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quantities whose values are known, and those whose values are unknown. We begin with relations which they are stated to have to each other, and we proceed step by step to other relations, till we arrive at a result or a property previously known. As it is of the utmost importance that the pupil should possess clear views of the nature of these operations, I shall add an example of a problem investigated both by synthesis and analysis.
Problem: Divide 30 into two parts so that the first may be to the second as 2 to 3.
If we suppose the number divided into 30 equal parts, we may pick out 2 of them with the left hand, and 3 at the same time with the right, and place them in separate heaps. We may continue the same operation till the whole is exhausted. The two heaps will obviously be in the ratio of 2 to 3. Now it is evident that each time we went to the original number or heap of units we removed 5, or the sum of 2 and 3. Hence if 30 be divided by 5 it will give us the number of times we went to the original heap, which is 6 times.
But each time we went to the heap we carried off 2 with the left hand, hence the heap formed by the left hand must be 12, and that formed by the right hand 18. The same reasoning would obviously apply to more numbers than two. Hence the following rule for all questions of this kind. Divide the given number by the sum of the numbers which denote the ratio of the parts, and multiply the quotient successively by each of those numbers; the products will give the numbers required.
first part: second part; and since the second part is equal to 30 diminished by the first part, we have 23: first part: 30 diminished by the first
part. Again, when four numbers are in proportion, the sum of the first and second is to the first, as the sum of third and fourth to the third; therefore
The sum of 2 and 3 : 2 :: 30 : first part,
Also the sum of 2 and 3: 3 :: 30 second part. Hence the following rule: as the sum of the numbers which denote the ratio of the parts, is to each of those numbers, so is the number to be divided to each of the parts. This rule is obviously the same as the preceding though differently expressed.
I would strongly recommend to every scientific teacher of youth the necessity of familiarizing the minds of his pupils with the various methods of investigation which may be employed in the discovery of truth. There is generally one method by the application of common sense, without the parade of science, which ought to be the first on which the pupil should try his strength. Before he runs for assistance to his a's and b's, his x's and y's, his pluses and minuses, on the "slightest provocation," let him first put the shoulder of common sense to the wheel, and he will frequently find that he has been able to extricate his chariot without the necessity of employing any of the mechanical powers.
If he succeed, it will give him some idea of the native strength of his own mind; if he fail by this method, and succeed by using a, b, x, plus, minus, &c., it will show him the powers of the mighty instrument which he is now learning to wield.
SECT. I. On the principal definitions and elementary operations, with illustrations.
1. In the solution of problems by Algebra, quantities whose values are known are represented by the first letters of the alphabet a, b, c, &c., and those whose
values are yet unknown or which are to be found, are denoted by the last letters x, y, z.
2. The arithmetical operations to which numbers are subjected are denoted by certain signs, which by the consent of all nations are now universally adopted. The sign, called plus, denotes that the number before which it is placed is to be added. Quantities having plus before them are generally called positive quantities; the term additive would have been more appropriate. The sign called minus, denotes that the number or quantity before which it is placed is to be subtracted. Quantities having minus before them are generally called negative quartities; the term subtractive would have been more significant. The sign, denominated equality, and read equal to, denotes that the result of the combination of quantities placed before it is equal to that placed after it, thus, 2+3=7+1 − 3
An expression of this kind is called an equation, the quantities which are placed on the left hand being called the first side of the equation, those on the other the second. The sign × placed between two numbers, and read "multiplied into," denotes that the numbers are to be multiplied together. If the quantities are expressed by letters, multiplication is denoted by placing them after each other or with a dot between them. Thus if a denote 3, and if b denote 5, then a b or a. b = 3 x 5 =
read "divided by," denotes that the quantity before it is to be divided by that which follows. Division is also expressed by placing the dividend above a short line, and the divisor under it. Thus if a denote 12 and
denote 3, then x = a÷ b = = 4.
3. When a line called a vinculum is drawn over a
combination of quantities, or when they are inclosed by a parenthesis, the result of the quantities so circumstanced is to be used as a simple quantity. Thus, if
also a + b
(a+c—b) (a + b) = 4 X 8=32
4. When a quantity denoted by a single letter is to be multiplied by a known number, whether expressed by figures or letters, the multiplier is placed before the quantity and is called its coefficient, thus a × 3 is denoted by 3 a. Also (a+b) × 3, is denoted by 3 (a+b), the number 3 being called the co-efficient of a + b.
5. When a number, whether denoted by figures or letters, is to be multiplied a certain number of times by itself, the operation is denoted by placing a small character or figure, denoting the number of times it enters into the operation, a little above the number towards the right hand. Thus 2 × 2 × 2 = 23 = 8. The figure or character thus placed above the number is called its index or exponent, and the number itself is said to be raised to the power denoted by the exponent. Thus in the example 329, the number 3 is said to be raised to the second power or square, and 9 is called the second power or square of 3. third power or cube of 3.
In like manner 27 is the
The term square has been employed from the cir. cumstance that the area of a square is found by multiplying the length of the side by itself. The term cube has been introduced because the number of cubic inches in a given cube is found by multiplying the length of the side twice in succession by itself. As we have frequently numbers multiplied successively by themselves without any reference to squares or cubes, the terms, second, third, fourth power, &c., are used.
6. The converse of the last operation, or that of a number, which, when multiplied a certain number of times by itself will produce a given number, is denoted by placing the reciprocal of the power in the same situation as the power itself, the fractional number thus employed being called the index or exponent. Thus 32-9 and 9*= 3; 23 — 8 and 8a— 2.
The number found by this operation is called the second, third, fourth, &c. root of the given number. Thus 3 is the second or square root of 9; 2 is the third or cube root of 8.
Instead of this elegant and symmetrical notation, the old characters, V, denoting the square and cube roots, are too often found disgracing the pages of most works on algebra. We would recommend the pupil to avoid using them, as the others will be found much easier of application.
7. When the result of a compound quantity is to be raised to a given power or have a given root extracted, the index is placed above the compound quantity when a vinculum has been drawn over it or when it has been inclosed in a parenthesis.
Thus if a 4 and b =9281.
Also a +
old expression for (a + b)
5 then a + b2 or (a + b)2
We would strongly recommend the teacher to avoid using the radical sign ✔, and vinculum, the parenthesis and fractional index being much more compact and elegant.
8. The learner will now easily perceive that all the arithmetical rules which he has acquired are equally applicable to numbers represented by letters; thus if 3 times 17. or 31. be added to 5 times 17. or 51. the sum is 81.