In languages, there are rules that tell how sentences must be put together and how they may be changed. These rules are called grammar. The same is true in mathematics. These rules in mathematics are called algebra, and they apply to equations the way grammar applies to sentences. Algebra is therefore the grammar of math. An equation that is true in mathematics is one in which two quantities are equal, as in 3 + 2 = 5. The rules of algebra tell how this equation can be formed and changed. One of the rules of algebra says that any number can be subtracted equally from both sides of an equation. The equation 3 + 2 = 5, therefore, can be changed into 3 = 5 −2, which is also true. Algebra is a way of thinking about arithmetic in a general way. Instead of using specific numbers, as in the example above, mathematicians have found that the easiest way to write down the general rules of algebra is to use what are called variables. Variables are symbols (usually letters like a, b, x, y) that represent either any number or an unknown number. For example, one of the rules of algebra says that a + b = b + a, where a and b are variables that represent any numbers. The rule is true no matter what numbers the letters represent. If a = 3 and b = 2, then 3 + 2 = 2 + 3, or if a =246 and b=912, then 246 + 912 = 912 + 246, and so on. UsesAlgebra and variables make solving math problems easy. Whenever a math problem needs to be solved, simply write the problem as an equation that contains a letter, or variable, for the number that needs to be figured out. It is then possible to read the equation as a question with the words “what number” in the place of the variable. The rules of algebra can then be used to change the equation until it tells what number the variable stands for. Algebra can be useful for solving problems in real life. For example, suppose that Anne has 2 dollars and she wants to buy a book that costs 5 dollars. In order to find out how much more she needs to be able to buy the book, Anne can use algebra. The problem can be written as x +2=5 and can be read in English as “What number plus two equals five?” By using the rules of algebra, the answer can be gotten by starting with the original equation, x +2=5, then subtracting 2 from both sides of the equation: x + 2 − 2 = 5 − 2, or x = 3. In this case, the answer may have been obvious without using algebra, but the same rules can be used to solve problems that involve very complicated numbers. Using the rules of algebra and basic arithmetic—addition, subtraction, multiplication and division—almost any equation that has a variable can be solved to find out what number the variable stands for. HistoryThe word algebra comes from the title of a book on mathematics written in the early 9th century by an Arab astronomer and mathematician named al-Khwarizmi. The rules of algebra are older than that, however. The ancient Greeks wrote down some of the rules that make up algebra, but others came later. In the 6th century AD Hindu mathematicians in India added the idea of 0. One of the final steps in the development of modern algebra came in the 17th century, when mathematicians developed the idea of negative numbers. Although the ancient Chinese and others had a way to indicate negative numbers, it was not until the 1600s that they were properly understood. |