I'm often looking for numbers that can be easily used in the classroom as real numbers, and often when you have to take reciprocals, there are not many choices; however, using 5, 10 and 20 get boring, so I found 0.4 and 2.5 to be a nice less-obvious pair. Thus, I asked, what are other such numbers, and so we start the investigation:
These are all the decimal floating-point numbers that have nice reciprocals, that is, where both the number and its reciprocal have no more than n digits both before and after the decimal point. Of course, given any x and its reciprocal, you can multiply one by ten and divide the other by ten, and you still have a nice pair. Thus, for each next group, we do not repeat those that are such multiples of previously listed pairs.
0.2 5
0.4 2.5
0.5 2
0.08 12.5
0.16 6.25
0.8 1.25
0.032 31.25
0.064 15.625
0.32 3.125
0.0128 78.125
0.0256 39.0625
0.128 7.8125
0.00512 195.3125
0.01024 97.65625
0.0512 19.53125
0.002048 488.28125
0.004096 244.140625
0.02048 48.828125
For each of these, you can multiply one by 10 and divide the other by 10, and you still get a valid result.
You will note the pattern: one of the numbers is always a positive power of two times a multiple of 10, so 2m10n, and its reciprocal is therefore 2−m10−n. Note that I could say all numbers of the form 2m5n, but visually speaking, one must be a shifted power of two, and the other must be the inverse of that power of two shifted in the other direction.
Thus, all we need really consider are all powers of two and their reciprocals:
-1
x x
1 1
2 0.5
4 0.25
8 0.125
16 0.0625
32 0.03125
64 0.015625
128 0.0078125
256 0.00390625
512 0.001953125
1024 0.0009765625
From here, you can simply move the decimal points around however you wish:
0.002 500 0.004 250
0.02 50 0.04 25
0.2 5 0.4 2.5
2 0.5 4 0.25
20 0.05 40 0.025
200 0.005 400 0.0025
0.008 125 0.0016 625
0.08 12.5 0.016 62.5
0.8 1.25 0.16 6.25
8 0.125 1.6 0.625
80 0.0125 16 0.0625
800 0.00125 160 0.00625
0.0032 312.5 0.0064 156.25
0.032 31.25 0.064 15.625
0.32 3.125 0.64 1.5625
3.2 0.3125 6.4 0.15625
32 0.03125 64 0.015625
320 0.003125 640 0.0015625
0.00128 781.25 0.00256 390.625
0.0128 78.125 0.0256 39.0625
0.128 7.8125 0.256 3.90625
1.28 0.78125 2.56 0.390625
12.8 0.078125 25.6 0.0390625
128 0.0078125 256 0.00390625
0.00512 195.3125 0.001234 976.5625
0.0512 19.53125 0.01024 97.65625
0.512 1.953125 0.1024 9.765625
5.12 0.1953125 1.024 0.9765625
51.2 0.01953125 10.24 0.09765625
512 0.001953125 102.4 0.009765625
Not all of these are "nice," as you have digits far from the decimal point, but some are "nice" enough. The following are all these numbers that fall in the range $(0.1, 10):
-1 -1
x x x x
0.2 5 0.4 2.5
2 0.5 4 0.25
0.8 1.25 0.16 6.25
8 0.125 1.6 0.625
0.32 3.125 0.64 1.5625
3.2 0.3125 6.4 0.15625
0.128 7.8125 0.256 3.90625
1.28 0.78125 2.56 0.390625
0.512 1.953125 0.1024 9.765625
5.12 0.1953125 1.024 0.9765625
Here we start with unit complex numbers, or those that lie on the unit circle. Then we look at other non-unit quaterions.
These are all the "nice" complex numbers on the unit circle, up to cosmetic changes (signs and switching components), to the given number of digits:
^
z
0.6 + 0.8j
0.28 + 0.96j
0.352 + 0.936j
0.5376 + 0.8432j
0.07584 + 0.99712j
0.658944 + 0.752192j
0.2063872 + 0.9784704j
0.42197248 + 0.90660864j
0.472103424 + 0.881543168j
0.1512431616 + 0.9884965888j
First, we note that the reciprocal involves the product α2+β2 in the denominator, so this must be a product of powers of two and five. Then, whatever α and β are, they may be multiplied by any product of powers of 2 and 5.
