Run the Math

Real Numbers, Absolute Values, and Intervals
By Bryan Bartlett
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Real Numbers

Real Numbers (R) are any numbers that can be expressed in decimal form. This includes

Natural Numbers (N): Also known as counting numbers, this would be positive integers {1,2,3,4,5...}
Integers (Z): Which include all of N plus all negative Integers plus 0.
Rational Numbers (Q), which are any numbers that can be written as p/q, where p,q are integers
and Irrational Numbers, which are all non-terminating and non-repeating decimals, like the square root of 2, pi, or e.

Note: For the rest of the lectures, we will use N⁰ to denote the set of positive integers, with 0. When we want to talk about just the positive integers, we will use Z⁺. This is due to cultural differences on what the definition of N is, and we are all about keeping things simple

Real numbers are ordered, meaning they have a set, never changing order to them. 2 is always bigger than 1, and pi is always between 3 and 4. Given that real numbers are ordered, we can state the following rules when dealing with any 3 real numbers a,b and c.

If a < b and b < c, then a < c
if a < b, then a ± c < b ± c
if a < b and 0 < c, then a * c < b * c
if a < b and c < 0, then b * c < a * c
if 0 < a < b or a < b < 0, then 1 / b < 1 / a

Note: the rules are the same for >,≤ and ≥

Absolute Values

The absolute value of a number |a| is its magnitude: that is how far away from 0 it is.. regardless of the sign. This means |a| = a if a ≥ 0, and |a| = -a if a < 0. In practice, it means removing the sign.

Extra Credit: Think of the absolute value as the length of a line that goes from 0 to a. This becomes important later because the | | will come up again when dealing with vectors of higher dimensions. If you wanted to find |(x,y)|, you would use √(x² + y²), which is a positive number! We are doing the same thing in here but in one dimension.. √(a²)
Note: We will assume √ references the positive square root. When I want the negative square root, the convention would be -√

The following are rules that apply to the use of absolute numbers, given that a and b are real numbers

|a| > 0 if a ≠ 0, |0| = 0
|-a| = |a|
|a * b| = |a| * |b|
|a / b| = |a| / |b|, b ≠ 0
|a + b| ≤ |a| + |b| (Triangular Inequality)
|a - b| ≤ |a| + |b|
|a| - |b| ≤ |a - b|

Reminder: If you end up multiplying or dividing by a negative number, the inequality flips around!

Intervals and Neighborhoods

When we write intervals, we use a combination of parenthesis ( ) and Bracket [ ]. They allow us to set a specific group of numbers. Let us assume a ≤ b. If we wanted every number between a and b, but not including a and b, we would use parenthesis to show this, in the form (a,b). If we wanted every number between a and b, including a and b, we would use brackets, in the form [a,b]. You can combine them as well. [a,b) is the interval between a and b, including a but not including b. This mixed format is seen a lot when you have an infinite on one side or another. [a,∞) for example, would be all numbers greater or equal to a.

We can also write the interval in a set notation: (a,b) = {x|a < x < b}, which simply states All x, where x is greater than a and less than b.

Neighborhoods work a bit differently. In this, the segment is centered around a number a, instead of having a at one of the endpoints. An example of this would be {x|a - b< x < a + b} = (a - b,a + b). Otherwise, it works the same.

*Extra credit: A deleted neighborhood is one where a itself is removed from the interval.. [a - b,a) U (a, a + b]. U means union of two sets. since neither of the sets contains a, due to the parenthesis, this is an example of a deleted neighborhood.

Assignment:

1. Show that 0.125 is a real number using definition of Q
2. Using the definitions of a real number, show why √-1 is not a real number
3. -4x < 8 Using bracket notation, what is x?
4. Write the following in bracket notation: {x | 3a² - 4 ≤ x < a - b}
5. Write the following in set notation: (-∞,a + 5)