At the back of this unit are some sheets of multiplication tables. The first is simply
eight copies of a basic 1 to 9 multiplication square, which makes it easy.
Multiplication Methods A collection of some methods for doing multiplication that have been devised and used in previous times.
Notes on Multiplication Methods ~ 1 All of the methods given here are written methods. For an additional account of some mechanical methods see under 'Artifacts for Mathematics' (in the trol menu) where the material for making, together with instructions for using: Napier's Rods, Genaille's Rods and a Slide Rule, are to be found. At the back of this unit are some sheets of multiplication tables. The first is simply eight copies of a basic 1 to 9 multiplication square, which makes it easy to provide everyone with their own personal copy when those facts are needed. Needing to “know their tables” should not be made an issue in this work. The second, ‘A Book of Multiplication Tables’ is much more complicated to make. First of all it requires some care with the double-sided photocopying needed. Check that the crosses in the middle of the two sheets match (within 2 mm if possible), and also that the 94× table does back the 89× table (and NOT the 65×). The sheet has then to be folded in half three times. Do not rush this stage. Press the folds down well with some hard object (ruler or pen). The dotted lines are only a guide, the important thing is to match up the edges of the paper each time, and see that the dotted lines are on the inside of the fold. The titled front cover should always be on the outside of the fold. A staple through the last fold (the hinged edge) and three cuts along the outer edges with a guillotine will complete the process. It is nowhere near as difficult as it sounds! An alternative cover for the booklet is provided. This will have to be cut and pasted onto the master copy before the photocopying is done. All of this would only be worthwhile if the tables might be used in some other work. These same tables are also available in a straight sheet format if wanted. They can be found in the ‘Division’ unit (in the trol menu). It must be emphasied that there are many more ways of doing multiplication than those given here, though most of them are merely variations on a general theme. The ones suggested in this unit are those considered to offer the most useful insights into the historic development of this particular piece of arithmetic so some might appreciate that “No, it hasn’t always been like that!” Exactly how any or all of this material is used will be, as always, a matter for individual teachers to decide and plan for. It can be surprising to see the lengths to which mathematicians and other professional users of mathematics have gone to over the centuries to simplify the multiplication process. But remember that even so great a mathematician as Leibnitz (1646 to 1716) who invented the calculus said “It is unworthy of excellent men to lose hours like slaves in the labour of calculation.” (A feeling which many of our pupils share!)
© Frank Tapson 2004 [trolFG:2]
Notes on Multiplication Methods ~ 2 Using Doubling and Halving 1. The Ancient Egyptian Method This is probably the oldest of all the written methods. It is of some interest because of its links with the binary form of a number, and also because of the simplicity of the idea - it needs only an ability to be able to multiply by 2 and add. However it should also be appreciated that in those days (1000 BC or so) such things were not for 'ordinary folk' who would have had no need of such a skill. It was very much the preserve of professionals such as astrologers, architects and mathematicians. 2. The Russian Peasant Method This method was more widely used than its name might imply. It was in fact taught and practised throughout Europe for quite some time. It is a less demanding algorithm than the previous one since the numbers to be crossed off are easier to identify. It is interesting to write them side by side and see how they are doing the same thing. The link of course is the binary system underlying both. 1 2 4 8 16 32 43
268 536 1072 2144 4288 8576 11524
43 21 10 5 2 1
268 536 1072 2144 4288 8576 11524
Using the Gelosia Method (Pronounced jee - low - see - uh)
2
Any analysis or explanation of why this method works is probably best linked to looking at the place values of the different figures involved.
6 8
The worked example from the sheet is shown on the right.
2
4
1
It is easy to extend the diagram to cover tens of thousands and upwards or, tenths, hundredths, thousandths and so on downwards, and thus show why placing the decimal point works as it does.
8
H
T Th Th
4
4
3
×
H H
H H
2
U T
T
U
U
T T
This method almost certainly originated in India, as did so much of our arithmetic. From there it spread outwards to China, Persia and the Arab world. It reached Europe through Italy where it first appeared in the 14th century. The name 'gelosia' was given to it in Italy since the necessary grid resembled the lattices or gratings (= gelosia) which were made in metal and fastened in front of the windows of the houses for security. It has been claimed that Napier got the idea for his rods from this method. Whether or not he did, the possibility can clearly be seen. In fact, multiplication tables and a knowledge of place value could be dispensed with by using the two methods together. Once the grid has been drawn, a set of Napier's Rods would provide all the needed answers to complete it, and the final addition would be the only arithmetic necessary. One practical point. Do not slow things up by drawing precisely ruled grids. It is quite sufficient to write down the two numbers (adequately spaced) and sketch the grid around them free-hand. © Frank Tapson 2004 [trolFG:3]
3 2
6 The lower diagram replaces each of the the figures in the top diagram with its place value, using the usual notation of H T U, with the addition of Th for thousands. The neat way each place value falls into line along the diagonals makes it very clear why the method works. It also makes it very apparent why zeros must be included in the numbers when necessary.
×
8
Notes on Multiplication Methods ~ 3 Using Quarter Squares The reason why this method works can only be justified by the use of algebra. It can be shown that
(a + b) 2 ab ≡ 4
(a – b) 2 – 4
Given a and b are the two numbers to be multiplied together, and the two terms on the right are quarter squares (as defined on the sheet) then the algorithm follows from the above identity. And (provided the requisite tables are available) multiplication has been replaced by one addition and two subtractions without need for any multiplication facts to be known or found. In passing it is of interest to consider (and prove) why the remainder can be disregarded. Note that this method is completely accurate in giving all the figures possible in the final answer. Its big limitations are that (a) a set of tables is needed and, (b) the size of the numbers it can handle is constrained by the size of the tables (which has to cover the sum of the two numbers being multiplied). The second limitation is not too bad. The sample sheet included here goes to 600, so it is easy to visualize the smallish book which would be needed to reach 100,000. That means an accuracy of 4 significant figures (or better) which should cover most practical needs. Approximations allow the tables to be used for larger numbers. The method was used (but how widely is not known) and the necessary tables to enable this were published in the late 1800’s.
Using Logarithms This is a much simplified version of what was once taught in the classrooms of every secondary school. The words ‘characteristic’ and ‘mantissa’ do not appear, and only numbers greater than 1 are dealt with to avoid that very troublesome negative characteristic. Also only 3-figure tables are used and the customary difference table does not appear. However, the essentials required to give a glimpse of what was done are there. In modern terms this could be called “Logarithms Lite”. The characteristic which is so necessary to control the place value has been replaced here by the e-notation. This is the notation introduced by the IEEE (= Institute of Electrical and Electronic Engineers) and used in all computer work. Gaining an acquaintance with this notation might be regarded as a valuable by-product of this piece of work. One of the inadequancies of 3-figure tables is very apparent once the numbers go beyond 5.0 when many of the logarithmic values are repeated. This cannot be avoided without moving on to 4 or 5-figure tables. Sheet 1 does not require any knowledge of the characteristic as the examples are chosen so as not to go beyond the range of the table. Also there are no ambiguous values for the logarithms. Sheet 2 introduces the idea of the e-notation. Some help might be needed with the business of using the tables, first in finding the logarithm from the number, and then the much more difficult matter of reversing that process. Further work could be done with numbers less than 1 and division.
© Frank Tapson 2004 [trolFG:4]
Notes on Multiplication Methods ~ 4 Using Triangle Numbers This method has the big advantage that it can be easily explained or justified diagrammatically, without any recourse to algebra, except for the generalisation at the end. For this purpose it is better to change the conventionally drawn triangle numbers (as equilateral triangles) into rightangled triangles. The appropriate drawing for T4 is shown on the right. We will do the sum 8 × 3 using five diagrams. 1.
2.
3.
The sum 8 × 3 requires the number of objects making this rectangle to be counted.
First make the triangle number which matches the longest edge. In this case it is T8
Cut away the triangle (shown in black) which is above the size of the rectangle. In this case it is T5
8
5
3
3 8
8
8
5.
4. Triangle removed was T5 or T (8 - 3) leaving this incomplete rectangle
Complete the rectangle with smallest possible triangle (shown in black). In this case T2 or T(3 - 1)
3 8
Physically we have shown that 8 × 3 = 36 –
15
+
3
= T8 –
T5
+
T2
= T8 –
T(8 – 3)
+
T(3 – 1)
Which generalises to
a × b = Ta –
T(a – b)
+
T(b – 1)
Rearranged for the algorithm
a × b = Ta +
T(b – 1)
–
T(a – b)
Another approach to using triangle numbers is implied by this diagram which leads to a different algorithm.
3 8
a b
a+b
a+b
a b
a b
A similar thing can be done using square numbers (or indeed any polygon numbers). The identity (a + b)2 = a2 + b2 + 2ab which is illustrated on the left can be rearranged to give ab = and the required algorithm can be produced from that. A table giving the values of square numbers Sn is available.
