1.1: Real Numbers

Real Numbers

Let's review what the real numbers are.

The natural numbers look like $1, 2, 3, 4, 5, \dots$.

Integers comprise of the natural numbers, their negatives, and zero: \[\dots, -3, -2, -1, 0, 1, 2, 3, \dots\]

Rational numbers are all possible ratios of integers. They look like \[r = \dfrac{m}{n}\] where $m$ and $n$ are integers, $n \neq 0$.

Irrational numbers are numbers that cannot be written as a ratio of integers. For example \[\sqrt{2}, \qquad \sqrt{3}, \qquad \pi, \qquad \dfrac{3}{\pi},\] are all irrational.

Real numbers are numbers that are either rational or irrational.

If a number is rational, its decimal representation repeats. If it is irrational, the decimal representation will never repeat.

For each number, decide if it is natural, integer, rational, or irrational. It can be more than one.

Properties of Real Numbers

Let's discuss the mathematical operation of addition.

If I give you 4 apples, then 6 apples later, you end up with 10 apples.

Now if I give you 6 apples first, then four, you still end up with 10 apples.

This shows in addition, order does not matter.

Now for multiplication. Suppose I would like to add up the number 5 sixteen times. Here it is: \[5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5\]

That is quite cumbersome. Multiplication is used as a shorthand to communicate how many times you would like to add.

Adding up 5 sixteen times is simply denoted $5\cdot 16$.

Here are common properties of real numbers.

Let $a, b, c$ be real numbers.
Which statements are true? Point out which property was used.
  1. $3 + 2 + 4 = 3 + 4 + 2$
  2. $3 + 2 \cdot 4 = 3 + 4 \cdot 2$
  3. $5 \cdot 2 + 4 = 5 \cdot 4 + 2$
  4. $4(x + y) = 4x + y$
  5. $5(x + y + 2) = 5x + 5y + 5\cdot2$

Tip: In practice, when you are solving math problems and you write the equals sign $=$, you need to be able to point out which property or fact you used.

You can see in problem 4 we were unable to point out a valid mathematical property that was used. This means it is incorrect.

Use the distributive property on the following:
  1. $2(x + 3)$
  2. $(a + b)(x + y)$

Addition and Subtraction

Suppose $a$ is a real number.

The number 0 is called the additive identity since $a + 0 = a$; the number $a$ is left unchanged.

The number $a$ has a negative, denoted $-a$. This negative has the special property that \[a + (-a) = 0\]

Subtraction is the operation that undoes addition. So technically subtraction doesn't exist as an operation; it is just addition on a negative number: \[a - b = a + (-b)\]

Properties of Negatives
Suppose $a,b$ are real numbers.
  1. $(-1)a = -a$
  2. $-(-a) = a$
  3. $(-a)b = a(-b) = -(ab)$
  4. $(-a)(-b) = ab$
  5. $-(a+b) = -a-b$

Property 1 says whenever you see a number $-a$, it's really $-1$ times $a$.

Use the properties of negatives to simplify the expression:
  1. $-(x+2)$
  2. $-(-(-3))$
  3. $-(x+y-z)$

The Approach

Some of mathematics works like this: you get some rules and properties, and you are expected to see when and how to use them accurately.

Problems can be extremely complicated and it can be hard to identify when to use what property.

To accurately do this in calculus, you need to be able to identify terms and factors.

Terms and Factors

Terms are entities separated by subtraction and addition.

Factors are entities separated by multiplication and division.

When identifying terms and factors, you need to specify which context you are working with.

Many common mistakes made by students are due to not doing this, therefore applying the wrong property or even applying the property incorrectly.

Identify the terms and factors in the global context.
  1. $x + 2$
  2. $(x+2)(x+4)$
  3. $(x+2)(x+4) - 3$

For fractional expressions, only consider the global context of just the numerator and denominator when labeling terms/factors.

Identify the terms and factors and state the context.
  1. $\dfrac{(x^2 + 1)(x + 3) + 4}{x + 3}$

Multiplication and Division

The number 1 is called the multiplicative identity because it is the only real number where $a \cdot 1 = a$.

