4236-42 9900 which brings it to the rule as above stated, and the same may be shown of every fraction of this kind. 74. It may be worth while, before taking leave of this subject, to show the method of reducing quantities of lower denominations to equivalent decimals of a higher denomination, and conversely. The nature of these processes will be readily understood from examples. Let it be required to reduce 17s. 6d. to the decimal of a £. This may obviously be done by reducing 17s. 6d. to the fraction of a £, and then that fraction to a decimal. Thus 17s. 6d. (No. 49.) 0.875 (No. 71.) Or it may be done as follows: 6d. of a shilling.5, consequently 17s. 6d. 17.5, which, when divided by 20 to reduce it to £, will become 0.875. = 6 =12 4 3.00 12 9.75 2014.8125 Ex. 2. Reduce 14s. 9 d. to the decimal of a L. Conversely, given a decimal to find its value. .478125= 478125 1000000 now the numerator will be multiplied by 20, to reduce the pounds to shillings, after which the product must be divided by the denominator, which will be done by pointing off 6 figures, since there are 6 ciphers in the denominator, and so on to the lowest denomination. 740625 .478125 20 9.562500 12 6.750000 3.000000 EXERCISES IN DECIMAL FRACTIONS. 1. Reduce to a decimal fraction. Ans. 0.9375. 2. Reduce to decimals, and then add together the fractions 3 4. Multiply by 18 after reducing them to decimals. 400 Ans, 0.00084375. 5. Divide by 7, after reducing them to decimals. 6. Reduce 0.625 to a vulgar fraction. 10. Reduce 15s. 6d. to the decimal of a pound. Ans. 0.1. Ans. . Ans. 76 Ans. 0.778125. 11. Reduce 6 d. to the decimal of a shilling. Ans. 0.5625. 12. Required the value of 0.678125 of a pound. Ans. 13s. 6 d. PART II. ALGEBRA. 1. ALGEBRA is that branch of Mathematical Science in which quantity is made the subject of calculation by means of certain general signs and symbols. Algebra differs from Arithmetic chiefly in this; that, in the latter, every figure has a determinate and individual value peculiar to itself; whereas, in Algebra, the characters which are employed to denote quantities are general, or independant of any particular signification, and may, therefore, represent all sorts of numbers or quantities, according to the nature of the question to which they are applied. 2. In questions connected with this science, the object is fo discover the value of unknown quantities, from having their relation specified to such as are given or known. And, with a view to a more complete distinction, known or given quantities are represented by the first letters of the alphabet, a, b, c, d, &c.; and those which are unknown by the last, x, y, x, &c. 3. To express the relations which quantities bear to each other in respect of composition, certain signs are employed, which are as follow: 4. The sign (plus) signifies that the quantity to which it is prefixed is to be added. Thus a+b signifies that the quantity represented by b is to be added to that represented by a. If a represent 5 and 6 4, then will a+b represent 5+4 or 9. If no sign is prefixed to a quantity, the sign + is understood; thus a signifies +a. which it is placed must be subtracted. Thus ab signifies that b is to be taken from a; if a be 12, and b, 7, then a -b will express 12-7, or 5. 6. Quantities to which the sign + is prefixed, are called positive or affirmative quantities; and those having the sign prefixed, are called negative quantities. Instead of positive and negative, the terms additive and subtractive are sometimes employed. Thus, in the expression a x-y+b—z, a a and b are the positive or additive quantities, and -y-x the negative or subtractive. 7. The sign x (into) signifies that the quantities between which it stands are to be multiplied together. Thus, a × b signifies that the quantity represented by a is to be multiplied by that represented by b; if a be 8 and b, 7, then axb will be 8×7, or 56. A point is sometimes interposed between the factors instead of the sign x; thus a. b signifies a xb. Both of these signs, however, are frequently omitted; and then the product is indicated by the mere juxtaposition of the quantities. Thus, the product of a by b is expressed by a b; of a, b and x by a b x, &c. 8. The sign (divided by) placed between two quantities, signifies that the former of those quantities is to be divided by the latter. Thus, ab signifies that the quantity represented by a is to be divided by that represented by b; if a be 18, and b, 6, then a÷b will be equal to 186, or 3. This sign is seldom employed. The quotient arising from the division of one quantity by another, is indicated by placing the dividend above the divisor, in the form of a fraction, a thus, signifies the quotient of a divided by b. 9. The sign (equal to) signifies that the quantities between which it is placed are equal to one another; thus, a=b signifies that the quantity represented by a is equal to that represented by b. 10. The coefficient of a quantity is that which multiplies it; thus, in the expressions 5 and a y, 5 and a are respectively the coefficients of x and y. And if a quantity have no coefficient expressed, it is understood to be unity; thus a is the same as 1 x. 11. When a quantity a is multiplied any number of times by itself, the products are called powers of the original quantity a, which, in reference to them, is denominated the root. If the quantity be multiplied once by itself, the product is called the square, or second power; if twice, the cube, or third 1 power, &c. The number of factors which enter into the power is indicated by a small figure on the right of the quantity. Thus the product of a xa or a a is expressed by a2. In like manner, of a xa xa xa or aa aa by a*. The small figure that indicates the number of factors of which the power is composed, is called the index or exponent. The student must be careful to distinguish between the terms coefficient and index, or exponent. In the expression 3 a3, for example, 3 is the coefficient, and 5 the index, or exponent. In ax", a is the coefficient, and n the index or exponent. 12. The roots of quantities are expressed by the sign √ together with the proper index, thus: aora expresses the square root of a, or such a quantity as, when multiplied once by itself, will produce a. a expresses the cube root of a, or such a quantity as, when multiplied twice into itself, will produce a. In the same manner, a+a expresses the fourth root of a + x, &c. 13. A simple quantity is that which consists of a single term, as a, bxy, с d' &c. A compound quantity is that which consists of two or more terms; if it consist of two, it is called a binomial; if of three, a trinomial, &c. Thus, a+b is a binomial; a + bx-c, a trinomial. That particular species of binomial which expresses the difference between two quantities, is called a residual, as a — b. 14. When a line is drawn over any number of algebraic quantities, it indicates that they are to be taken collectively, or as one whole quantity. Thus a + bx c + d, means that the sum of a and b is to be multiplied by the sum of c and d If the quantities stood thus, a+bx c +d, then this signifies that the sum of a and b is to be multiplied by c; and that d is to be added to the product. The line drawn over the quantities is called a vinculum. Instead of the vinculum, a parenthesis is frequently used. Thus, (ab) (c-d,) means that the difference of a and b is to be multiplied by the difference of c and d; (a + b) = (cd) signifies that the sum of a and b is to be divided by the difference of c and d. 15. Like quantities are those which consist of the same letter, or combination of letters; thus, a, 2 a, 7 a, 3 a b, 5 ab, a b x2, 3 a b x2; &c. are called like quantities. Unlike quan |