The Algebra of Real Numbers

Algebra begins with a systematic study of the operations and rules of arithmetic. The operations of addition, subtraction, multiplication and division serves as a basis for all arithmetic calculations. In order to achieve generality, letters of the alphabet are used in algebra to represent numbers. A letter such as $x,y,a,$ or $b$ can stand for a particular number (known or unknown), or it can stand for any number at all. A letter that represents an arbitrary number is called a variable.

The sum, difference, product and quotient of two numbers, $x$ and $y$, can be written as $x + y$, $x – y$, $x \times y$ and $x \div y$.

In algebra, the notation $x \times y$ for the product of $x$ and $y$ is not often used because of the possible confusion of the letter $x$ with the multiplication sign $\times$. The preferred notation is $x \cdot y$ or simply $xy$. Similarly, the notation $x \div y$ is usually avoided in favor of the fraction $\frac{x}{y}$ or $x/y$.

Algebraic notation (the shorthand of mathematics) is designed to clarify ideas and simplify calculation by permitting us to write expressions compactly and efficiently. For instance, $x + x + x + x + x$ can be written simply as $5x$. The use of exponents provides an economy of notation for products; for instance, $x \cdot x$ can be written simply as ${x^2}$ and $x \cdot x \cdot x$ as ${x^3}$. In general, if $n$ is a positive integer, ${x^n}$ means $x \cdot x \cdot x \cdots x$ ($n$ times) and ${x^{ – n}}$means$\frac{1}{{{x^n}}}$.

In using the exponential notation ${x^n}$, we refer to $x$ as a base and $n$ as the exponent, or the power to which the base is raised. When the exponent is negative, we must assume that the base is nonzero to avoid a zero in the denominator. By writing it with an equal sign ($=$) between two algebraic expressions, we obtain an equation, or formula, stating that two expressions represents the same number. Using equations and formulas, we can express mathematical facts in compact, easily remembered forms. Formulas are used to express relationships among various quantities in such fields as geometry, physics, engineering, statistics, geology, business, medicine, economics, and life sciences. Calculating the numerical value expressed by a formula when particular numbers are assigned to letters is known as evaluation.

Example 1 :
(a) Write a formula for the volume $V$ of a cube that has edges of the length $x$ units.
(b) Evaluate $V$ when $x = 5.23$ centimeters.

Solution:
(a) $V = x \cdot x \cdot x = {x^3}$ cubic centimeters.
(b) When $x = 5.23$ centimeters, $V = {(5.23)^2} = 143.055667$ cubic centimeters.

Example 2:
A certain type of living cell divides every hour. Starting with one such cell in a culture, the number $N$ of cells present at the end of $t$ hours is given by the formula $N = {2^t}$. Find the number of cells in the culture after $6$ hours.

Solution:
Substituting $t = 6$ in the formula $N = {2^t}$, we find that $N = {2^6} = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64$ cells.