"Laws" of Algebra

by  Jess H. Brewer

In Algebra we learn to "solve" equations. What does that mean? Usually it means that we are to take a (relatively) complicated equation that has the "unknown" (often but not always called "$x$") scattered all over the place and turn it into a (relatively) simple equation with $x$ on the left-hand side by itself and a bunch of other symbols (not including $x$) on the right-hand side of the "=" sign. Obviously this particular format is "just" a convention. But the idea is independent of the representation: we want to "solve" for the "unknown" quantity, in this case $x$.

Operations and Notation

Most algebra involves only a few simple operations:

• Equality: If we write $a = b$ we are saying that $a$ and $b$ are the same kind of thing and are exactly the same size.


• Equivalence: If we write $a \equiv b$, we are saying that $a$ and $b$ are the same thing. This may sound like the same thing as equality, but it's actually much stronger. Below it will be applied to equivalent notations.


• Addition: If $a$ and $b$ are entities of the same type (usually just numbers), we can add them together as $a+b$ to get a new number or entity of the same type, called their sum. Example: $1+2=3$. Let's try a few others. Click "Next Addition" to get your question, then enter your answer in the textbox and click "GO" (repeat as many times as you like):      
  Next Addition         GO


• Subtraction: By the same token, we can subtract $b$ from $a$ to get their difference, $a-b$. Example: $3-1 = 2$. Let's try a few of these too. Click "Next Subtraction" to get your question, then enter your answer in the textbox and click "GO":      
  Next Subtraction         GO


• Multiplication: The product of $a$ and $b$ is written as either $a \times b$ or $a \cdot b$ or just $ab$, with the understanding that each entity is represented by a single character. In virtually all computer languages the symbol for multiplication is "*". Example: $2 \times 3 = 6$ or 2*3 = 6. Let's try a few others. Click "Next Multiplication" to get your question, then enter your answer in the textbox and click "GO":      
  Next Multiplication         GO


• Division: Just as subtraction is sort of the opposite of addition, division (written $\frac{a}{b}$ or $a/b$ or $a \div b$) is sort of the opposite of multiplication. Example: $\frac{6}{2} \equiv 6/2 \equiv 6 \div 2 = 3$. Let's try som of these. Click "Next Division" to get your question, then enter your answer in the textbox and click "GO":      
  Next Division         GO


• Powers: We can multiply $a$ by itself $n$ times (also called "raising $a$ to the power $n$) to get $a^n$. (In most computer languages we write xⁿ as "x**n".) Example: $2^5 = 16$. Let's try a few other powers. Click "Next Power" to get your question, then enter your answer in the textbox and click "GO":      
  Next Power         GO


• Roots: In something like the opposite of raising $a$ to the power $n$, we can find the $n^{\rm th}$ root of $b$, written $\sqrt[n]{b} \equiv b^{1/n}$. Computer languages have various notations for roots. The square root of x is almost always written "sqrt(x)" but the n^th root of x usually has to be expressed as "x**(1/n)". Example: $16^{1/5} \equiv \sqrt[5]{16} = 2$ = 16**(1/5). Let's try a few. Click "Next Root" to get your question, then enter your answer in the textbox and click "GO":      
  Next Root         GO

Laws

There are a few basic rules we use to "solve" problems in Algebra; these are called "laws" by Mathematicians who want to emphasize that you are not to question their content or representation.

• Definition of Zero:

  $$a - a = 0$$ (1)

  
  



• Negative Values: Along with the definition of zero, the subtraction operation allows us to assign a negative value to the expression $-a$. That means if we add $-a$ to $a$ we get zero again. Note that if $a$ itself has a negative value, $-a$ is positive, and $a - a = 0$ still holds.¹ Does this seem a bit circular? Right you are! It is!


