Some facts seldom taught in schools


1.) The invisible number 1.

Exponent of 1:

When the exponent on a number or variable is not written, it is understood to be 1.

There are two other positions where there is a 1 that is normally is not written.

In front of the number (1 times the number) and divided by 1 (the number over 1)

(so every number or variable can be treated as a fraction).

Example: Take the number 3. You can view this number the following way: 1(3)¹/1.

All numbers and variables have them. Take X. It can be written as 1(X)¹/1.

In case of a fraction, say 2/3. It can be written as (1)2¹/1(3)¹.

(1)A means 1 times A.


2) The three positions of a fraction.

a) out in front even with the division symbol

b) in front of the numerator (top)

c) in front of the denominator (bottom).

Changing the sign of any two does not affect the value of the fraction.

Example: 2/3: - (-2/3) (changed sign in front and on top), -(2/-3) (changed sign in front and

on bottom). -2/-3 (changed sign on top and on bottom).


3) The - sign.

In mathematics, this means -1, so when you see -X, it means (-1) times X.

On your TI-83 calculator there is a - sign at the bottom and one on the right column.

The one on the far right is used to connect two terms (see terms below) under the

operation of subtraction. The one at the bottom does not.

Example: When coding in the equation: Y= X-X², you would use the - sign on the right.

However, when coding in the equation: Y= -X², you would start with the - sign on the

bottom.


4) Terms: This is difficult for beginners to understand and often leads to many mistakes in

calculations. Look at the expression under consideration. If it is a single letter or a single

number, then it is one term. If the ENTIRE expression is a sum or difference (i.e., addition

or subtraction), each piece of the addition or subtraction are separate terms.

Example: x + y (two terms), x-y+z (three terms). If the entire expression is composed of

multiplication or division, it is ONE term no matter how complicated.

Example: 5x is one term. 5x - 3y (two terms). Here is the tricky one: 5(x+y) is ONE term

Even though a piece consists of two terms. But the ENTIRE expression written is

composed with multiplication, hence ONE term. If you multiplied the 5 across the

parenthesis to get 5x +5y, you have just made a one-term expression into a two-term

expression.

Examples: 5xy- 6xy-15 (three terms). 2x²y³[z/7(x + y)] is one term

Example: When solving equations, one often says "take a given term to the other side of

the equation". That will change the sign of the term moved.

When moving a term from one side of the equation to the other side, it will be added/subtracted from the

terms on that side depending on its sign change.

Example: Solve for a: 3b - a = 52.

I would tell my students, "Take the 2nd term on the left side (-a) to the right side (becomes

+a or just a) and take the 52 on the right side to the left side (becomes -52 on the left side)

to get: 3b -52 = a (done). Solving for a variable means getting it on one side alone (does not matter which side)

with a coefficient (number in front of it) of +1 and it does not occur anywhere else in the equation.


5. The invisible number 10.

During lesson #10 of our course, I will introduce and develop the exponential and logarithmic functions. So, don't worry

about them at this present time.

The logarithmic functions are written as log_bA where b will be the positive base of the function.

Example: log₃7 reads "log of 7 to the base 3"
Example: log₂x reads "log of x to the base 2"
Example: log₁₀x reads "log of x to the base 10"
Example: log_ex reads "log of x to the base e"

After lesson #10 you will have a good understanding of their meaning and will realize the importance of the positive number e.

Since most students never get to Calculus (unfortunately), the only logarithm they encounter (if any) will be the base 10 log.

Therefore, they are referred to as "the COMMON logs" and the 10 is not written.

Example: log x reads "log x base 10". There is a button on your calculators for log. This is exclusively for base 10.

Students of higher mathematics where calculus dominates primarily use the base e log. (will explain when we get to lesson #10).

It is so special in calculus that it's given its own special symbol, namely ln

Example: ln x reads "log x base e" and rarely written as log_ex.

There is a button on your calulator for this as well.

These logs are referred to as "NATURAL" logs and are involved in many calculus based problems.