## Forward and Backward

Algebra involves numbers, unknown numbers and operations between the numbers and unknown numbers. Algebraic expressions and equations are built with a series of operations. Each operation comes in pairs with a "forward" operation and a "backward" operation. The forward operations in algebra are addition and multiplication because they are easier than subtraction and division. The progression of operations both forward and backward in high school math is as follows:

Addition/Subtraction

Multiplication/Division

Powers/Roots

Exponentials/Logarithms

Trig operations/Arc Trig operations

Differentiation/Integration

The first 2 groups are covered in Algebra 1; the third group in Algebra 2; the fourth and fifth groups in Pre-calculus and the last group in Calculus.

For example the equation 3x + 2 = 14 is a series of operations applied to x. The order of operations requires the first step to be multiply x by 3. The second step is to add 2 to the result which gives 14. To solve for the unknown x we need to reverse the steps
in reverse order, just like reversing our steps: subtract 2 and then divide by 3 is the reverse of multiply by 3 and add 2. Algebraically (with no arithmetic) we get:

3x = 14 - 2, and then

x = (14-2)/3

Think of taking two steps; first with the right foot and then with the left. To reverse that, step backwards with the left foot followed by a backward step with the right foot: the reverse steps in the reverse order.

A more complicated example: if ln (sin x) = 0, then applying reverse operations in reverse order gives x = arcsin (e^0), which of course is arcsin (1) = pi/2.

There you have it, math in a nutshell. Any questions?