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.
$\dfrac{1}{2}$
$3.5474747\dots = 3.5\overline{47}$
$5$
$\sqrt{2} = 1.414213562373095\dots$
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 $b\neq 0$, \[a\div b = a \cdot \frac{1}{b} = \dfrac{a}{b}\]
Hey look: above we had fractions! Here are the five ways to deal with fractions.
Properties of Fractions
$\dfrac{a}{b}\cdot \dfrac{c}{d} = \dfrac{ac}{bd}$
Meaning Multiplying fractions requires multiplying global context of the numerators and global context of the denominators.
Meaning Dividing fractions requires taking the reciprocal of the right fraction, then multiplying.
$\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}$
Meaning Adding fractions with the same denominator requires adding the global context of the numerators together.
$\dfrac{a}{b} + \dfrac{c}{d}\qquad $ Find the LCD, then use Fraction Law 3.
$\dfrac{ac}{bc} = \dfrac{a}{b}$
Meaning Cancelling an entity $c$ requires the entity to be a global factor in both the numerator and denominator.
Can I cancel out the $x^2$ in \[\dfrac{x^2 + 1}{x^2 + 2}\]
Why or why not?
Cancelling requires global factors.
$x^2$ is a global term in numerator/denominator. You cannot cancel.
Can you cross out the $x-1$ in \[\dfrac{(x-1)(x+2) + 3x^2}{(x-1)(x+3)}\]
Why or why not?
Cancelling requires global factors.
$(x-1)$ is a global factor in denominator but $(x-1)$ is a local factor in the numerator. You cannot cancel.
Can you cross out the $\color{pink}x$ in \[\dfrac{ {\color{pink}x} (x-1)(x+1)}{2 {\color{pink}x} + 1}\]
Why or why not?
Cancelling requires global factors.
$x$ is a global factor in numerator but $x$ is a local factor in the denominator. You cannot cancel.
We see importance of terms and factors.
In Fraction Law #5, $c$ can be cancelled out only when $c$ is a global (L1) factor in both numerator/denominator.
Simplify. State the law you are using every time you write the $=$ sign.
$\dfrac{2}{3} \cdot \dfrac{5}{7}$
$\dfrac{2}{5} + \dfrac{7}{5}$
$\dfrac{2}{3} \div 2$
$\dfrac{2}{3} \div \dfrac{5}{7}$
$\dfrac{5}{36} + \dfrac{7}{120}$
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.
Properties of Real Numbers
Let $a, b, c$ be real numbers.
Commutative Properties
$a + b = b + a$
$ab = ba$
Associative Properties
$(a + b) + c = a + (b + c)$
$(ab)c = a(bc)$
Tip: in English, the commutative/associative properties only interact within the same context level (e.g. only in L3)!
Distributive Property
$a(b + c) = ab + ac$
$(b+c)a = ab + ac$
Tip: in English, the distributive law describes how terms and factors interact. This is the only way they can interact; anything else is false.
Which statements are true? Point out which property was used (possibly incorrectly).
Tip: In practice, when 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.
Practicing writing down the laws will help you use properties correctly.
When you feel comfortable skipping writing down the property, do so.
Apply the distributive property once:
$y(x-2)$
$(a + b + c)(x + y)$
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.
$(-1)a = -a$
$-(-a) = a$
$(-a)(-b) = ab$
$-(a+b) = -a-b$
Property 1 says whenever you see a number $-a$, it's really $(-1)\cdot a$.
Negatives are really a factor of $(-1)$.
Use the properties of negatives to simplify the expression:
$-(x+2)$
$-(-(-3))$
$-(-x + 4)-(x - 3)$
The Real Line
The real numbers can be represented with a horizontal line. See in class...
Interval Notation
Sets of real numbers are called intervals. I will describe them in class.
Describe in English what each set means:
$(1,3) \cup [4,7]$
$(-\infty,-3) \cup (3, \infty)$
$(-\infty, -2) \cup (0, 1) \cup (1, \infty)$
Absolute Value and Distance
If $a$ is a real number, then the absolute value of $a$ is \[\lvert a \rvert = \begin{cases} a & a \geq 0 \\ -a & a < 0 \end{cases}\]