Though interesting, and illuminating, to see how these work they were never more than mathematical curiosities, and (unlike the method of quarter squares) there appears to be no evidence that they were ever used in any practical way. One advantage this method does have over quarter squares is that, in this case the largest number in the tables is also the largest possible multiplier.
© Frank Tapson 2004 [trolFG:5]
Multiplication using Doubling and Halving 1. The Ancient Egyptian Method Consider the sum 268 × 43 First write out two columns of numbers side by side. The left-hand column starts with 1. The right-hand colum starts with the larger of the two numbers being multiplied (268). The columns are formed by doubling each of the previous numbers, stopping only when the figure in the left-hand column would become bigger than (or equal to) the other number being multiplied (43). The final result would be as shown on the right. Take care to keep the columns lined up, both vertically and horizontally. The next stage is to look at the left-hand column and find the numbers which are needed to add up to the value of the smaller multiplier (43). In this case the necessary set is 1 + 2 + 8 + 32 = 43 (Note that the bottom number should always be needed, and it is best working backwards from that, subtracting as you go.)
Cross out all the numbers in that column not needed together with the numbers in the right-hand column which are on the same level. Finally, add up the numbers in the right-hand column which have not been crossed out. That is the answer. The completed sum is shown on the right. So, 268 × 43 = 11524
1 2 4 8 16 32
268 536 1072 2144 4288 8576
1 2 4 8 16 32 43
268 536 1072 2144 4288 8576 11524
Try these, using the Ancient Egyptian method. 26 × 314
41 × 61
71 × 213
55 × 97
32 × 104
2. The Russian Peasant Method Consider the sum 268 × 43 First write out two columns of numbers side by side. Each column starts with one of the two numbers to be multiplied. (43 and 268) Make one column by doubling each of the previous numbers. Make the other column by halving each of the previous numbers, ignoring any remainders, and stopping at 1. (It is best to make the halving column under the smaller of the two multipliers)
251 × 21
43 21 10 5 2 1
268 536 1072 2144 4288 8576
43 21 10 5 2 1
268 536 1072 2144 4288 8576 11524
The final result would be as shown on the right. Take care to keep the columns lined up, both vertically and horizontally. The next stage is to look at the halving column and find all the even numbers. In this case they are 10 and 2 Cross out the even numbers in that column together with the numbers in the right-hand column which are on the same level. Finally, add up the numbers in the right-hand column which have not been crossed out. That is the answer. The completed sum is shown on the right. So, 268 × 43 = 11524 Try these, using the Russian Peasant method. 24 × 316 © Frank Tapson 2004 [trolFG:6]
30 × 54
63 × 81
128 × 231
71 × 483
327 × 45
Multiplication using the Gelosia Method Consider the sum 268 × 43 (That is 3 digits × 2 digits) Sketch a grid like that on the right, and place the two numbers as shown.
2
6
×
8
4 3
Next, do the six single-digit multiplications formed by matching the separate numbers across the top with those on the right-hand edge. In this case, it means doing the six sums shown on the right and the answers to those multiplications are written in the appropriate boxes of the grid for each number-pair.
2×4 2×3
6×4 6×3
8×4 8×3
2
6
8
No answer can have more than two digits. (tens and units) Any tens (L.H. digit) are written above the diagonal line, The units (R.H. digit) are written below the diagonal line. (It may be zero) The completed grid should look like that on the right.
2
8
3
4
1 8
2
6 2
8
1
5
4
4
3
×
8 3
4
1 1
2
2
6
The final answer is produced by adding (the results of the multiplication only) along the diagonals, working from right to left, and carrying figures as necessary, and this is shown on the right.
×
2
4
4
3
2
6
8
2
4
So, 268 × 43 = 11524 Try these, using the gelosia method. 386 × 24
758 × 34
76 × 89
Decimal points are dealt with very simply. Consider the sum 2.68 × 4.3 Work exactly as before with 268 × 43 and then put the two decimal points in those numbers on the vertical and horizontal lines respectively. Follow those two lines into the grid until they meet, and then, moving down the diagonal line from there, put the decimal point in the answer. All of this can be seen in the diagram on the right by following the arrows. So, 2.68 × 4.3 = 11.524
402 × 31
999 × 78
478 × 219
2
6 2
8
3
4
1 6 1
1
.
5
×
8 2
4
4
3
2 8
2
4
Try these, using the gelosia method. First, by using the grids already created in the previous work, write down the answers to these: 3.86 × 24
75.8 × 3.4
7.6 × 89
40.2 × 31
9.99 × 78
47.8 × 2.19
7.32 × 18
508 × 3.7
0.617 × 3.8
0.059 × 23
0.074 × 0.038
and then do these: 12.3 × 4.6
© Frank Tapson 2004 [trolFG:7]
Multiplication using Quarter Squares The sheet headed Values of Quarter Squares Qn is needed
The quarter square of a number, is the value of that number multiplied by itself and then divided by 4, ignoring any remainder (if there is one). Examples The quarter square of 2 is: 2 × 2 ÷ 4 = 4 ÷ 4 = 1 3 is: 3 × 3 ÷ 4 = 9 ÷ 4 = 2 rem 1 which becomes 2 5 is: 5 × 5 ÷ 4 = 25 ÷ 4 = 6 rem 1 which becomes 6 and so on. Quarter squares would be of little use for the purpose of simplifying multiplication if they had to be worked out each time they were needed and so a table is used to look up their values. Look at the sheet: Values of Quarter Squares and confirm that the quarter square of 413 is 42 642 (or 42,642 if preferred). Having such a table enables us to change a multiplication sum into a much simpler matter of addition and subtraction. Consider the sum 268 × 43 First add them: 268 + 43 = 311 and find the quarter square of that = 24 180 Then subtract them: 268 – 43 = 255 and find the quarter square of that = 12 656 Finally, subtract the quarter squares to get the final answer = 11 524 and 268 × 43 = 11 524 It could be set out something like this: × + –
268 43 311 225
Qn 24 180 12 656 – 11 524
Try these, using the quarter squares method. 245 × 32
© Frank Tapson 2004 [trolFG:8]
286 × 45
364 × 123
74 × 36
376 × 21
443 × 157
Multiplication using Triangle Numbers The sheet headed Values of Triangle Numbers Tn is needed
T1 = 1
T2 = 3
T3 = 6
T4 = 10
T5 = 15
Triangle numbers are those that can be made in the way shown above where the first five are drawn. The actual value of any triangle number can be worked out using Tn = n × (n + 1) ÷ 2 Example The value of the triangle number for 8 is: 8 × (8 + 1) ÷ 2 = 8 × 9 ÷ 2 = 36 Triangle Numbers would be of little use for the purpose of simplifying multiplication if they had to be worked out each time they were needed and so a table is used to look up their values. Look at the sheet: Values of Triangle Numbers and confirm that the triangle number corresponding to 413 is 85 491 (or 85,491 if preferred). Having such a table enables us to change a multiplication sum into a much simpler matter of addition and subtraction. Consider the sum 268 × 43 Find the triangle number corresponding to the larger (268) Subtract 1 from the smaller (43 - 1 = 42) and find its triangle number Add those two triangle numbers Subtract the two multiplying numbers (268 - 43 = 225) and find the triangle number for that Subtract that from the previous result to get the final answer and 268 × 43 = 11 524
= 36 046 = 12 656 = 36 949 = 25 425 = 11 524
It could be set out something like this: Tn 268 36 046 43 – 1 = 42 903 + 36 949 – 225 25 425 – 11 524
Try these, using the triangle numbers method. 213 × 87
© Frank Tapson 2004 [trolFG:9]
367 × 42
179 × 53
485 × 132
593 × 341
259 × 34
Multiplication using Logarithms ~ 1 The sheet headed Values of Logarithms Ln is needed
We will not concern ourselves here with what logarithms are, but only how to use them for multiplication. (They can also be used for division.) First of all, every number has a logarithm associated with it and that association is unique. In other words, given any number it has only one logarithm and, just as importantly, given any logarithm it can only represent one number. We will start with a very simple example of how we use logarithms. Consider the sum 2 × 4 To use the table of logarithms we need to see this as 2.00 × 4.00 From the table we can find that 2.00 has a logarithm of and that 4.00 has a logarithm of Then we ADD the two logarithms to get That is the answer to the sum but, it is a logarithm, so we need to find what number it represents. Using the table (in reverse) shows that 0.903 is the logarithm of 8.00 So 2.00 × 4.00 = 8.00 (Hardly a surprise, but good to see it works.)
0.301 0.602 0.903
Now consider the much harder sum 2.15 × 3.62 We find their logarithms are 0.332 and 0.559 and these add to 0.891 The logarithm 0.891 matches the number 7.78 So 2.15 × 3.62 = 7.78 It could be set out something like this: N 2.15 × 3.62 7.78
Ln 0.332 0.559 + 0.891
The full answer should have been 7.783 but it must be recognised that the very simplified table we are using is not capable of that sort of accuracy. We wish only to establish how logarithms are used and, for most practical purposes, this accuracy is adequate.
