Algebra - The Gateway to Higher Mathematics
Algebra is the gateway to all higher mathematics. No two ways about it: if you do not master algebra, kiss your mathematical pursuits goodbye. Yet struggles with algebra often are encountered because of common pitfalls which can be avoided. Here we discuss how to overcome some common mistakes and how to avoid some nasty pitfalls. If you avoid just these two pitfalls, you will be way on your way to mastering this discipline.
Pitfall 1: Know the Power of the Negative Sign Outside Parentheses -( )
This has to be the foremost pitfall in all of algebra, and this error ripples all the way up to calculus and beyond. Indeed I remember a time back in level two calculus when this error reared its ugly head. I remember one particular problem that my professor was doing in class on the blackboard (back in the 80´s we did not have whiteboards).
My professor had covered the entire board with numbers and other algebraic symbols, a full four panels worth, only to realize that the answer he arrived at did not match the answer given in the book (this was a homework problem for which a student had requested the solution).
When he backtracked to try and find the error, he realized that at one point he had failed to distribute the negative sign over the parentheses -( ). This one mistake rippled through the whole problem and led to the wrong answer. Once the distribution of the negative sign was carried through, the correct answer was obtained. This story illustrates the importance of obeying this rule. Simply put, when there is a negative sign outside parentheses, make sure that you change the sign of every term inside parentheses upon their removal. Thus if you have something like -(2x – 3y – 4), make sure that you change the sign of every term to its opposite to obtain -2x + 3y +4. If you do this, you will have eliminated this trap forever.
Pitfall 2: Never Cancel a Term from a Fraction Unless Every Term in the Numerator Contains That Term or a Factor of That Term
This is a killer pitfall. I have seen this mistake so many times that I could write a book on it. This pitfall occurs because students are never shown why cancellation is valid. Cancellation is a property of the field of real numbers, within which most of algebra occurs. If you forgot, the real numbers are those which include the whole numbers, the negative integers, the fractions, and the irrationals. The cancellation property basically says that if you have ab/a and a is not equal to 0, then this fraction becomes b. For example, 2x/2 = 2, and 5x/x = 5, provided x does not equal 0.
In the above examples, little difficulty is encountered. The problem occurs when the numerator contains more than one term joined by addition or subtraction. For example, take the expression (3x + 2y)/3. The pitfall here is to cancel the 3 from the 3x term and say this is equal to (x + 2y). This is not true because the 2y term does not contain a factor of 3. However, if we change the expression slightly to (3x + 3y)/3, then we can cancel the 3 and write that this equals x + y. To see why this works, let us factor out the 3 and write this expression as 3(x +y)/3. We can then apply the cancellation property for the field of real numbers, letting a = 3 and b = x + y. Since a is not 0, we can cancel it to obtain x + y.
If you remember to avoid just these two common pitfalls, you will be far ahead of the class. Remember, the difference between success and failure in anything in life is often not that much. Be aware of these two pitfalls every time you see parenthetic expressions or a fraction. You will then be well on your way to mastering algebra and being able to advance to the higher realms. Good luck.