Basics of algebra

Algebra is the study of relations between quantities. Focusing on the relations rather than the exact values enables us to generalize our understanding. In algebra, we represent quantities with placeholders, which are symbols of any sort, but we typically use letters.

Constants versus variables

• A constant is a quantity that has a fixed value within its context. A number is always a constant.
• A variable is a quantity that can assume one of many predefined values within its context.
• The set of allowable (or valid) values of a variable is typically called the domain of the variable.

Notation

• We use a single letter as a placeholder for any variable or constant; e.g., we use \(A\) for ‘area’, not \(Ar\) or \(Area\).
• Every letter corresponds to one and only one quantity, and every quantity is represented by one and only one letter in its context; e.g. \(c\) may mean either ‘count’ or ‘cost’, but not both; and ‘distance’ is represented by either \(d\) or \(s\), but not both.
• Uppercase letters are considered distinct from lowercase letters; e.g. \(A\) and \(a\) are understood to mean two different quantities.
• Multiplication
• We may use the ‘centre dot’ \(\cdot\) or parentheses \((\ )\) instead of the cross sign \(\times\).
  For instance, instead of \(a\times b\) we may write \(a\cdot b\) or \(a(b)\).
• We may drop the multiplication sign altogether when we are multiplying placeholders.
  For instance, instead of \(a\times b\) we just write \(ab\).
• We do not drop it if we are multiplying numbers, because we would this would be confusing.
  For instance, if we were allowed to drop it, we would not know whether \(245\) is the number ‘two-hunderd forty five’, \(2\times 45\), \(24\times 5\) or \(2\times 4\times 5\).
• It is advisable not to use the centre dot for multiplication when only numbers are being multiplied. The reason is to avoid confusion with the decimal point.
  Note, for example, how \(3\cdot 4\) and \(3.4\) look very much like each other. This is especially so in handwriting rather than typeset mathematical writing.
• When placeholders are multiplied by a single number, we write the number first and drop the multiplication sign.
  For instance, instead of \(a\times b\times 5\), we write \(5ab\).
• We tend to write constants before variables, and we tend to write letters in their alphabetic order. This makes it easier to work with placeholders.

Terms and expressions

• An algebraic term is formed of constants and/or variables by mathematical operations of other than addition and subtraction. Raising to a power (i.e., exponentiation) is essentially repeated multiplication, so it is also allowed in a term.
• The coefficient of a term is its constant part. If no constant part is written explicitly, the coefficient is \(1\). The variable part of a term is its non-constant part, formed only of variables.
• The degree of a term is the sum of the powers of all its variables. If the term has no variables, its degree is zero, and it is called a constant term. Remember that a variable with no exponent written explicitly has a ‘hidden’ exponent of \(1\).
• Like terms are terms that have the same variable part; i.e., the exact same variables to the exact same powers. When adding or subtracting like terms, we just add or subtract their constant parts.
  For instance \(4ab^2\) and \(3a^2b\) are not like terms, but \(3xy^2\) and \(5xy^2\) are like terms. Also \(5xy^2+3xy^2=8xy^2\) and \(5xy^2-3xy^2=2xy^2\).
• When we multiply or divide terms, it helps to proceed in the following order
• Sign: Determine whether the result is positive or negative;
• Numbers: Determine the constant part of the result by multiplying or dividing the constants (usually explicit numbers) in the terms;
• Variables, one at a time: Multiply similar variables. Proceed with one at a time to avoid confusion.
• When two terms are multiplied, the degree of the result is the sum of the degrees of those terms.

• An algebraic expression is formed of one or more terms by the operations of addition and/or subtraction.
• An expression is often named according the number of terms in it. Common names are
• Monomial: one term;
• Binomial: two terms;
• Trinomial: three terms;
• Quadrinomial: four terms;
• Polynomial ‘many’ terms. The term can also refer to an expression with any number of terms.

Value versus form

• The form of an expression is its shape, or more simply how it is written.
• The form \(2+4\) is a sum, which is the result of addition.
• The form \(9-3\) is a difference, which is the result of subtraction.
• The form \(2\times 3\) is a product, which is the result of multiplication.
• The form \(30\div 5\) is a quotient, which the result of division.
• The form \(\sqrt{36}\) is a root.
• The value of an expression is what it is equal to after all the operations have been done and the expression is reduced to a single quantity. All the previous forms evaluate to \(6\).
• Two expressions may have
• the same form and the same value, like \(2+5\) and \(3+4\); both are sums and both evaluate to \(7\);
• different forms but the same value, like \(1+8\) and \(3\times 3\); the first is a sum, the second is a product, but both evaluate to \(9\);
• the same form but different values like \(2\times 3\) and \(3\times 4\); both are products but the first evaluates to \(6\) and the second to \(12\); or
• different forms and different values like \(3+2\) and \(9\div 3\); the first is a sum that evaluates to \(5\) and the second is a quotient that evaluates to \(3\).
• As you will learn later, certain forms are equivalent to one another. For example, \(7-3\) is the same as \(7+(-3)\), and \(\frac{8}{2}\) is the same as \(8\times\frac{1}{2}\).

Expressions versus equations

• Given any expression, we aim to preserve its value, while we usually try to change its form. This means that
  
  • if we are adding or subtracting, we can only add or subtract the equivalent of zero; and
  • if we are multiplying or dividing, we can only multiply or divide by the equivalent of one.
  While this sounds ridiculous, it may have the effect of changing the form of the expression, which can be very useful in certain situations as you will learn later.
• An equation is about the balance between one expression on the left side and another on the right side of the equality sign \(=\). Given any equation, we must preserve the balance it represents. This means we have one and only one rule for handling an equation
  whatever we do to one whole side of an equation, we must do it exactly to the other whole side.