Algebra Chater 1 Section 1: Some Basics of Algebra

In this first section from Chapter 1, I'll cover some basics of Algebra, including basic terminology.

Basic Terms:
When a letter is used for a stand-in for a value which can take up any range of numbers, we call it a variable.  When a letter is used to represent a specific number (typically when we don't know what it is, but there can only be one), it is said to be a constant.

Our Constants
An example of this is the gross pay; the pay on your pay check before taxes is given by PG=H×W, where H is the hours worked and the W is your hourly wage.  Since hourly wage doesn't change from week to week, it is considered to be constant.  If you work part time, than your hours in any given week is considered a variable, because they do tend to change.

An algebraic expression consists of a combination of constants, variables, and operation signs.  For example, H×W from the pay equation above is considered an expression.  When an equals sign is placed between two expressions, we now have an equation of two expressions, such as the total pay equation PG=H×W.

When we are dealing with word problems (after all, we are humans who deal with words, not machines who deal with code), it is a good idea to be able to translate certain words into mathematics formulas.  Part of this is the notion that certain words and phrases translate into operators.  Here's a table of a few of those phrases:


Addition (+)
Subtraction (-)
Multiplication (×)
Division (÷)
Add
Subtract
Multiply
Divide
Sum
Difference
Product
Divided by
Plus
Minus
Times
Quotient
Increased by
Decreased By
Twice/Thrice, etc.
Ratio
More than
Less Than
Of
Per

When we are making a substitution, like in sports, we are replacing what's already in the equation with a value that holds true for the particular case at that particular time.  In this case, we're substituting variables and constants for numbers.  The process of substitution and running through the operations to get to a final answer is the process of evaluating the expression or equation.

One form of multiplication is the exponent.  The expression an, where n is a counting number, is an exponent expression and x is the exponent. This is a compact form of saying you want to multiply together n copies of a. For example, 54=5×5×5×5.

Now we may talk about the order of operations.  Here are the steps to be taken in this specific order:
  1. Calculate inside any parentheses or brackets from the inside out.
  2. Simplify all exponents.
  3. Perform all multiplications and divisions from left to right.
  4. Perform all additions and subtractions from left to right.
The reason for this particular order is to take into account that first we must "collect like terms" (the parentheses step), which will be explained later in this chapter.  An example of this is calculating the mean or the standard deviation in Statistics.  The rest of the ordering of the order of operations is merely trying to get to the point of adding or subtracting numbers of "similar magnitude", which is fancy-talk for same number of digits.  Exponents change magnitudes by the largest amount, and multiplication and division changes the magnitude by less, but still by much larger amount than adding or subtracting.

There are things in Mathematics called "Sets", and it is a good idea to get an idea of what they represent.  Here's a list of the sets
  • The Natural Numbers are the counting numbers, which begins with 1, ℕ={1, 2, 3, 4, ...}.
  • The Whole Numbers, which includes 0 with the natural numbers, ℕ0={0, 1, 2, 3, 4, ...}.
  • The Integers are all of the Whole Numbers as well as their negative counterparts, ℤ ={-4, -3, -2, -1, 0, 1, 2, 3, 4, ...}.
  • The Rational Numbers are all numbers which can be represented exactly by whole-number fractions, ℚ ={p/q}, where p and q are in ℤ and q≠0
  • The Irrational Numbers are those numbers which cannot be represented exactly by whole-number ratios. Examples of this are π and √2.
  • The Real Numbers ℝ are all numbers in both the Rational Numbers and the Irrational Numbers.
As a side note, when all members of one set are completely within another set, the first set is said to be a subset of the second.

Until next time, stay curious.

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