Try doing these, using logarithms 2×3
© Frank Tapson 2004 [trolFG:10]
3×3
1.92 × 3.36
5.18 × 1.45
3.55 × 1.29
4.07 × 1.93
Multiplication using Logarithms ~ 2 Notice that the table of logarithms covers only numbers from 1.00 to 9.99 and that is true for all tables of logarithms. For greater accuracy it might be 1.0000 to 9.9999 but the principle is the same. So now we consider how to adapt all numbers to fit the tables and see how powerful this tool is in handling all multplications. For this we need to be familiar with a particular way of writing numbers which first sets them down as a value between 1 and 10 and then puts enough × 10's after them so they would be restored to their correct value. For example 43 can be written as 4.3 × 10 268 can be written as 2.68 × 10 × 10 There is a special notation used for this which writes the above numbers as 4.3e1 and 2.68e2 respectively where the number after the 'e' shows how many 10's are needed. Or it can be thought of as the amount that the decimal point must be moved to the right to restore the correct value.
Now when we look up the logarithms of 4.3 and 2.68 we replace the 0 on the front with the e-number (if there is one). So that the logarithm of 43 (log 4.3 = 0.633 with e = 1) becomes 1.633 and the logarithm of 268 (log 2.68 = 0.428 with e = 2) becomes 2.428 Then we add the two logarithms to get 4.061 Remembering this stands for a logarithm of 0.061 with e = 4 We next find 0.061 in the logarithm tables and see that it corresponds to the number 1.15 With e = 4 we must multiply by 10, 4 times 1.15 × 10 × 10 × 10 × 10 = 11500 More accurately 43 × 268 = 11524 But we can write 43 × 268 = 11500 (to 3 significant figures)
It could be set out something like this: N 43 × 268 1.15
Ln 1.633 2.428 + 4.061
Try doing these, using logarithms 37 × 26
© Frank Tapson 2004 [trolFG:11]
45 × 78
274 × 63
517 × 842
65.4 × 173
3610 × 904
© Frank Tapson 2004 [trolFG:12]
Qn
0 1 2 4 6
9 12 16 20 25
30 36 42 49 56
64 72 81 90 100
110 121 132 144 156
169 182 196 210 225
240 256 272 289 306
324 342 361 380 400
420 441 462 484 506
529 552 576 600 625
n
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
26 27 28 29 30
31 32 33 34 35
36 37 38 39 40
41 42 43 44 45
46 47 48 49 50
96 97 98 99 100
91 92 93 94 95
86 87 88 89 90
81 82 83 84 85
76 77 78 79 80
71 72 73 74 75
66 67 68 69 70
61 62 63 64 65
56 57 58 59 60
51 52 53 54 55
n
2304 2352 2401 2450 2500
2070 2116 2162 2209 2256
1849 1892 1936 1980 2025
1640 1681 1722 1764 1806
1444 1482 1521 1560 1600
1260 1296 1332 1369 1406
1089 1122 1156 1190 1225
930 961 992 1024 1056
784 812 841 870 900
650 676 702 729 756
Qn
146 147 148 149 150
141 142 143 144 145
136 137 138 139 140
131 132 133 134 135
126 127 128 129 130
121 122 123 124 125
116 117 118 119 120
111 112 113 114 115
106 107 108 109 110
101 102 103 104 105
n
5329 5402 5476 5550 5625
4970 5041 5112 5184 5256
4624 4692 4761 4830 4900
4290 4356 4422 4489 4556
3969 4032 4096 4160 4225
3660 3721 3782 3844 3906
3364 3422 3481 3540 3600
3080 3136 3192 3249 3306
2809 2862 2916 2970 3025
2550 2601 2652 2704 2756
Qn
196 197 198 199 200
191 192 193 194 195
186 187 188 189 190
181 182 183 184 185
176 177 178 179 180
171 172 173 174 175
166 167 168 169 170
161 162 163 164 165
156 157 158 159 160
151 152 153 154 155
n
9604 9702 9801 9900 10 000
9120 9216 9312 9409 9506
8649 8742 8836 8930 9025
8190 8281 8372 8464 8556
7744 7832 7921 8010 8100
7310 7396 7482 7569 7656
6889 6972 7056 7140 7225
6480 6561 6642 6724 6806
6084 6162 6241 6320 6400
5700 5776 5852 5929 6006
Qn
246 247 248 249 250
241 242 243 244 245
236 237 238 239 240
231 232 233 234 235
226 227 228 229 230
221 222 223 224 225
216 217 218 219 220
211 212 213 214 215
206 207 208 209 210
201 202 203 204 205
n 100 201 302 404 506
210 321 432 544 656
15 15 15 15 15
14 14 14 14 15
13 14 14 14 14
13 13 13 13 13
129 252 376 500 625
520 641 762 884 006
924 042 161 280 400
340 456 572 689 806
12 769 12 882 12 996 13 110 13 225
12 12 12 12 12
11 664 11 772 11 881 11 990 12 100
11 130 11 236 11 342 11 449 11 556
10 609 10 712 10 816 10 920 11 025
10 10 10 10 10
Qn
296 297 298 299 300
291 292 293 294 295
286 287 288 289 290
281 282 283 284 285
276 277 278 279 280
271 272 273 274 275
266 267 268 269 270
261 262 263 264 265
256 257 258 259 260
251 252 253 254 255
n
21 22 22 22 22
21 21 21 21 21
20 20 20 20 21
19 19 20 20 20
19 19 19 19 19
18 18 18 18 18
17 17 17 18 18
17 17 17 17 17
16 16 16 16 16
15 15 16 16 16
904 052 201 350 500
170 316 462 609 756
449 592 736 880 025
740 881 022 164 306
044 182 321 460 600
360 496 632 769 906
689 822 956 090 225
030 161 292 424 556
384 512 641 770 900
750 876 002 129 256
Qn
346 347 348 349 350
341 342 343 344 345
336 337 338 339 340
331 332 333 334 335
326 327 328 329 330
321 322 323 324 325
316 317 318 319 320
311 312 313 314 315
306 307 308 309 310
301 302 303 304 305
n
Values of Quarter Squares
29 30 30 30 30
29 29 29 29 29
28 28 28 28 28
27 27 27 27 28
26 26 26 27 27
25 25 26 26 26
24 25 25 25 25
24 24 24 24 24
23 23 23 23 24
22 22 22 23 23
929 102 276 450 625
070 241 412 584 756
224 392 561 730 900
390 556 722 889 056
569 732 896 060 225
760 921 082 244 406
964 122 281 440 600
180 336 492 649 806
409 562 716 870 025
650 801 952 104 256
Qn
396 397 398 399 400
391 392 393 394 395
386 387 388 389 390
381 382 383 384 385
376 377 378 379 380
371 372 373 374 375
366 367 368 369 370
361 362 363 364 365
356 357 358 359 360
351 352 353 354 355
n
39 39 39 39 40
38 38 38 38 39
37 37 37 37 38
36 36 36 36 37
35 35 35 35 36
34 34 34 34 35
33 33 33 34 34
32 32 32 33 33
31 31 32 32 32
30 30 31 31 31
204 402 601 800 000
220 416 612 809 006
249 442 636 830 025
290 481 672 864 056
344 532 721 910 100
410 596 782 969 156
489 672 856 040 225
580 761 942 124 306
684 862 041 220 400
800 976 152 329 506
Qn
446 447 448 449 450
441 442 443 444 445
436 437 438 439 440
431 432 433 434 435
426 427 428 429 430
421 422 423 424 425
416 417 418 419 420
411 412 413 414 415
406 407 408 409 410
401 402 403 404 405
n
Q n for n = 1 to 600
49 49 50 50 50
48 48 49 49 49
47 47 47 48 48
46 46 46 47 47
45 45 45 46 46
44 44 44 44 45
43 43 43 43 44
42 42 42 42 43
41 41 41 41 42
40 40 40 40 41
729 952 176 400 625
620 841 062 284 506
524 742 961 180 400
440 656 872 089 306
369 582 796 010 225
310 521 732 944 156
264 472 681 890 100
230 436 642 849 056
209 412 616 820 025
200 401 602 804 006
Qn
496 497 498 499 500
491 492 493 494 495
486 487 488 489 490
481 482 483 484 485
476 477 478 479 480
