* * * * / - - - - IRRATIONAL QUANTITIES, or SURDS. 69 2. To find a multiplier that shall make v5+ v3 rational. Given surd v5+v3 Multiplier V5–V3 5. To find a multiplier that shall make V5-ya: rational. 6. To find a multiplier that shall make Va-Hyb rational. 7. To find multipliers that shall make a-Hvb rational. 8. It is required to find a multiplier that shall make . 1 – $22a rational. * 9. It is required to find a multiplier that shall make 3/3—# /2 rational. - ... ? -- Y Y- ". ; : + Vo CASE XIII. To reduce a fraction, whose denominator is either a simple or a compound surd, to gnother that shall have a rational denominatár. 1. When any simple fraction is of the form #, mul tiply each of its terms by va, and the resulting fraction - b V a will be a Or when it is of the form * multiply them by va’, and the result will be b{Za". ^ 0. And for the general form b o/a , multiply by Va", and 0. 2. If it be a compound surd, find such a multiplier, by the last rule, as will make the denominator rational ; and multiply both the numerator and denominator by it, and the result will be the fraction required. ExAMPLES. 1. Reduce the facion; and #. to others that shall have rational denominators. Here #=#xo-o: and #=#2.É.# =%2–;wiss the answer required. 2 - - 3. Reduce so 2 to a fraction, whose denominator 6 . Reduce —“– 4. Reduce v7+ v3 to a fraction, that shall have a • { rational denominator. 5. Reduce siz. to a fraction that shall have a rational denominator. ... 6. Reduce a—vb to a fraction, the denominator of a-H vb which shall be rational. OF ARITHMETICAL PROPORTION ARITHMETICAL PRoPortion, is the relation which two quantities, of the same kind, have to two others, when the difference of the first pair is equal to that of the second. Hence, three quantities are said to be in arithmetical proportion, when the difference of the first and second is equal to the difference of the second and third. Thus, 2, 4, 6, and a, a-Hb, a-H2b, are quantities in arithmetical proportion. And four quantities are said to be in arithmetical proportion, when the difference of the first and second is equal to the difference of the third and fourth. y Thus, 3, 7, 12, 16, and a, a-Hb, c, c-Fb, are quantities in arithmetical proportion. ARITHMETIcAL PRogREssion is when a series of quantities increase or decrease by the same common difference. Thus, 1, 3, 5, 7, 9, &c. and a, a--d, a-H2d, a-H3d, &c. are increasing series in arithmetical progression, the common differences of which are 2 and d. And 15, 12, 9, 6, &c. and a, a -d, a -2d, a -3d, &c. are decreasing series in arithmetical progression, the common differences of which are 3 and d. The most useful properties of arithmetical proportion and progression are contained in the following theorems : . 1. If four quantities are in arithmetical proportion, the sum of the two extremes will be equal to the sum of the two means. Thus, if, the proportionals be 2, 5, 7, 10, or a, b, c, d; then will 2+1 =5+7, and a--d-b-i-c. 2. And if three quantities be in arithmetical propor tion, the sum of the two extremes will be double the Ionean. will 3+9=2×6=12, and a-i-c-2b. 3. Hence an arithmetical mean between any two quan tities is equal to half the sum of those quantities. Thus, an arithmetical mean between 2 and 4 is 2+4 5-H6 2 And an arithmetical mean between a and b is *#. 5. The last term of any increasing arithmetical series is equal to the first term plus the product of the common difference by the number of terms less one ; and if the series be decreasing, it will be equal to the first term minus that product. Thus, the nth term of the series a, a-H d, a-H2d, a-H3d, a-H4d, &c. is a--(n-1)d. And the nth term of the series a, a -d, a -2d, a -3d, a—4d, &c. is a-(n-1)d. 6. The sum of any series of quantities in arithmetical rogression is equal to the sum of the two extremes multiplied by half the number of terms. Thus, the sum of 2, 4, 6, 8, 10, 12, is = (2+12) × |