Similar to negatives, every real number $a$ has an inverse, denoted $1/a$ where \[a \cdot \dfrac{1}{a} = 1\]

Division is the operation that undoes multiplication. To divide by $b$, we just multiply by $1/b$. If $b\neq 0$, \[a\div b = a \cdot \frac{1}{b} = \dfrac{a}{b}\]

The following properties describe how we interact with fractions:

Properties of Fractions
Suppose $a,b,c,d$ are real numbers.
  1. $\dfrac{a}{b}\cdot \dfrac{c}{d} = \dfrac{ac}{bd}$
  2. $\dfrac{a}{b}\div \dfrac{c}{d} = \dfrac{a}{b}\cdot \dfrac{d}{c}$
  3. $\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}$
  4. $\dfrac{a}{b} + \dfrac{c}{d}\qquad $ Find the LCD, then use Fraction Law 3.
  5. $\dfrac{ac}{bc} = \dfrac{a}{b}$
Simplify:
  1. $\dfrac{2}{3} \cdot \dfrac{5}{7}$
  2. $\dfrac{2}{5} + \dfrac{7}{5}$
  3. $\dfrac{1}{3} + \dfrac{1}{3} + \dfrac{4}{3}$
  4. $\dfrac{2}{3} \div \dfrac{5}{7}$
  5. $\dfrac{5}{36} + \dfrac{7}{120}$

Here we can already see the importance of terms and factors. In Fraction Law #5, the only time $c$ can be cancelled out is when $c$ is a factor in the global context of the numerator and denominator.

Use the proper fraction law to simplify the following:
  1. $\dfrac{1}{x + 1} + \dfrac{1}{x + 2}$
Can I cancel out the $x^2$ in \[\dfrac{x^2 + 1}{x^2 + 2}\] Why or why not?
Can you cross out the $x-1$ in \[\dfrac{(x-1)(x+2) + 3x^2}{(x-1)(x+3)}\] Why or why not?
Can you cross out the $x$ in \[\dfrac{x(x-1)(x+1)}{2x + 1}\] Why or why not?

The Real Line

The real numbers can be represented with a horizontal line. See in class...

Sets and Intervals

A set is a collection of objects. These objects are called elements of the set.

For example, the set of all real numbers is denoted $\mathbb{R}$. If $a$ is a real number, we write $a \in \mathbb{R}$ meaning $a$ is an element of the real numbers.

Other common sets are

  1. $\mathbb{N}$, the set of natural numbers
  2. $\mathbb{Z}$, the set of integers
  3. $\mathbb{Q}$, the set of all rational numbers

There are two ways to describe a set.

The first is to list out all the elements: \[A = \{1,2,3,4,5,6\}\]

The second is to use set-builder notation: \[A = \{\text{objects} \ : \ \text{condition}\}\]

Set $A$ above could also be described as \[A = \{x : x \in \mathbb{Z} \text{ and } 0 < x < 7\}\]

If $S$ and $T$ are sets, there are two operations:

  1. union, denoted $S \cup T$, which is the set that contains all elements that are in $S$ or $T$
  2. intersection, denoted $S \cap T$, which is the set that contains all elements that are in $S$ and $T$

If \[S = \{1,2,3,4,5\} \qquad T = \{4,5,6,7\} \qquad V = \{6,7,8\}\] find
  1. $S \cup T$
  2. $S \cap T$
  3. $S \cap V$

Sets of real numbers are called intervals. I will describe them in class.

Describe in English what each set means:
  1. $(1,3) \cup [4,7]$
  2. $(-\infty,-3) \cup (3, \infty)$
  3. $(-\infty, -2) \cup (0, 1) \cup (1, \infty)$
Use set builder notation to describe the set "all real numbers except for 1 and 4."

Absolute Value and Distance

If $a$ is a real number, then the absolute value of $a$ is \[\lvert a \rvert = \begin{cases} a & a \geq 0 \\ -a & a < 0 \end{cases}\]
Simplify:
  1. $\lvert 3 \rvert$
  2. $\lvert -3 \rvert$
  3. $\lvert-\lvert -3 \rvert\rvert$
  4. $\lvert 2 - \pi \rvert$