• Definition of Unity:

  $${a \over a} = 1$$ (2)

  
  



• Commutative Laws:²
  

  $$a + b = b + a$$ (3)

  $$\hbox{\rm and} \qquad \qquad ab = ba$$ (4)

  
  


• Distributive Law:

  $$a(b+c) = ab + bc$$ (5)

  
  



• Sum or Difference of Two Equations: Adding (or subtracting) the same thing from both sides of an equation gives a new equation that is still OK.

  $$\begin{array}[t]{c} ~ \\ + \; ( \\ ~ \end{array} \begin{array . . . . . . \\ b+a \end{array} \begin{array}[t]{c} ~ \\ ) \\ ~ \end{array}$$ (6)

  
  

  $$\begin{array}[t]{c} ~ \\ - \; ( \\ ~ \end . . . . . . ay}[t]{l} d \\ c \\ d-c \end{array} \begin{array}[t]{c} ~ \\ ) \\ ~ \end{array}$$ (7)

  
  



• Product or Ratio of Two Equations: Multiplying (or dividing) both sides of an equation by the same thing also gives a new equation that is still OK.

  $$\begin{array}[t]{c} ~ \\ \times \; ( \\ ~ \end{array} \begin{ . . . . . . a \\ ab \end{array} \begin{array}[t]{c} ~ \\ ) \\ ~ \end{array}$$ (8)

  
  

  $$\begin{array}[t]{c} ~ \\ \div \; ( \\ ~ \ . . . . . . ay}[t]{c} d \\ c \\ d/c \end{array} \begin{array}[t]{c} ~ \\ ) \\ ~ \end{array}$$ (9)

  
  

• Imaginary and Complex Numbers: So far we have limited ourselves to the real numbers. In that domain, the square root of -1 is undefined: there is no real number that will yield -1 when squared. One imagines a particularly persistent child insisting, "But what if there were such a number?" The teacher would grumble, "You certainly have an active imagination!" And the child would say, 'Fine. Let's call it an imaginary number, and call it "i" for short!' The inclusion of multiples of i more than doubles the domain of algebra, since it means we can also have combinations of real and imaginary numbers, z = a + i b. These are called complex numbers.

These "laws" may seem pretty trivial (especially the first two) but they define the rules of Algebra whereby we learn to manipulate the form of equations and "solve" Algebra "problems." We quickly learn equivalent shortcuts like "moving a factor from the bottom of the left-hand-side [often abbreviated LHS] to the top of the right-hand side [RHS]:"

$${x - a \over b} = c + d \quad \Rightarrow \quad x - a = b(c+d)$$ (10)

and so on; but each of these is just a well-justified concatenation of several of the fundamental steps.

You may ask, "Why go to so much trouble to express the obvious in such formal terms?" Well, as usual the obvious is not necessarily the truth. While the real, imaginary and complex numbers may all obey these simple rules, there are perfectly legitimate and useful fields of "things" (usually some sort of operators) that do not obey all these rules, as we may see later. It is generally a good idea to know your own assumptions; we haven't the time to keep reexamining them constantly, so we try to state them as plainly as we can and keep them around for reference "just in case . . . . "

"I'm thinking of a number, and its name is '$x$' . . . " So if

$$a x^2 + b x + c = 0,$$ (11)

what is $x$? Well, we can only say, "It depends." Namely, it depends on the values of $a, b$ and $c$, whatever they are. Let's suppose the dimensions of all these "parameters" are mutually consistent³ so that the equation makes sense. Then "it can be shown" (a classic phrase if there ever was one -- if you want to see the details, check out this link) that the "answer" is generally⁴

$$x = {-b \pm \sqrt{b^2 - 4ac} \over 2a} .$$ (12)

This formula (and the preceding equation that defines what we mean by $a, b$ and $c$) is known as the Quadratic Theorem, so called because it offers "the answer" to any quadratic equation (i.e. one containing powers of $x$ up to and including $x^2$). The power of such a general solution is prodigious.

Want to work out a few examples? Click "Next Quadratic Equation" to get your question, then enter your answer in the textbox and click "GO":      
Next Quadratic Equation         GO

This also suggests an interesting new way of looking at the relationship between $x$ and the parameters $a, b$ and $c$ that determine its value(s). Having $x$ all by itself on one side of the equation and no $x$'s anywhere on the other side is what we call a "solution" in Algebra.

Let's make a simpler version of this sort of equation: "I'm thinking of a number, and its name is 'y' . . . " So if y = f(x), what is y? The answer is again, "It depends!" [In this case, upon the value of x and the detailed form of the function f(x).] Time to move on into CALCULUS!