11. How will you mark off BD equal to AC?—It could be done by opening dividers to the length AC, and so transferring it. Would it help to have a second copy of the scale?—We could then place the point A of the second scale against B (Fig. 4), and C on the second scale would then lie against the point D that we want. Make two such scales on strips of paper, and use them to calculate 3.43 × 2·68. Use them also to calculate 3·43 ÷ 2.68. 12. Use your two scales to calculate 9.19 × 2·68.—B being the point marked 9.19 on the first scale, C the point 2.68 on the second, we put the beginning of the second scale against B (Fig. 5). We know that 1.0140 = 9.18 X = кс 1.014K × 1.01KC = 2.68; but C is beyond the end K of the first scale, that is, AC 1.0140 AC is greater than 231. 10 x 1.015, so we slide the first scale along till A lies where K was, to the position A'K'. The point on the first scale now opposite C is marked 2-46, so that 1.015 2·46, and 9.19 × 2.68 = 24.6. = The sliding of the first scale to its new position is likely to lead to inaccuracy. Can you suggest an improvement? -We could make the first scale on a strip of twice the length, and repeat the graduations on the added length. The added length would then be ready to hand in the position A'K' whenever it was needed. Your two strips of paper are now (Fig. 6) a simple form of the instrument called a slide-rule. Use your slide-rule to find all the products and quotients of Art. 6. = 1.0141 have fre 13. Such relations as 1.0111 x 1.0130 quently turned up. Show once more the truth of this relation and generalize it. Since 1.0111 means 1·01 multiplied by itself 11 times, the relation written merely says that the product of 11 factors multiplied by the product of 30 factors is the same as if we had taken all 41 factors at once and multiplied them together. What the particular numbers of factors in the two sets are matters nothing, so that we know and 1.0111 x 1.01a: = 1.0111+a, 1.01 x 1.01α = 1·01a+b. Nor does it matter whether the number that forms every factor is 101 or not. Any number k will do, so that we know that kα × kb = ka+b ̧ What kind of numbers may a, b, k be?—a and b are numbers of factors, and must be whole numbers. k may be fractional. 14. We had also such relations as 1.0110 x 1.0110 = 1.0120, Justify these relations, and give a shorter way of writing them. The first relation says that the product of 10 factors multiplied by the product of 10 factors is the same as the product of the 20 factors taken all together. The second relation says we get the same result by multiplying together 30 factors as by first separating them into batches of 10, finding the product of each batch, and then multiplying the three products together. Just as 1.01 x 1.01 is more shortly written 1.012, the 2 showing how many factors 1.01 there are; so 1.0110 × 1.0110 may be written (10110)2, the 2 showing how often the factor 1.0110 occurs. In the same way (1.0110)3 means the product of 3 factors, each factor being 1.010. So that we may write the relations in the form (1.0110)21-0120, (1.0110)31-0130. 15. Can you generalize these relations ?-We might have any number of factors instead of 2 or 3. If we have a factors, each being 1.0110, each of these a factors contains the factor 1.01 10 times, so that the factor 1.01 occurs altogether a x 10 times; so that It matters nothing that there are 10 factors in each batch. The reasoning applies equally to any number b, so that we know And nothing depended on the particular number 1.01 that forms each factor. Any number k, integral or fractional, does equally well, so that we know (kb) a = kbxa What sort of numbers must a and b be?—They are whole numbers, as they show the number of factors. We know what 1.017 means and what 1.016 means, but we have no meaning for 1.016. 16. Can you generalize 1.0130÷1∙0117 = 1·0113 ?—As before, we show that where of course a must be greater than 17. We show further that and 1.011.01 1.01a-b, a being greater than b. = ka÷kb = ka-b How would you express 1.0117÷1·0130 simply?—It is 17 factors multiplied together, and then divided by 30 factors. The result is unaltered if we omit the multiplica tion by the 17 factors, and at the same time omit the division by 17 of the 30 factors. There remains simply division by 13 factors, which we express by 1÷1·0113. What becomes of the relation 1.01" ÷ 1.0117 when a = = = a-17 1.01a 18, and when a = 17?-When a = 18 we have 18 factors divided by 17 factors, so that one is left, and the quotient is 1.01. When a 17 we have 17 factors divided by 17 factors, so that the quotient is 1. the relation 1.01"÷ 1.0117 1.01α-17, and moment that it has any meaning, and put does it become?-It becomes 1.0118 ÷ = If you take forget for a 18 for a, what 1·0117 1.011. = Hitherto, we have not met the index 1. Can you give it a meaning?- -The index shows how many factors there The index 1 says there is only 1 factor, so that 1.011 are. means simply 1.01. = 1.01o. ÷ = 17. Take again the relation 1.01 1.0117 1.01-17, and ignoring its meaning write 17 for a. -The relation becomes 1.0117÷1.0117 What do you make of this, with a zero for an index ?-An index zero would seem to say that there are no factors. We know that 1.0117 1.0117 is 1. Clearly, then, if 1010 is to have any meaning it must mean 1. Is it possible that 1.01o might turn up somewhere where its meaning had to be something else?—In expressions such as 1.01 ÷ 1·016, 1.01o can only turn up when a and b are equal, and then 1.01÷ 1·016 — 1, so that in all such cases 1 is the appropriate meaning of 1.010. What meaning would you give to ko?-This symbol will turn up from 17÷k17 or k"÷k", or such expressions, so its appropriate meaning is 1. = 18. Once more write a = 30 in the relation 1.0117÷ 1.01a 1.0117-a.. -It becomes 1.0117 1.0130 1.01-13. Here is another new kind of index, a negative index. Can you give it a meaning ?-We know that multiplying together 17 factors and then dividing by 30 factors amounts simply to dividing 1 by 13 factors. So, while 1.0113 means that 13 factors are to be multiplied together, 1-01-13 must mean that 1 is to be divided by 13 factors. Is it possible that 1.01-13 might turn up where it would have to have some other meaning?—It can only turn up when there are 13 more factors in the divisor than in the dividend, so that the meaning already given is appropriate. What meaning would you give to k-a? 19. We have sometimes wanted to find the square root of a number, that is, to find another number which, multi = plied by itself, gives the original number. Hitherto, our only means has been the graph of xxx (which may also be written xx or x2). Can you use the graph y=1.01% to find the square root of a number? Find, for instance, the square root of 7.48. At the point A that marks the power 1.01DE x 1.01DE = 1.014B = 7.48. 7.48 we draw the abscissa AB to the curve (Fig. 7). We bisect AB in C, and lay in the curve the abscissa DE, whose length is AC. The power 2.73 represented by D is the square root. For 2·73 = 1·01DE, so that 2.73 x 2.73 Could you use a complete table of powers of 1.01 to find a square root ?—The table shows that 7.48 lies between 1.01202 and 1.01203. Now 1.01101 × 1.01101 1.01202, so that 1.01101 is the square root of 1.01202 or 7·48, and we can look its value out in the table. We might have taken 1.01203 as the value of 7.48, but we could not separate the 203 factors into two equal batches. To find a square root we must use only even indices. = = 1.01α×2, 20. We had such relations as 1.01a × 1·01α which are true so long as a is a whole number. Let us for a moment forget the meaning of this relation and put instead of a. What does the relation become ?-It becomes 1.01 x 1.01 = 1·01. If 1.01 is to have any meaning it must mean the number which, multiplied by itself, makes 1.01, that is, the square root of 1.01. Calculate the squares of 1·001, 1·002, 1·003, . . Do these give you the square root of 1.01?-1.005 x 1.005 = 1.010025, so that 1.005 is pretty accurately the square root of 1'01. |