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(Sir Isaac Newton, 1642–1727) Discoverer of the law of gravitation and famous in algebra for his discovery of the binomial theorem. Inventor of the branch of higher mathematics called the Calculus, wherein rates of motion and other changing, or variable, quantities are extensively studied.
FIRST COURSE IN ALGEBRA
1. The Use of Letters. In arithmetic we often find it convenient to let a letter stand for a number. Thus, if we let I stand for the interest on $ 125 for 2 years at 5%, we may write
In Algebra the use of letters in this way is very common. Let us take another example. We know that the area of a rectangle may always be found by multiplying its length by its breadth. To express this rule by the use of letters we have only to let A stand for area, l for length, and b for breadth, and then the rule becomes simply
A=1Xb. Observe that this gives the right value for A in every case. Thus, if we suppose l=2 ft. and b=3 ft., it gives instantly
A=2X3, or 6 sq. ft., which we know by arithmetic is the right answer.
The advantage in using letters is twofold. It enables us to write general rules and it is of great assistance in solving problems, as we shall soon see.
2. Literal Numbers. Numbers that are represented by letters are called literal numbers.
1. State by use of letters the following rule : The volume of a box is equal to the product of its length, its width, and its height.
(Hint. Let V stand for volume, 1 for length, w for width, and h for height.]
2. Show that your result for Ex. 1 gives the correct answer when the length is 4 ft., the width 3 ft., and the height 2 ft.
3. Symbols. The symbols +, -, X, and · are used with literal numbers as with other numbers, and their meanings are as in arithmetic.
Just as 8+5 means the sum of 8 and 5, so a+b (read a plus b) means the sum of a and b. Likewise, just as 8-5 means the result of subtracting 5 from 8, so a-b (read a minus b) means the result of subtracting b from a. Again, just as 8X5 means the product of 8 and 5, so axb (read a times b) means the product of a and b. Finally, a:b (read a divided by b) means in all cases the quotient of a divided by b..
Besides the rules just stated, there are others in algebra which must now be carefully noted :
Instead of 2x a it is customary to write simply 2 a. In the same way, 3Xa is written 3a, and 4Xx is written 4x, etc. In fact, if a and b stand for any numbers, then a Xb is written in the simple form ab.
But notice that for 2x 3 we cannot write 23, since this means twenty-three. What we have said applies only to literal numbers. Sometimes axb is written in another form, namely a.b. Observe that the dot is here placed just above the line.
If a stands for 2, then it follows from what we have just said that 5 a and 5 · a mean the same thing, namely 5 x 2, or 10. In the same way, if x stands for 2 and y stands for 6, then xy and x· y both mean 2x6, or 12.
Again, we know from arithmetic that when any two numbers are multiplied together it makes no difference which is taken as the multiplier and which as the multiplicand. Thus, 2X3=3X2, and 7X9=9X7, etc. Since a is a number, it follows that 2Xa=aX2, and each of these expressions is written 2 a.
The quotient a = b is frequently written in the form , or a/b.
ORAL EXERCISES 1. If a tennis ball costs 35 cents, what do 5 such balls cost?
2. If a tennis ball costs c cents, what do 5 such balls cost? Ans. 50 cents.
3. If a tennis ball cost r cents last year and the price has advanced 10 cents, what is the present cost?
Ans. r+10 cents. 4. Since 3 feet make one yard, how many feet are there in 5 yards ? in 65 yards ? in n yards ?
5. Since there are 16 ounces in one pound, how many ounces in 7 pounds ? in n pounds ? in r pounds ? . 6. How many minutes in 4 hours ? in 30 hours ? in b hours ?
7. Give the expression that represents the number one greater than d.
Ans. d+1. 8. Give the expression that represents the number 10 less than c.
9. State all the ways in which the following can be written without the use of the sign X. (a) 7Xx (c) yX9. (e) mX9 (g) PXQ (b) xX7 (d) 9XY (1) mxn (h) 10XF
10. State the value of each of the following when a=2: (a) 1+a (6) 3-a. (c) 6 a (d) (e) 3a-2
11. If it takes x minutes to walk to school and 10 minutes to return, what is the total time occupied ?
12. A boy plans to go hunting. If it takes him x minutes to go to the station and 5 times as long on the train, how long will it take him to go where he expects to hunt ?
13. A man has d dollars invested in a farm and three times that amount in bonds. How much has he invested in all ?
14. If, in Ex. 13, d=$50,000, how much has he invested in bonds ?
15. Three bundles of shingles are required for every 100 square feet in a roof. How many bundles of shingles are required for a roof containing 900 square feet? How many bundles are required for a roof of n square feet?
16. The cost of a baseball bat is 6 times that of a ball. If the bat costs b cents, what is the price of the ball ?
17. If a man rides a certain distance in 10 hours, what part of the distance does he ride in 1 hour? in 7 hours ? in h hours ?
18. If A can do a piece of work in 3 hours, what part of it can he do in 1 hour? in 2 hours ? in r hours ? in r+5 hours ?
19. If A's age is 3r and B's age is 4 times A's, what represents B's age ?