471 472 473 474 475
466 467 468 469 470
461 462 463 464 465
456 457 458 459 460
451 452 453 454 455
n
61 61 62 62 62
60 60 60 61 61
59 59 59 59 60
57 58 58 58 58
56 56 57 57 57
55 55 55 56 56
54 54 54 54 55
53 53 53 53 54
51 52 52 52 52
50 51 51 51 51
504 752 001 250 500
270 516 762 009 256
049 292 536 780 025
840 081 322 564 806
644 882 121 360 600
460 696 932 169 406
289 522 756 990 225
130 361 592 824 056
984 212 441 670 900
850 076 302 529 756
Qn
546 547 548 549 550
541 542 543 544 545
536 537 538 539 540
531 532 533 534 535
526 527 528 529 530
521 522 523 524 525
516 517 518 519 520
511 512 513 514 515
506 507 508 509 510
501 502 503 504 505
n
74 74 75 75 75
73 73 73 73 74
71 72 72 72 72
70 70 71 71 71
69 69 69 69 70
67 68 68 68 68
66 66 67 67 67
65 65 65 66 66
64 64 64 64 65
62 63 63 63 63
529 802 076 350 625
170 441 712 984 256
824 092 361 630 900
490 756 022 289 556
169 432 696 960 225
860 121 382 644 906
564 822 081 340 600
280 536 792 049 306
009 262 516 770 025
750 001 252 504 756
Qn
596 597 598 599 600
591 592 593 594 595
586 587 588 589 590
581 582 583 584 585
576 577 578 579 580
571 572 573 574 575
566 567 568 569 570
561 562 563 564 565
556 557 558 559 560
551 552 553 554 555
n
88 89 89 89 90
87 87 87 88 88
85 86 86 86 87
84 84 84 85 85
82 83 83 83 84
81 81 82 82 82
80 80 80 80 81
78 78 79 79 79
77 77 77 78 78
75 76 76 76 77
804 102 401 700 000
320 616 912 209 506
849 142 436 730 025
390 681 972 264 556
944 232 521 810 100
510 796 082 369 656
089 372 656 940 225
680 961 242 524 806
284 562 841 120 400
900 176 452 729 006
Qn
© Frank Tapson 2004 [trolFG:13]
136 153 171 190 210
231 253 276 300 325
351 378 406 435 465
496 528 561 595 630
666 703 741 780 820
16 17 18 19 20
21 22 23 24 25
26 27 28 29 30
31 32 33 34 35
36 37 38 39 40
41 861 42 903 43 946 44 990 45 1035
1081 1128 1176 1225 1275
91 92 93 94 95
66 78 91 105 120
11 12 13 14 15
46 47 48 49 50
86 87 88 89 90
21 28 36 45 55
6 7 8 9 10
96 97 98 99 100
81 82 83 84 85
76 77 78 79 80
71 72 73 74 75
66 67 68 69 70
61 62 63 64 65
56 57 58 59 60
51 52 53 54 55
1 3 6 10 15
1 2 3 4 5
n
Tn
n
4656 4753 4851 4950 5050
4186 4278 4371 4465 4560
3741 3828 3916 4005 4095
3321 3403 3486 3570 3655
2926 3003 3081 3160 3240
2556 2628 2701 2775 2850
2211 2278 2346 2415 2485
1891 1953 2016 2080 2145
1596 1653 1711 1770 1830
1326 1378 1431 1485 1540
Tn
146 147 148 149 150
141 142 143 144 145
136 137 138 139 140
131 132 133 134 135
126 127 128 129 130
121 122 123 124 125
116 117 118 119 120
111 112 113 114 115
106 107 108 109 110
101 102 103 104 105
n
10 731 10 878 11 026 11 175 11 325
10 011 10 153 10 296 10 440 10 585
9316 9453 9591 9730 9870
8646 8778 8911 9045 9180
8001 8128 8256 8385 8515
7381 7503 7626 7750 7875
6786 6903 7021 7140 7260
6216 6328 6441 6555 6670
5671 5778 5886 5995 6105
5151 5253 5356 5460 5565
Tn
196 197 198 199 200
191 192 193 194 195
186 187 188 189 190
181 182 183 184 185
176 177 178 179 180
171 172 173 174 175
166 167 168 169 170
161 162 163 164 165
156 157 158 159 160
151 152 153 154 155
n
706 878 051 225 400
861 028 196 365 535
041 203 366 530 695
246 403 561 720 880
391 578 766 955 145
471 653 836 020 205
19 19 19 19 20
306 503 701 900 100
18 336 18 528 18 721 18 915 19 110
17 17 17 17 18
16 16 16 17 17
15 576 15 753 15 931 16 110 16 290
14 14 15 15 15
13 14 14 14 14
13 13 13 13 13
12 12 12 12 12
11 476 11 628 11 781 11 935 12 090
Tn
246 247 248 249 250
241 242 243 244 245
236 237 238 239 240
231 232 233 234 235
226 227 228 229 230
221 222 223 224 225
216 217 218 219 220
211 212 213 214 215
206 207 208 209 210
201 202 203 204 205
n
30 30 30 31 31
29 29 29 29 30
27 28 28 28 28
26 27 27 27 27
25 25 26 26 26
24 24 24 25 25
23 23 23 24 24
22 22 22 23 23
21 21 21 21 22
381 628 876 125 375
161 403 646 890 135
966 203 441 680 920
796 028 261 495 730
651 878 106 335 565
531 753 976 200 425
436 653 871 090 310
366 578 791 005 220
321 528 736 945 155
20 301 20 503 20 706 20 910 21 115
Tn
296 297 298 299 300
291 292 293 294 295
286 287 288 289 290
281 282 283 284 285
276 277 278 279 280
271 272 273 274 275
266 267 268 269 270
261 262 263 264 265
256 257 258 259 260
251 252 253 254 255
n 626 878 131 385 640
191 453 716 980 245
43 44 44 44 45
42 42 43 43 43
41 41 41 41 42
39 39 40 40 40
38 38 38 39 39
36 37 37 37 37
956 253 551 850 150
486 778 071 365 660
041 328 616 905 195
621 903 186 470 755
226 503 781 060 340
856 128 401 675 950
35 511 35 778 36 046 36 315 36 585
34 34 34 34 35
32 896 33 153 33 411 33 670 33 930
31 31 32 32 32
Tn
48 48 49 49 49 50 50 50 51 51 51 52 52 52 52 53 53 53 54 54 54 946 55 278 55 611 55 945 56 280 56 56 57 57 57
58 311 58 653 58 996 59 340 59 685 60 60 60 61 61
311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350
031 378 726 075 425
616 953 291 630 970
301 628 956 285 615
681 003 326 650 975
086 403 721 040 360
516 828 141 455 770
971 278 586 895 205
46 47 47 47 48
306 307 308 309 310
451 753 056 360 665
45 45 46 46 46
Tn
396 397 398 399 400
391 392 393 394 395
386 387 388 389 390
381 382 383 384 385
376 377 378 379 380
371 372 373 374 375
366 367 368 369 370
361 362 363 364 365
356 357 358 359 360
351 352 353 354 355
n
78 79 79 79 80
76 77 77 77 78
74 75 75 75 76
72 73 73 73 74
70 71 71 72 72
69 69 69 70 70
67 67 67 68 68
65 65 66 66 66
63 63 64 64 64
61 62 62 62 63
606 003 401 800 200
636 028 421 815 210
691 078 466 855 245
771 153 536 920 305
876 253 631 010 390
006 378 751 125 500
161 528 896 265 635
341 703 066 430 795
546 903 261 620 980
776 128 481 835 190
Tn
446 447 448 449 450
441 442 443 444 445
436 437 438 439 440
431 432 433 434 435
426 427 428 429 430
421 422 423 424 425
416 417 418 419 420
411 412 413 414 415
406 407 408 409 410
401 402 403 404 405
n
99 100 100 101 101
97 97 98 98 99
95 95 96 96 97
93 93 93 94 94
90 91 91 92 92
88 89 89 90 90
86 87 87 87 88
84 85 85 85 86
82 83 83 83 84
80 81 81 81 82
Tn for n = 1 to 600
301 302 303 304 305
n
Values of Triangle Numbers
681 128 576 025 475
461 903 346 790 235
266 703 141 580 020
096 528 961 395 830
951 378 806 235 665
831 253 676 100 525
736 153 571 990 410
666 078 491 905 320
621 028 436 845 255
601 003 406 810 215
Tn 926 378 831 285 740
106 106 107 107 108
491 953 416 880 345
104 196 104 653 105 111 105 570 106 030
101 102 102 103 103
Tn
496 497 498 499 500
491 492 493 494 495
526 003 481 960 440
123 123 124 124 125
120 121 121 122 122
256 753 251 750 250
786 278 771 265 760
341 828 316 805 295
115 921 116 403 116 886 117 370 117 855
113 114 114 114 115
111 156 111 628 112 101 112 575 113 050