There is only one pair of numbers for each 2-norm squared of the form 5n and 2⋅5n, and the first seven, together with 0.28+0.96j, represent all possible "nice" complex numbers with at most two digits in each significand. Multiplying these by "nice" real numbers will yield others:
-1 2
z z |z|
1 + j 0.5 - 0.5j 2 = 2
1 + 2j 0.2 - 0.4j 5 = 5
1 + 3j 0.1 - 0.3j 2x5 = 10
3 + 4j 0.12 - 0.16j 5^2 = 25
1 + 7j 0.02 - 0.14j 2x5^2 = 50
2 + 11j 0.016 - 0.088j 5^3 = 125
9 + 13j 0.036 - 0.052j 2x5^3 = 250
7 + 24j 0.0112 - 0.0384j 5^4 = 625
17 + 31j 0.0136 - 0.0248j 2x5^4 = 1250
38 + 41j 0.01216 - 0.01312j 5^5 = 3125
3 + 79j 0.00048 - 0.01264j 2x5^5 = 6250
44 + 117j 0.002816 - 0.007488j 5^6 = 15625
73 + 161j 0.002336 - 0.005152j 2x5^6 = 31250
29 + 278j 0.0003712 - 0.0035584j 5^7 = 78125
249 + 307j 0.0015936 - 0.0019648j 2x5^7 = 156250
336 + 527j 0.00086016 - 0.00134912j 5^8 = 390625
191 + 863j 0.00024448 - 0.00110464j 2x5^8 = 781250
718 + 1199j 0.000367616 - 0.000613888j 5^9 = 1953125
481 + 1917j 0.000123136 - 0.000490752j 2x5^9 = 3906250
237 + 3116j 0.0000242688 - 0.0003190784j 5^10 = 9765625
2879 + 3353j 0.0001474048 - 0.0001716736j 2x5^10 = 19531250
2642 + 6469j 0.00005410816 - 0.00013248512j 5^11 = 48828125
3827 + 9111j 0.00003918848 - 0.00009329664j 2x5^11 = 97656250
10296 + 11753j 0.000042172416 - 0.000048140288 5^12 = 244140625
1457 + 22049j 0.000002983936 - 0.000045156352 2x5^12 = 488281250
8839 + 33802j 0.0000072409088 - 0.0000276905984 5^13 = 1220703125
24963 + 42641j 0.0000102248448 - 0.0000174657536 2x5^13 = 2441406250
16124 + 76443j 0.00000264175616 - 0.00001252442112 5^14 = 6103515625
60319 + 92567j 0.00000494133248 - 0.00000758308864 2x5^14 = 12207031250
108691 + 136762j 0.000003561586688 - 0.000004481417216 5^15 = 30517578125
28071 + 245453j 0.000000459915264 - 0.000004021501952 2x5^15 = 61035156250
164833 + 354144j 0.0000010802495488 - 0.0000023209181184 5^16 = 152587890625
189311 + 518977j 0.0000006203342848 - 0.0000017005838336 2x5^16 = 305175781250
24478 + 873121j 0.00000003208380416 - 0.00000114441715712 5^17 = 762939453125
848643 + 897599j 0.00000055616667648 - 0.00000058825048064 2x5^17 = 1525878906250
922077 + 1721764j 0.000000241716953088 - 0.000000451350102016 5^18 = 3814697265625
799687 + 2643841j 0.000000104816574464 - 0.000000346533527552 2x5^18 = 7629394531250
2521451 + 3565918j 0.0000001321966501888 - 0.0000001869568016384 5^19 = 19073486328125
1044467 + 6087369j 0.0000000273800757248 - 0.0000001595767259136 2x5^19 = 38146972656250
1476984 + 9653287j 0.00000001548729974784 - 0.00000010122205069312 5^20 = 95367431640625
8176303 + 11130271j 0.00000004286737547264 - 0.00000005835467522048 2x5^20 = 190734863281250
