, 3, 4, \ldots \} \).
Definition: The integers consist of the natural numbers together with the negatives and zero. We denote the set of integers by \( \mathbb{Z} \). So \( \mathbb{Z} = \{\ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots \} \). .
Definition: The rational numbers consist of all numbers of the form \( \dfrac{m}{n} \) where \(m\) and \(n\) are integers and \(n \neq 0\). We denote the set of integers by \( \mathbb{Q} \).
The above definition is saying that a rational number is just a ratio of two integers. Here are some examples of rational numbers: \( \frac{1}{2}, \frac{9}{10}, 0.61 = \frac{61}{100}, - 0.23 = - \frac{23}{100} \).
However, not all real numbers can be expressed as a ratio of two integers. For instance, although we do not prove this here, note that \( \sqrt{2} \) cannot be expressed as a ratio of two integers. Additionally, \( \pi \) cannot be expressed as a ratio of two integers. These numbers are called irrational numbers . Putting the rational numbers and irrational numbers together yields what we call the set of real numbers which we denote by \( \mathbb{R} \).
The real numbers follow certain rules which we are familiar with. These rules are the presented below.
Suppose \(a, b\) and \(c\) are real numbers. Then the following are true:
1) \( a + b = b + a \) (this is called the commutative property of addition)
2) \( a \times b = b \times a \) (this is called the commutative property of multiplication)
3) \( (a + b) + c = a + (b + c) \) (this is called the associative property of addition)
4) \( (a \times b) \times c = a \times (b \times c) \) (this is called the associative property of multiplication)
5) \( a(b+c) = ab + ac \) (this is called the distributive property)
In words, the above list is saying when we add two numbers, the order does not matter. The same is true for multiplication. Additionally, when we add three numbers, it does not matter which two we add first. The same can be said for multiplication. And finally, property 5 is saying that when we multiply a number by a sum of two numbers, we can multiply the outside number to each inside number and add the resulting products.