486 118 487 118 488 119 489 119 490 120
481 482 483 484 485
476 477 478 479 480
471 472 473 474 475
466 108 811 467 109 278 468 109 746 469 110 215 470 110 685
461 462 463 464 465
456 457 458 459 460
451 452 453 454 455
n
546 547 548 549 550
541 542 543 544 545
536 537 538 539 540
531 532 533 534 535
526 527 528 529 530
521 522 523 524 525
516 517 518 519 520
511 512 513 514 515
506 507 508 509 510
501 502 503 504 505
n
601 128 656 185 715
981 503 026 550 075
386 903 421 940 460
816 328 841 355 870
271 778 286 795 305
751 253 756 260 765
916 453 991 530 070
149 149 150 150 151
331 878 426 975 525
146 611 147 153 147 696 148 240 148 785
143 144 144 145 146
141 246 141 778 142 311 142 845 143 380
138 139 139 140 140
135 136 137 137 138
133 133 134 134 135
130 131 131 132 132
128 128 129 129 130
125 126 126 127 127
Tn
596 597 598 599 600
591 592 593 594 595
586 587 588 589 590
581 582 583 584 585
576 577 578 579 580
571 572 573 574 575
566 567 568 569 570
561 562 563 564 565
556 557 558 559 560
551 552 553 554 555
n
177 178 179 179 180
174 175 176 176 177
171 172 173 173 174
169 169 170 170 171
166 166 167 167 168
163 163 164 165 165
160 161 161 162 162
157 158 158 159 159
154 155 155 156 157
152 152 153 153 154
906 503 101 700 300
936 528 121 715 310
991 578 166 755 345
071 653 236 820 405
176 753 331 910 490
306 878 451 025 600
461 028 596 165 735
641 203 766 330 895
846 403 961 520 080
076 628 181 735 290
Tn
© Frank Tapson 2004 [trolFG:14]
51 52 53 54 55
1 4 9 16 25
36 49 64 81 100
121 144 169 196 225
256 289 324 361 400
441 484 529 576 625
676 729 784 841 900
961 1024 1089 1156 1225
1296 1369 1444 1521 1600
1681 1764 1849 1936 2025
2116 2209 2304 2401 2500
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
26 27 28 29 30
31 32 33 34 35
36 37 38 39 40
41 42 43 44 45
46 47 48 49 50
8281 8464 8649 8836 9025
7396 7569 7744 7921 8100
6561 6724 6889 7056 7225
5776 5929 6084 6241 6400
5041 5184 5329 5476 5625
4356 4489 4624 4761 4900
3721 3844 3969 4096 4225
3136 3249 3364 3481 3600
2601 2704 2809 2916 3025
Sn
96 9216 97 9409 98 9604 99 9801 100 10 000
91 92 93 94 95
86 87 88 89 90
81 82 83 84 85
76 77 78 79 80
71 72 73 74 75
66 67 68 69 70
61 62 63 64 65
56 57 58 59 60
n
n Sn
146 147 148 149 150
141 142 143 144 145
136 137 138 139 140
131 132 133 134 135
126 127 128 129 130
121 122 123 124 125
116 117 118 119 120
111 112 113 114 115
106 107 108 109 110
101 102 103 104 105
n
21 21 21 22 22
19 20 20 20 21
18 18 19 19 19
17 17 17 17 18
15 16 16 16 16
14 14 15 15 15
13 13 13 14 14
12 12 12 12 13
316 609 904 201 500
881 164 449 736 025
496 769 044 321 600
161 424 689 956 225
876 129 384 641 900
641 884 129 376 625
456 689 924 161 400
321 544 769 996 225
11 236 11 449 11 664 11 881 12 100
10 201 10 404 10 609 10 816 11 025
Sn
196 197 198 199 200
191 192 193 194 195
186 187 188 189 190
181 182 183 184 185
176 177 178 179 180
171 172 173 174 175
166 167 168 169 170
161 162 163 164 165
156 157 158 159 160
151 152 153 154 155
n
38 38 39 39 40
36 36 37 37 38
34 34 35 35 36
32 33 33 33 34
30 31 31 32 32
29 29 29 30 30
27 27 28 28 28
25 26 26 26 27
24 24 24 25 25
22 23 23 23 24
416 809 204 601 000
481 864 249 636 025
596 969 344 721 100
761 124 489 856 225
976 329 684 041 400
241 584 929 276 625
556 889 224 561 900
921 244 569 896 225
336 649 964 281 600
801 104 409 716 025
Sn
246 247 248 249 250
241 242 243 244 245
236 237 238 239 240
231 232 233 234 235
226 227 228 229 230
221 222 223 224 225
216 217 218 219 220
211 212 213 214 215
206 207 208 209 210
201 202 203 204 205
n
60 61 61 62 62
58 58 59 59 60
55 56 56 57 57
53 53 54 54 55
51 51 51 52 52
48 49 49 50 50
46 47 47 47 48
44 44 45 45 46
42 42 43 43 44
40 40 41 41 42
516 009 504 001 500
081 564 049 536 025
696 169 644 121 600
361 824 289 756 225
076 529 984 441 900
841 284 729 176 625
656 089 524 961 400
521 944 369 796 225
436 849 264 681 100
401 804 209 616 025
Sn
296 297 298 299 300
291 292 293 294 295
286 287 288 289 290
281 282 283 284 285
276 277 278 279 280
271 272 273 274 275
266 267 268 269 270
261 262 263 264 265
256 257 258 259 260
251 252 253 254 255
n
87 88 88 89 90
84 85 85 86 87
81 82 82 83 84
78 79 80 80 81
76 76 77 77 78
73 73 74 75 75
70 71 71 72 72
68 68 69 69 70
65 66 66 67 67
63 63 64 64 65
616 209 804 401 000
681 264 849 436 025
796 369 944 521 100
961 524 089 656 225
176 729 284 841 400
441 984 529 076 625
756 289 824 361 900
121 644 169 696 225
536 049 564 081 600
001 504 009 516 025
Sn
346 347 348 349 350
341 342 343 344 345
336 337 338 339 340
331 332 333 334 335
326 327 328 329 330
321 322 323 324 325
316 317 318 319 320
311 312 313 314 315
306 307 308 309 310
301 302 303 304 305
n
96 97 97 98 99
93 94 94 95 96
90 91 91 92 93
276 929 584 241 900
041 684 329 976 625
856 489 124 761 400
721 344 969 596 225
636 249 864 481 100
601 204 809 416 025
Sn
119 120 121 121 122
116 116 117 118 119
112 113 114 114 115
716 409 104 801 500
281 964 649 336 025
896 569 244 921 600
396 397 398 399 400
391 392 393 394 395
386 387 388 389 390
381 382 383 384 385
376 377 378 379 380
371 372 373 374 375
366 367 368 369 370
361 362 363 364 365
356 357 358 359 360
351 352 353 354 355
n
156 157 158 159 160
152 153 154 155 156
148 149 150 151 152
145 145 146 147 148
141 142 142 143 144
137 138 139 139 140
133 134 135 136 136
130 131 131 132 133
126 127 128 128 129
123 123 124 125 126
816 609 404 201 000
881 664 449 236 025
996 769 544 321 100
161 924 689 456 225
376 129 884 641 400
641 384 129 876 625
956 689 424 161 900
321 044 769 496 225
736 449 164 881 600
201 904 609 316 025
Sn
446 447 448 449 450
441 442 443 444 445
436 437 438 439 440
431 432 433 434 435
426 427 428 429 430
421 422 423 424 425
416 417 418 419 420
411 412 413 414 415
406 407 408 409 410
401 402 403 404 405
n
Sn for n = 1 to 600
109 561 110 224 110 889 111 556 112 225
106 106 107 108 108
103 103 104 104 105
99 100 101 101 102
Values of Square Numbers
198 199 200 201 202
194 195 196 197 198
190 190 191 192 193
185 186 187 188 189
181 182 183 184 184
177 178 178 179 180
173 173 174 175 176
168 169 170 171 172
164 165 166 167 168
160 161 162 163 164
916 809 704 601 500
481 364 249 136 025
096 969 844 721 600
761 624 489 356 225
476 329 184 041 900
241 084 929 776 625
056 889 724 561 400
921 744 569 396 225
836 649 464 281 100
801 604 409 216 025
Sn
496 497 498 499 500
491 492 493 494 495
486 487 488 489 490
481 482 483 484 485
476 477 478 479 480
471 472 473 474 475
466 467 468 469 470
461 462 463 464 465
456 457 458 459 460
