Symbols In Algebra (1961)

[Music] [Applause] [Music] the branch of mathematics which we call algebra has developed into the form we study today through contributions by people of Many Lands over hundreds of years Arabs living about 800 ad contributed the word aljabar from which modern algebra derives its name Arabic writings described many of the processes of this branch of mathematics yet it was not until the 16th century that Scholars began using some of the mathematical symbols that provide the basis for present day algebra for some of us the symbols of algebra may seem unfamiliar and difficult to understand yet algebra is not new to us we have already used algebra to help us in arithmetic you'll probably recognize these formulas although we might not have known it when we were using them these formulas are algebraic statements algebraic statements are general rules that tell us how to solve problems in arithmetic and other fields the first algebraic statement tells us the general rule for finding the area of any rectangle the area equals the length times the width the second algebraic statement tells us the general rule for solving problems of motion in which the speed is uniform the distance traveled always equals the speed or rate of travel times the time another algebraic statement which we are familiar with is this general rule we use it to solve problems involving simple interest on a loan the simple interest equals the principal time the rate of interest time the time establishing general rules such as these is the basic purpose of algebra notice that in writing these general rules instead of using numbers we use letters to stand for the numbers for example consider the letters in the general rule for the area of a rectangle suppose the length of the rectangle is 10 units and the width is 7 units then in in this case the letter L stands for 10 units the letter W stands for seven units while the letter A stands for 10 * 7 or 70 square units here we have a smaller rectangle whose length is 6 units and whose width is 5 units the letter L stands for six units the letter W stands for five unit units the letter A stands for 6 * 5 or 30 square units so we can represent different quantities with each letter when the letters are arranged in this algebraic statement they apply to any rectangle in arithmetic we know how to show the basic operations of addition subtraction multiplication and division using letters we show these operations in much the same way for example in a problem in addition we show that to the number three we add the number two with a plus sign as we know this may also be written in reverse order 2 + 3 in either order the sum is equal to the same number five similarly using letters we can write a plus b B this may be written in reverse order b + a the letters Express a general rule of addition numbers have the same sum even if we reverse the order in which they are added in a problem in subtraction we show that from the number three we subtract the number two with a minus sign but the reverse 2 - 3 is not equal to 3 - 2 we indicate this with a symbol called an unequal sign using letters we write a minus B the reverse B minus a is not the same we indicate this by using the unequal sign the letters Express a general rule of subtraction numbers do not always have the same difference if we reverse the order in which they are subtracted when writing multiplication using numbers we normally show it using a sign that looks like the letter X because this sign might be confusing in algebra where the letter X may be used to stand for a number the x is usually replaced by a DOT above the line of writing we can also show multiplication of the numbers by inserting each in parentheses when writing multiplication with letters we may also use an X or a DOT or parentheses but more commonly the letters are simply written together if we reverse the order in which the numbers are multiplied they will have the same product using letters we express this as a general rule for all numbers AB equals ba when writing a problem in division using numbers we can show division by a fraction bar a division sign or by a ratio sign similarly using letters to stand for numbers we can show division in the same way although in algebra the fraction bar is the method most commonly used if we reverse the order in which the numbers are divided we use an unequal sign to show that the quotients are not generally the same you using letters we express this as a general rule for all numbers so in algebra we see that we write operations using letter symbols in much the same way that we use number symbols while the main purpose of algebraic symbols is to state general mathematical rules algebra can also be used to help us solve problems for example consider these three glass containers of different sizes first the water from the small container is poured into the large container next the water from the mediumsized container is poured into the large container now we'll repeat this three more times so that the total quantity of water in the large container will be equal to the contents of the small container plus the contents of Four of the medium-sized containers now that the large container is full this is the problem the small container holds 4 oz of water the large container holds 32 oz of water how much water does the mediumsized container hold some of you may be able to figure out the answer in your heads but let's see how we would solve this problem using Al albra first let the number four stand for the known capacity of the small container meaning 4 o of water let 32 stand for the known capacity of the large container 32 oun of water then use a symbol such as C to stand for the unknown capacity of the medium-sized container now we can write a complete algebraic statement of the problem four the number of ounces of water held by the small container plus 4 C or four times the number of ounces held by the medium-sized container equals 32 the number of ounces held by the large container a statement like this showing quantities connected by an equal sign is an equation we can solve an equation by performing like operations on both sides for example first subtract four from both sides of the equation this tells us that 4 C equals 32 - 4 or 4 C = 28 next divide both sides of the equation by 4 this tells us that c = 28 / 4 so c = 7 the unknown capacity of the mediumsized container is 7 o of water we can see that that letting a letter stand for an unknown number makes a problem easy to solve using algebra remember that the main purpose of algebra is to write general rules that help us solve many problems these general rules are written with letters Each of which may stand for any one of many numbers a letter may also be used to stand for a single unknown number to help us solve a specific problem we will learn Lear many new ways to use letters to help us solve specific problems and to state general rules as we learn more about algebra