## Basics & Number Properties

All equations have various parts. The variable represents the unknown. We can tell what "x" represents in this particular equation, almost by guessing. WHAT + 23 = 45?

The VARIABLE here is "x"

The CONTANT is 23, because it is "constantly" there, and we have already identified its value as 23.

Mathematically, we can obtain what "WHAT" equals, by subtracting 23 from 45.

45 - 23 = 22

Therefore, x = 22

{module [79]}

A negative number is any number to the left of 0 on the number line.

By contrast, a positive number is any number to the right of 0 on the number line.

The technical rule to remember here is that for any real number a:

(-1)(-a) = a.

Absolute value is the distance on the real number line between any given number and zero.

Absolute value can never be negative, because technically distance can NEVER be negative.

Distance is always positive.

Absolute value is denoted between | number | - so to the reader, it will appear as: |-3| = 3.

**Absolute value is the distance on a number line between any given point and zero.**

Therefore, because distance is always POSITIVE (you cannot have a negative distance), absolute value is also always POSITIVE.

For example, if you have |x| = 2

This means that x = -2 or 2

WHY? Because distance is always POSITIVE.

Therefore, remember: Any variable in brackets could have TWO values, one positive and one negative.

A **set **is a collection of "elements," or "members," and each set is entirely determined by its members. If \[ x \]is a member of a set \[ U, \] we write \[ x \in U. \] In basic algebra, the elements of a set are usually numbers.

We can also designate a set by enclosing its members in braces , { }.

Counting Numbers:

### {1, 2, 3, 4, 5, 6, ...}

BY THE WAY, "..." notates continuation going up infinitely.

The set of **Natural Numbers** \( \mathbb{N}\ \ \) = {0, 1, 2 3, 4, 5, 6...}.

The set of **Integers**, \( \mathbb{Z}\ \ \) = {..., -3, -2, -1, 0, 1, 2, 3}.

The set of **Rational numbers**, \( \mathbb{Q}\ \ \) is the set of ratios of integers.

That is, every \(q \in \mathbb{Q}\ \)

is of the form:

\( q = \frac{n}{m} \ \)

With \( n, m \in \mathbb{Z}\ \) .

The set of **Real Numbers** \( \mathbb{R}\ \ \) is the set of limits of sequences of rational numbers.