451 452 453 454 455
n
246 247 248 249 250
241 242 243 244 245
236 237 238 239 240
231 232 233 234 235
226 227 228 229 230
221 222 223 224 225
217 218 219 219 220
212 213 214 215 216
016 009 004 001 000
081 064 049 036 025
196 169 144 121 100
361 324 289 256 225
576 529 484 441 400
841 784 729 676 625
156 089 024 961 900
521 444 369 296 225
207 936 208 849 209 764 210 681 211 600
203 401 204 304 205 209 206 116 207 025
Sn
546 547 548 549 550
541 542 543 544 545
536 537 538 539 540
531 532 533 534 535
526 527 528 529 530
521 522 523 524 525
516 517 518 519 520
511 512 513 514 515
506 507 508 509 510
501 502 503 504 505
n
681 764 849 936 025
296 369 444 521 600
961 024 089 156 225
676 729 784 841 900
441 484 529 576 625
256 289 324 361 400
121 144 169 196 225
036 049 064 081 100
001 004 009 016 025
298 116 299 209 300 304 301 401 302 500
292 293 294 295 297
287 288 289 290 291
281 283 284 285 286
276 277 278 279 280
271 272 273 274 275
266 267 268 269 270
261 262 263 264 265
256 257 258 259 260
251 252 253 254 255
Sn
596 597 598 599 600
591 592 593 594 595
586 587 588 589 590
581 582 583 584 585
576 577 578 579 580
571 572 573 574 575
566 567 568 569 570
561 562 563 564 565
556 557 558 559 560
551 552 553 554 555
n
355 356 357 358 360
349 350 351 352 354
343 344 345 346 348
337 338 339 341 342
331 332 334 335 336
326 327 328 329 330
320 321 322 323 324
314 315 316 318 319
309 310 311 312 313
303 304 305 306 308
216 409 604 801 000
281 464 649 836 025
396 569 744 921 100
561 724 889 056 225
776 929 084 241 400
041 184 329 476 625
356 489 624 761 900
721 844 969 096 225
136 249 364 481 600
601 704 809 916 025
Sn
© Frank Tapson 2004 [trolFG:15]
0.176 0.204 0.230 0.255 0.279
0.301 0.322 0.342 0.362 0.380
0.398 0.415 0.431 0.447 0.462
0.477 0.491 0.505 0.519 0.531
0.544 0.556 0.568 0.580 0.591
0.602 0.613 0.623 0.633 0.643
0.653 0.663 0.672 0.681 0.690
1.5 1.6 1.7 1.8 1.9
2.0 2.1 2.2 2.3 2.4
2.5 2.6 2.7 2.8 2.9
3.0 3.1 3.2 3.3 3.4
3.5 3.6 3.7 3.8 3.9
4.0 4.1 4.2 4.3 4.4
4.5 4.6 4.7 4.8 4.9
0.654 0.664 0.673 0.682 0.691
0.603 0.614 0.624 0.634 0.644
0.545 0.558 0.569 0.581 0.592
0.479 0.493 0.507 0.520 0.533
0.400 0.417 0.433 0.449 0.464
0.303 0.324 0.344 0.364 0.382
0.179 0.207 0.233 0.258 0.281
1 0.004 0.045 0.083 0.117 0.149
0.655 0.665 0.674 0.683 0.692
0.604 0.615 0.625 0.635 0.645
0.547 0.559 0.571 0.582 0.593
0.480 0.494 0.508 0.521 0.534
0.401 0.418 0.435 0.450 0.465
0.305 0.326 0.346 0.365 0.384
0.182 0.210 0.236 0.260 0.283
2 0.009 0.049 0.086 0.121 0.152
0.656 0.666 0.675 0.684 0.693
0.605 0.616 0.626 0.636 0.646
0.548 0.560 0.572 0.583 0.594
0.481 0.496 0.509 0.522 0.535
0.403 0.420 0.436 0.452 0.467
0.308 0.328 0.348 0.367 0.386
0.185 0.212 0.238 0.262 0.286
3 0.013 0.053 0.090 0.124 0.155
0.657 0.667 0.676 0.685 0.694
0.606 0.617 0.627 0.637 0.647
0.549 0.561 0.573 0.584 0.596
0.483 0.497 0.510 0.524 0.537
0.405 0.422 0.438 0.453 0.468
0.310 0.330 0.350 0.369 0.387
0.188 0.215 0.241 0.265 0.288
4 0.017 0.057 0.093 0.127 0.158
0.658 0.667 0.677 0.686 0.695
0.607 0.618 0.628 0.638 0.648
0.550 0.562 0.574 0.585 0.597
0.484 0.498 0.512 0.525 0.538
0.407 0.423 0.439 0.455 0.470
0.312 0.332 0.352 0.371 0.389
0.190 0.217 0.243 0.267 0.290
5 0.021 0.061 0.097 0.130 0.161
0.659 0.668 0.678 0.687 0.695
0.609 0.619 0.629 0.639 0.649
0.551 0.563 0.575 0.587 0.598
0.486 0.500 0.513 0.526 0.539
0.408 0.425 0.441 0.456 0.471
0.314 0.334 0.354 0.373 0.391
0.193 0.220 0.245 0.270 0.292
6 0.025 0.064 0.100 0.134 0.164
0.660 0.669 0.679 0.688 0.696
0.610 0.620 0.630 0.640 0.650
0.553 0.565 0.576 0.588 0.599
0.487 0.501 0.515 0.528 0.540
0.410 0.427 0.442 0.458 0.473
0.316 0.336 0.356 0.375 0.393
0.196 0.223 0.248 0.272 0.294
7 0.029 0.068 0.104 0.137 0.167
0.661 0.670 0.679 0.688 0.697
0.611 0.621 0.631 0.641 0.651
0.554 0.566 0.577 0.589 0.600
0.489 0.502 0.516 0.529 0.542
0.412 0.428 0.444 0.459 0.474
0.318 0.338 0.358 0.377 0.394
0.199 0.225 0.250 0.274 0.297
8 0.033 0.072 0.107 0.140 0.170
To find the value of the logarithm of n Match the first two digits of n (containing the decimal point) with those in the left-hand column of the table. Go along that line of the table to the three figure decimal number which lies directly under the value at the head of the table matching the third digit of n. That will be the logarithm of n.
0 0.000 0.041 0.079 0.114 0.146
1.0 1.1 1.2 1.3 1.4
0.662 0.671 0.680 0.689 0.698
0.612 0.622 0.632 0.642 0.652
0.555 0.567 0.579 0.590 0.601
0.490 0.504 0.517 0.530 0.543
0.413 0.430 0.446 0.461 0.476
0.320 0.340 0.360 0.378 0.396
0.201 0.228 0.253 0.276 0.299
9 0.037 0.076 0.111 0.143 0.173
Values of Logarithms 0
0.929 0.935 0.940 0.944 0.949 0.954 0.959 0.964 0.968 0.973 0.978 0.982 0.987 0.991 0.996
9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9
0.903 0.908 0.914 0.919 0.924
0.875 0.881 0.886 0.892 0.898
0.845 0.851 0.857 0.863 0.869
0.813 0.820 0.826 0.833 0.839
0.778 0.785 0.792 0.799 0.806
0.740 0.748 0.756 0.763 0.771
0.699 0.708 0.716 0.724 0.732
8.5 8.6 8.7 8.8 8.9
8.0 8.1 8.2 8.3 8.4
7.5 7.6 7.7 7.8 7.9
7.0 7.1 7.2 7.3 7.4
6.5 6.6 6.7 6.8 6.9
6.0 6.1 6.2 6.3 6.4
5.5 5.6 5.7 5.8 5.9
5.0 5.1 5.2 5.3 5.4
Ln for n = 1.0
0.978 0.983 0.987 0.992 0.996
0.955 0.960 0.964 0.969 0.974
0.930 0.935 0.940 0.945 0.950
0.904 0.909 0.914 0.920 0.925
0.876 0.881 0.887 0.893 0.898
0.846 0.852 0.858 0.864 0.870
0.814 0.820 0.827 0.833 0.839
0.779 0.786 0.793 0.800 0.807
0.741 0.749 0.757 0.764 0.772
0.700 0.708 0.717 0.725 0.733
1
0.979 0.983 0.988 0.992 0.997
0.955 0.960 0.965 0.969 0.974
0.930 0.936 0.941 0.945 0.950
0.904 0.910 0.915 0.920 0.925
0.876 0.882 0.888 0.893 0.899
0.846 0.852 0.859 0.865 0.870
0.814 0.821 0.827 0.834 0.840
0.780 0.787 0.794 0.801 0.808
0.742 0.750 0.757 0.765 0.772
0.700 0.709 0.718 0.726 0.734
2
to 9.99 3
0.979 0.984 0.988 0.993 0.997
0.956 0.960 0.965 0.970 0.975
0.931 0.936 0.941 0.946 0.951
0.905 0.910 0.915 0.921 0.926
0.877 0.883 0.888 0.894 0.899
0.847 0.853 0.859 0.865 0.871
0.815 0.822 0.828 0.834 0.841
0.780 0.787 0.794 0.801 0.808
0.743 0.751 0.758 0.766 0.773
0.702 0.711 0.719 0.727 0.735
4
0.980 0.984 0.989 0.993 0.997
0.956 0.961 0.966 0.970 0.975
0.931 0.937 0.942 0.946 0.951
0.905 0.911 0.916 0.921 0.926
0.877 0.883 0.889 0.894 0.900
0.848 0.854 0.860 0.866 0.872
0.816 0.822 0.829 0.835 0.841
0.781 0.788 0.795 0.802 0.809
0.744 0.751 0.759 0.766 0.774
0.702 0.710 0.719 0.728 0.736
5
0.980 0.985 0.989 0.993 0.998
0.957 0.961 0.966 0.971 0.975
0.932 0.937 0.942 0.947 0.952
0.906 0.911 0.916 0.922 0.927
