Variables = Letters = Unknowns
In algebra, a variable is the use of a letter, usually x and y, to represent an unknown quantity. For instance, if you have 7 cookies, and you're left with 3 after your friends leave, algebra and the use of a variable can help you find out how many were eaten: 7 - x = 3, or x + 3 = 7.
Typically, the first symbol in an algebra problem is the letter, or variable. However, either form of the equation shown above can be used to translate a word problem into an algebraic equation.
Symbols of Equality
Most of you are probably familiar with basic equals sign, which means 'the same as': =.
For instance, 3x - 2 = 4 means three times an unknown minus 2 is the same as 4.
The equal sign is important because it forms the balancing point or fulcrum of the equation. Whatever mathematical operation you perform on side of the equation, you must also perform on the other side.
In solving algebraic equations, you'll also come across an altered equal sign with three bars:
In algebra, an altered equivalence symbol means: 'is identical to' or 'is equal by definition'. Another symbol that indicates a number or quantity is almost equal to another number or quantity is:
As used here, the squiggly lines indicate that the numbers on either side of the equal sign are close to, but not exactly the same as, each other. So, if you wanted to find the value of the square root of 2 = 1.41421356…, you'd say the square root of 2 is about 1.4. When working with approximations, this equivalence symbol is the best one to use.
Symbols of Inequality
We use inequality symbols when the solution has a range of answers. For instance, the difference between x = 4 and x > 4, is that x > 4 indicates all the numbers larger than 4 through infinity, while x = 4 only refers to the number 4.
Inequality symbols are read from left to right. When the smaller side appears first, it's the less than symbol. When the larger side appears first, it's the greater than symbol. For example, you'd use
Let's look at another example, if you owned a business and wanted to work no more than 50 hours per week, you'd use the 'less than or equal to' symbol and write it this way:
Symbols of Organization
Parenthesis, brackets and braces help to keep algebraic equations organized and show which operation to perform first. Remember that, in algebra, we begin solving problems from the inside out, staring with the parentheses and then moving onto the brackets and braces. Let's use the order of operations to simplify this equation:
3{5(2-3) (4+1) - (6-7)} + 3: First, perform the calculations shown in parentheses.
3{5(-1) (5) - (-1)} + 3: Multiply the numbers inside the brackets or 5(-1).
3{5-5+1} + 3: Add the numbers inside the brackets or 5 + 1.
3{5} + 3: Multiply the numbers inside the braces or 5*(-4).
3{-20} + 3: Multiply 3 and -20.
-60+3: Add the remaining numbers.
-57: Solution
Factorials
Factorials are a great time saver in algebra and are indicated by an exclamation point (!). A factorial symbol tells you to multiply with a consecutive series of descending numbers. For instance, 5!, or 'five factorial', means 5 * 4 * 3* 2* 1. That's it! Just take the number given, and multiply by all the consecutive natural numbers of lesser value until you reach 1.
Greek Letters in Algebra
Greek letters are often used in algebra to represent certain operations or concepts. The first one, sigma, looks an odd-shaped, oversized 'E' and represents the addition of a series of numbers, as show below.
The sigma symbol is just a short way of saying: 'Let's multiply all the numbers from 1 to 5 by 3 and then add them together'. Let's use the sigma symbol to solve this equation:
The delta symbol is used for finding the change or difference in two variables.
For instance, the slope formula, or (y-y)/(x-x), can also be written as:
Euler's Number
Euler's number is an irrational number expressed as: e = 2.7182818…
You can use Euler's number whenever exponential growth or decay is involved. For example, if you wanted to calculate the population of bacteria, you'd use the following formula:
Lesson Summary
One of the most common algebraic symbols is the use of x, y or another letter of the alphabet to indicate a variable or unknown quantity. Common symbols used in algebra also include those related to equality, inequality, factorials and organization, like braces, brackets and parentheses. Greek letters, such as delta and sigma, are useful when working with specific mathematical ideas or operations, while Euler's number helps to solve problems related to exponential growth or decay. Below is a chart summarizing the most commonly used symbols in algebra.
