## Absolute Value: Notation, Expressions, Equations

### What Does Absolute Value Mean?

The term “Absolute Value” refers to the magnitude of a quantity without regard to sign. In other words, its distance from zero expressed as a positive number.

The notation used to indicate absolute value is a pair of vertical bars surrounding the quantity, sort of like a straight set of parentheses. These bars mean: evaluate what is inside and, if the final result (once the entire expression inside the absolute value signs has been evaluated) is negative, change its sign to make it positive and drop the bars; if the final result inside the bars is zero or positive, you may drop the bars without making any changes:

$\lvert ~1-4~ \rvert\\*~\\*=~\lvert ~-3~ \rvert\\*~\\*=~3$

Another example is:

$\lvert ~4-1~ \rvert\\*~\\*=~\lvert ~3~ \rvert\\*~\\*=~3$

Note that absolute value signs do not instruct you to make “all” quantities inside them positive. Only the final result, after evaluating the entire expression inside the absolute value signs, should be made positive.

$\lvert ~1-4~ \rvert~\ne~\lvert ~1+4~\rvert~~\text{ Do not make this mistake!}$

### Absolute Value expressions that contain variables

Just as with parentheses, absolute value symbols serve as grouping symbols: the expression inside the bars must be evaluated and expressed as either Continue reading Absolute Value: Notation, Expressions, Equations

## Combining or Collecting Like Terms

The phrases “combine like terms” or “collect like terms” are used a lot in algebra, and for good reason. The process they describe is used a lot in solving algebra problems. Two approaches, one intuitive and the other algebraic, can help in understanding why some terms are “like” terms, and others are not.

### Quantities With Units

Suppose you are sitting in front of a table that holds three piles of fruit:
– five apples
– three oranges
– four apples
If someone asks you “What do you see on the table?”, how would you answer the question?

Chances are you answered “nine apples and three oranges”. Why did you combine the two piles of apples with one another, but not with the oranges? How did you know that you could do that?

The quantities of apples may be combined because addition or subtraction only work with  Continue reading Combining or Collecting Like Terms

## Simplifying Fractions

Three concepts help explain the process of simplifying fractions:

1. Multiplying a quantity by 1 has no effect
2. A fraction whose numerator is exactly the same as its denominator is equal to 1 (unless the denominator equals zero)
$\dfrac{17a^2b}{17a^2b}~~=~~1~~~~a\ne 0,~~b\ne 0$
3. A product of two fractions can be rewritten as a fraction of two products (and vice versa)
$\dfrac{a}{b} \cdot \dfrac{c}{d}~~=~~\dfrac{ac}{bd}\\*~\\*\dfrac{ac}{bd}~~=~~\dfrac{a}{b} \cdot \dfrac{c}{d}$

To simplify a fraction:

• Rewrite both numerator and denominator as products of factors (if they are not already factored)
• Examine both numerator and denominator to see if they share any factors
• If they do share factors, use concept (3) above to move the shared factors into a separate fraction
• That separate fraction should now have a numerator that is exactly the same as its denominator, which by concept (2) above means that it must equal 1, therefore by concept (1) above we can drop it from the expression

Consider the following fraction… can it be simplified? Continue reading Simplifying Fractions

## Negative Differences

Algebra is a set of rules that allow us to change the appearance of an expression without changing the quantitative relationship that it represents. Sometimes the changes in appearance are greater than expected, causing us to doubt whether two expressions really do represent the same quantitative relationship.  The ways in which negative differences can be rewritten seem to surprise people until they become accustomed to them.

Consider a difference that is being subtracted:

$b-(a-3)$

If we wish to eventually drop the parentheses, we’ll have to distribute the negative sign that is in front of them first.  Leaving the parentheses in place while Continue reading Negative Differences

## Negative Fractions

Question: Where should I put the negative sign when I am writing a fraction like negative two thirds?

Answer: As long as you write only one negative sign, it does not matter where you put it.

Two ideas are useful to keep in mind during the explanation that follows:
– Subtraction is the same thing as the addition of a negative.
– The negative of a number can be created by multiplying the number by negative one.

These principles apply to fractions as well, so:

$-\dfrac{3}{5}\\*~\\*~\\*=(-1)(\dfrac{3}{5})\\*~\\*~\\*=(\dfrac{-1}{~1})(\dfrac{3}{5})\\*~\\*~\\*=\dfrac{-3}{~5}$

Placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to Continue reading Negative Fractions

## Geometric Sequences and Geometric Series

### Geometric Sequences / Progressions

The terms “sequence” and “progression” are interchangeable. A “geometric sequence” is the same thing as a “geometric progression”. This post uses the term “sequence”… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create a geometric sequence (also known as a geometric progression).

Pick a number, any number, and write it down.  For example:

$5$

Now pick a second number, any number (I’ll choose 3), which we will call the common ratio. Now multiply the first number by the common ratio, then write their product down to the right of the first number:

$5,~15$

Now, continue multiplying each product by the common ratio (3 in my example) and writing the result down… over, and over, and over:

$5,~15,~45,~135,~405,~1,215, ...$

By following this process, you have created a “Geometric Sequence”, a sequence of numbers in which the ratio of every two successive terms is the same.

### Vocabulary and Notation

In the example above, 5 is the first term (also called the starting term) of the sequence or progression. To refer to the first term of a sequence in a generic way that applies to any sequence, mathematicians use the notation

$a_1$

This notation is  Continue reading Geometric Sequences and Geometric Series

## Arithmetic Sequences and Arithmetic Series

### Arithmetic Sequences / Progressions

The terms “sequence” and “progression” are interchangeable. An “arithmetic sequence” is the same thing as an “arithmetic progression”. This post uses the term “sequence”… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create an arithmetic sequence (also known as an arithmetic progression).

Pick a number, any number, and write it down.  For example:

$5$

Now pick a second number, any number (I’ll choose 3), which we will call the common difference. Now add the common difference to the first number, then write their sum down to the right of the first number:

$5,~8$

Now, continue adding the common difference to the sum and writing the result down… over, and over, and over:

$5,~8,~11,~14,~17,~20,~23,~26,~29, ...$

By following this process, you have created an “Arithmetic Sequence”, a sequence of numbers that are all the same distance apart when graphed on a number line:

### Vocabulary and Notation

In the example above 5 is the first term, or starting term, of the sequence. To refer to the starting term of a sequence in a generic way that applies to any sequence, mathematicians use the notation

$a_1$

This notation is Continue reading Arithmetic Sequences and Arithmetic Series