0.878 0.884 0.889 0.895 0.900
0.848 0.854 0.860 0.866 0.872
0.816 0.823 0.829 0.836 0.842
0.782 0.789 0.796 0.803 0.810
0.744 0.752 0.760 0.767 0.775
0.703 0.712 0.720 0.728 0.736
6
0.980 0.985 0.989 0.994 0.998
0.957 0.962 0.967 0.971 0.976
0.932 0.938 0.943 0.947 0.952
0.906 0.912 0.917 0.922 0.927
0.879 0.884 0.890 0.895 0.901
0.849 0.855 0.861 0.867 0.873
0.817 0.823 0.830 0.836 0.843
0.782 0.790 0.797 0.803 0.810
0.745 0.753 0.760 0.768 0.775
0.704 0.713 0.721 0.729 0.737
7
0.981 0.985 0.990 0.994 0.999
0.958 0.962 0.967 0.972 0.976
0.933 0.938 0.943 0.948 0.953
0.907 0.912 0.918 0.923 0.928
0.879 0.885 0.890 0.896 0.901
0.849 0.856 0.861 0.867 0.873
0.818 0.824 0.831 0.837 0.843
0.783 0.790 0.797 0.804 0.811
0.746 0.754 0.761 0.769 0.776
0.705 0.713 0.722 0.730 0.738
8
0.981 0.986 0.990 0.995 0.999
0.958 0.963 0.968 0.972 0.977
0.933 0.939 0.943 0.948 0.953
0.907 0.913 0.918 0.923 0.928
0.880 0.885 0.891 0.897 0.902
0.850 0.856 0.862 0.868 0.874
0.818 0.825 0.831 0.838 0.844
0.784 0.791 0.798 0.805 0.812
0.747 0.754 0.762 0.769 0.777
0.706 0.714 0.723 0.731 0.739
9
0.982 0.986 0.991 0.995 1.000
0.959 0.963 0.968 0.973 0.977
0.934 0.939 0.944 0.949 0.954
0.908 0.913 0.919 0.924 0.929
0.880 0.886 0.892 0.897 0.903
0.851 0.857 0.863 0.869 0.874
0.819 0.825 0.832 0.838 0.844
0.785 0.792 0.799 0.806 0.812
0.747 0.755 0.763 0.770 0.777
0.707 0.715 0.723 0.732 0.740
×
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
4
6
8 10 12 14 16 18
2
4
6
8 10 12 14 16 18
3
6
9 12 15 18 21 24 27
3
6
9 12 15 18 21 24 27
4
8 12 16 20 24 28 32 36
4
8 12 16 20 24 28 32 36
9 18 27 36 45 54 63 72 81
1 2 3 4 5 6 7 8 9
×
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
4
6
8 10 12 14 16 18
2
4
6
8 10 12 14 16 18
3
6
9 12 15 18 21 24 27
3
6
9 12 15 18 21 24 27
4
8 12 16 20 24 28 32 36
4
8 12 16 20 24 28 32 36
9 18 27 36 45 54 63 72 81
1 2 3 4 5 6 7 8 9
×
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
4
6
8 10 12 14 16 18
2
4
6
8 10 12 14 16 18
3
6
9 12 15 18 21 24 27
3
6
9 12 15 18 21 24 27
4
8 12 16 20 24 28 32 36
4
8 12 16 20 24 28 32 36
9 18 27 36 45 54 63 72 81
1 2 3 4 5 6 7 8 9
×
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
4
6
8 10 12 14 16 18
2
4
6
8 10 12 14 16 18
3
6
9 12 15 18 21 24 27
3
6
9 12 15 18 21 24 27
4
8 12 16 20 24 28 32 36
1 2 3 4 5 6 7 8 9
4
8 12 16 20 24 28 32 36
5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72
5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72
5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72
5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81
© Frank Tapson 2004 [trolFG:16]
5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81
5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81
5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81
5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81
× × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
216 288 360 432 504 576 648
213 284 355 426 497 568 639
× × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
219 292 365 438 511 584 657 64 96 128 160 192 224 256 288
32 × 2 = 66 99 132 165 198 231 264 297
33 × 2 = = = = = = = =
3 4 5 6 7 8 9
3 4 5 6 7 8 9
= = = = = = = =
158 237 316 395 474 553 632 711
154 231 308 385 462 539 616 693
× × × × × × ×
= = = = = = =
2 3 4 5 6 7 8 9
= = = = = = = =
33 33 33 33 33 33 33
= = = = = = =
3 4 5 6 7 8 9
77 77 77 77 77 77 77
2 3 4 5 6 7 8 9
× × × × × × ×
32 32 32 32 32 32 32
225 300 375 450 525 600 675
= = = = = = =
75 75 75 75 75 75 75
222 296 370 444 518 592 666
79 × × × × × × × ×
= = = = = = =
152 228 304 380 456 532 608 684
79 79 79 79 79 79 79
3 4 5 6 7 8 9
= = = = = = = =
156 234 312 390 468 546 624 702
162 243 324 405 486 567 648 729
× × × × × × ×
2 3 4 5 6 7 8 9
= = = = = = = =
= = = = = = = =
31 31 31 31 31 31 31
76 76 76 76 76 76 76
2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
3 4 5 6 7 8 9
62 93 124 155 186 217 248 279
31 × 2 =
78 × × × × × × × ×
81 × × × × × × × ×
× × × × × × ×
60 90 120 150 180 210 240 270
46 69 92 115 138 161 184 207
78 78 78 78 78 78 78
81 81 81 81 81 81 81
= = = = = = =
= = = = = = =
= = = = = = = =
50 75 100 125 150 175 200 225
160 240 320 400 480 560 640 720
3 4 5 6 7 8 9
3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
= = = = = = = =
= = = = = = = =
× × × × × × ×
× × × × × × ×
58 87 116 145 174 203 232 261
29 29 29 29 29 29 29
29 × 2 =
23 23 23 23 23 23 23
2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
30 30 30 30 30 30 30
= = = = = = =
30 × 2 =
3 4 5 6 7 8 9
25 × × × × × × × ×
80 × × × × × × × ×
75 × 2 = 150
73 73 73 73 73 73 73
× × × × × × ×
56 84 112 140 168 196 224 252
28 28 28 28 28 28 28
28 × 2 =
44 66 88 110 132 154 176 198
25 25 25 25 25 25 25
80 80 80 80 80 80 80
× × × × × × ×
= = = = = = =
= = = = = = = =
48 72 96 120 144 168 192 216
54 81 108 135 162 189 216 243
74 74 74 74 74 74 74
3 4 5 6 7 8 9
73 × 2 = 146
× × × × × × ×
2 3 4 5 6 7 8 9
= = = = = = = =
= = = = = = = =
77 × × × × × × × ×
22 22 22 22 22 22 22
2 3 4 5 6 7 8 9
27 × × × × × × × ×
76 × × × × × × × ×
24 × × × × × × × ×
52 78 104 130 156 182 208 234
23 × × × × × × × ×
24 24 24 24 24 24 24
= = = = = = = =
22 × × × × × × × ×
26 × × × × × × × ×
74 × 2 = 148
72 72 72 72 72 72 72
72 × 2 = 144
71 71 71 71 71 71 71 98 147 196 245 294 343 392 441
49 × 2 = × × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
× × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
150 200 250 300 350 400 450
× × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
153 204 255 306 357 408 459
51 51 51 51 51 51 51
49 49 49 49 49 49 49
= = = = = = =
51 × 2 = 102
= = = = = = =
3 4 5 6 7 8 9
50 50 50 50 50 50 50
3 4 5 6 7 8 9
× × × × × × ×
94 141 188 235 282 329 376 423
47 47 47 47 47 47 47
47 × 2 =
50 × 2 = 100
× × × × × × ×
= = = = = = = 96 144 192 240 288 336 384 432
48 48 48 48 48 48 48
3 4 5 6 7 8 9
48 × 2 =
× × × × × × ×
92 138 184 230 276 322 368 414
46 46 46 46 46 46 46
46 × 2 = × × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
156 208 260 312 364 416 468
× × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
162 216 270 324 378 432 486
56 56 56 56 56 56 56
× × × × × × ×
3 4 5 6 7 8 9
× × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
159 212 265 318 371 424 477
× × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
165 220 275 330 385 440 495
57 57 57 57 57 57 57
168 224 280 336 392 448 504
× × × × × × ×
171 228 285 342 399 456 513
= = = = = = =
3 4 5 6 7 8 9
57 × 2 = 114
55 55 55 55 55 55 55
55 × 2 = 110
53 53 53 53 53 53 53
53 × 2 = 106
= = = = = = =
56 × 2 = 112
54 54 54 54 54 54 54
54 × 2 = 108
52 52 52 52 52 52 52
52 × 2 = 104
190 285 380 475 570 665 760 855
2 3 4 5 6 7 8 9
71 × 2 = 142
= = = = = = = =
27 27 27 27 27 27 27
210 280 350 420 490 560 630
2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
= = = = = = =
95 × × × × × × × ×
95 95 95 95 95 95 95
26 26 26 26 26 26 26
3 4 5 6 7 8 9
188 282 376 470 564 658 752 846
194 291 388 485 582 679 776 873
198 297 396 495 594 693 792 891
× × × × × × ×
= = = = = = = =
= = = = = = = =
= = = = = = = =
70 70 70 70 70 70 70
2 3 4 5 6 7 8 9
97 × × × × × × × ×
97 97 97 97 97 97 97
2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
192 288 384 480 576 672 768 864
99 99 99 99 99 99 99
99 × × × × × × × ×
196 294 392 490 588 686 784 882
70 × 2 = 140
94 × × × × × × × ×
94 94 94 94 94 94 94
96 × × × × × × × ×
96 96 96 96 96 96 96
= = = = = = = =
= = = = = = = =
× 1 2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
98 × × × × × × × ×
98 98 98 98 98 98 98
© Frank Tapson 2004 [trolFG:17]
Multiplication Tables 1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 3 6 9 12 15 18 21 24 27 4 8 12 16 20 24 28 32 36 5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81
= = = = = = =
246 328 410 492 574 656 738
× × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
252 336 420 504 588 672 756
= = = = = = =
249 332 415 498 581 664 747
× × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
255 340 425 510 595 680 765 40 60 80 100 120 140 160 180
20 × 2 = 42 63 84 105 126 147 168 189
21 × 2 =
270 360 450 540 630 720 810
3 4 5 6 7 8 9
= = = = = = =
273 364 455 546 637 728 819
279 372 465 558 651 744 837
= = = = = = =
3 4 5 6 7 8 9
258 344 430 516 602 688 774
67 × × × × × × × ×
65 65 65 65 65 65 65
= = = = = = =
2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
3 4 5 6 7 8 9
= = = = = = = =
= = = = = = = =
134 201 268 335 402 469 536 603
130 195 260 325 390 455 520 585
× × × × × × ×
93 93 93 93 93 93 93
276 368 460 552 644 736 828
= = = = = = =
3 4 5 6 7 8 9
92 92 92 92 92 92 92
45 60 75 90 105 120 135
15 15 15 15 15 15 15
42 56 70 84 98 112 126
= = = = = = =
3 4 5 6 7 8 9
× × × × × × ×
20 20 20 20 20 20 20
261 348 435 522 609 696 783
= = = = = = =
3 4 5 6 7 8 9
87 87 87 87 87 87 87
128 192 256 320 384 448 512 576
67 67 67 67 67 67 67
93 × 2 = 186
× × × × × × ×
= = = = = = = =
132 198 264 330 396 462 528 594
× × × × × × ×
91 91 91 91 91 91 91
91 × 2 = 182
267 356 445 534 623 712 801
2 3 4 5 6 7 8 9
= = = = = = = =
138 207 276 345 414 483 552 621
92 × 2 = 184
= = = = = = =
= = = = = = =
64 64 64 64 64 64 64
2 3 4 5 6 7 8 9
= = = = = = = =
= = = = = = =
3 4 5 6 7 8 9
3 4 5 6 7 8 9
66 × × × × × × × ×
2 3 4 5 6 7 8 9
3 4 5 6 7 8 9
× × × × × × ×
× × × × × × ×
70 105 140 175 210 245 280 315
66 66 66 66 66 66 66
69 × × × × × × × ×
× × × × × × ×
90 90 90 90 90 90 90
90 × 2 = 180
89 89 89 89 89 89 89
89 × 2 = 178
= = = = = = = =
74 111 148 185 222 259 296 333
69 69 69 69 69 69 69
15 × 2 = 30
39 52 65 78 91 104 117
264 352 440 528 616 704 792
2 3 4 5 6 7 8 9
= = = = = = = =
136 204 272 340 408 476 544 612
14 14 14 14 14 14 14
= = = = = = =
= = = = = = =
35 35 35 35 35 35 35
2 3 4 5 6 7 8 9
= = = = = = = =
14 × 2 = 28
3 4 5 6 7 8 9
3 4 5 6 7 8 9
37 × × × × × × × ×
2 3 4 5 6 7 8 9
= = = = = = =
× × × × × × ×
× × × × × × ×
68 102 136 170 204 238 272 306
37 37 37 37 37 37 37
68 × × × × × × × ×
3 4 5 6 7 8 9
13 13 13 13 13 13 13
13 × 2 = 26
88 88 88 88 88 88 88
88 × 2 = 176
= = = = = = = =
72 108 144 180 216 252 288 324
68 68 68 68 68 68 68
× × × × × × ×
= 36 = 48 = 60 = 72 = 84 = 96 = 108
33 44 55 66 77 88 99
2 3 4 5 6 7 8 9
= = = = = = = =
78 117 156 195 234 273 312 351
21 21 21 21 21 21 21
3 4 5 6 7 8 9
= = = = = = =
34 34 34 34 34 34 34
2 3 4 5 6 7 8 9
= = = = = = = =
= = = = = = =
× × × × × × ×
3 4 5 6 7 8 9
36 × × × × × × × ×
39 × × × × × × × ×
3 4 5 6 7 8 9
12 12 12 12 12 12 12
12 × 2 = 24
× × × × × × ×
82 123 164 205 246 287 328 369
36 36 36 36 36 36 36
76 114 152 190 228 266 304 342
= = = = = = =
11 11 11 11 11 11 11
11 × 2 = 22
= = = = = = = =
86 129 172 215 258 301 344 387
= = = = = = = =
3 4 5 6 7 8 9
30 40 50 60 70 80 90
2 3 4 5 6 7 8 9
= = = = = = = =
38 × × × × × × × ×
× × × × × × ×
= = = = = = =
41 41 41 41 41 41 41
2 3 4 5 6 7 8 9
90 135 180 225 270 315 360 405
19 19 19 19 19 19 19
3 4 5 6 7 8 9
43 × × × × × × × ×
= = = = = = = =
× × × × × × ×
38 57 76 95 114 133 152 171
19 × 2 =
× × × × × × ×
80 120 160 200 240 280 320 360
43 43 43 43 43 43 43
2 3 4 5 6 7 8 9
= = = = = = =
10 10 10 10 10 10 10
10 × 2 = 20
= = = = = = = =
84 126 168 210 252 294 336 378
45 × × × × × × × ×
3 4 5 6 7 8 9
= = = = = = =
2 3 4 5 6 7 8 9
= = = = = = = =
45 45 45 45 45 45 45
× × × × × × ×
3 4 5 6 7 8 9
40 40 40 40 40 40 40
2 3 4 5 6 7 8 9
88 132 176 220 264 308 352 396
18 18 18 18 18 18 18
× × × × × × ×
34 51 68 85 102 119 136 153
17 17 17 17 17 17 17
17 × 2 =
42 × × × × × × × ×
= = = = = = = =
× × × × × × ×
= 48 = 64 = 80 = 96 = 112 = 128 = 144 36 54 72 90 108 126 144 162
3 4 5 6 7 8 9
18 × 2 =
× × × × × × ×
118 177 236 295 354 413 472 531
42 42 42 42 42 42 42
2 3 4 5 6 7 8 9
87 × 2 = 174
85 85 85 85 85 85 85
16 16 16 16 16 16 16
16 × 2 = 32
= = = = = = = =
122 183 244 305 366 427 488 549
44 × × × × × × × ×
× × × × × × ×
3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
= = = = = = = =
44 44 44 44 44 44 44
86 86 86 86 86 86 86
× × × × × × ×
85 × 2 = 170
83 83 83 83 83 83 83
83 × 2 = 166
59 59 59 59 59 59 59
2 3 4 5 6 7 8 9
126 189 252 315 378 441 504 567
65 × × × × × × × ×
61 × × × × × × × ×
= = = = = = = =
64 × × × × × × × ×
116 174 232 290 348 406 464 522
61 61 61 61 61 61 61
2 3 4 5 6 7 8 9
35 × × × × × × × ×
= = = = = = = =
120 180 240 300 360 420 480 540
63 × × × × × × × ×
34 × × × × × × × ×
2 3 4 5 6 7 8 9
= = = = = = = =
63 63 63 63 63 63 63
41 × × × × × × × ×
58 58 58 58 58 58 58
2 3 4 5 6 7 8 9
124 186 248 310 372 434 496 558
40 × × × × × × × ×
60 × × × × × × × ×
= = = = = = = =
59 × × × × × × × ×
60 60 60 60 60 60 60
2 3 4 5 6 7 8 9
58 × × × × × × × ×
62 × × × × × × × ×
86 × 2 = 172
84 84 84 84 84 84 84
84 × 2 = 168
3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
× × × × × × ×
39 39 39 39 39 39 39
82 82 82 82 82 82 82
2 3 4 5 6 7 8 9
82 × 2 = 164
38 38 38 38 38 38 38
62 62 62 62 62 62 62
© Frank Tapson 2004 [trolFG:18]
An alternative front cover for 'A Book of Multiplication Tables 10 to 99 Cut out and paste over the existing front cover, before making multiple copies.
A Book of Multiplication Tables 10 to 99
© Frank Tapson 2004 [trolFG:19]