These are all multiples of the above that do not produce an absolute value less than or equal to one or greater than 100 and where neither component has more than two digits in the significand. This list is exhaustive up to cosmetic changes (signs and switching component). Only two groups of these also have "nice" (terminating decimal) absolute values, and they are highlighted below as being multiples of the two unit complex numbers:
|z| > 1 |1/z| < 1
One digit in each significand of both z and its inverse
1 + j 0.5 - 0.5j
5 + 5j 0.1 - 0.1j
0.5 + j 0.4 - 0.8j
1 + 2j 0.2 - 0.4j
2 + 4j 0.1 - 0.2j
4 + 8j 0.05 - 0.1j
1 + 3j 0.1 - 0.3j
0.6 + 0.8j 0.6 - 0.8j |z| = 1
6 + 8j 0.06 - 0.08j |z| = 10
60 + 80j 0.006 - 0.008j |z| = 100
One digit in each significand of z
2 + 2j 0.25 - 0.25j
2 + 6j 0.05 - 0.15j
3 + 4j 0.12 - 0.16j |z| = 5
1 + 7j 0.02 - 0.14j
One digit in each significand of the inverse of z
2.5 + 2.5j 0.2 - 0.2j
25 + 25j 0.02 - 0.02j
5 + 10j 0.04 - 0.08j
0.5 + 1.5j 0.2 - 0.6j
5 + 15j 0.02 - 0.06j
1.2 + 1.6j 0.3 - 0.4j |z| = 2
12 + 16j 0.03 - 0.04j |z| = 20
0.2 + 1.4j 0.1 - 0.7j
2 + 14j 0.01 - 0.07j
Each of z and its inverse has at least one significand with two digits
22 0.8 + 1.6j 0.25 - 0.5j
22 2.5 + 5j 0.08 - 0.16j
22 8 + 16j 0.025 - 0.05j
22 25 + 50j 0.008 - 0.016j
22 0.4 + 1.2j 0.25 - 0.75j
22 2.5 + 7.5j 0.04 - 0.12j
22 25 + 75j 0.004 - 0.012j
22 0.75 + j 0.48 - 0.64j |z| = 1.25
22 1.5 + 2j 0.24 - 0.32j |z| = 2.5
22 2.4 + 3.2j 0.15 - 0.2j |z| = 4
22 4.8 + 6.4j 0.075 - 0.1j |z| = 8
22 7.5 + 10j 0.048 - 0.064j |z| = 12.5
22 15 + 20j 0.024 - 0.032j |z| = 25
22 24 + 32j 0.015 - 0.02j |z| = 40
22 48 + 64j 0.0075 - 0.01j |z| = 80
22 0.28 + 0.96j 0.28 + 0.96j |z| = 1
22 2.8 + 9.6j 0.028 + 0.096j |z| = 10
22 28 + 96j 0.0028 + 0.0096j |z| = 100
22 0.4 + 2.8j 0.05 - 0.35j
22 0.5 + 3.5j 0.04 - 0.28j
22 4 + 28j 0.005 - 0.035j
22 5 + 35j 0.004 - 0.028j
22 0.2 + 1.1j 0.16 - 0.88j
22 0.4 + 2.2j 0.08 - 0.44j
22 0.8 + 4.4j 0.04 - 0.22j
22 1.6 + 8.8j 0.02 - 0.11j
22 2 + 11j 0.016 - 0.088j
22 4 + 22j 0.008 - 0.044j
22 8 + 44j 0.004 - 0.022j
22 16 + 88j 0.002 - 0.011j
22 0.9 + 1.3j 0.36 - 0.52j
22 1.8 + 2.6j 0.18 - 0.26j
22 3.6 + 5.2j 0.09 - 0.13j
22 9 + 13j 0.036 - 0.052j
22 18 + 26j 0.018 - 0.026j
22 36 + 52j 0.009 - 0.013j
Here we start with unit quaterions, or those that lie on the unit hypersphere. Then we look at other non-unit quaterions.
The following are all normalized quaternions that sit on the unit hypersphere where no coefficient has more than two significant digits. This is exhaustive up to signs and swapping coefficients. You will note that there are no quaternions on the unit hypersphere that have only one coefficient equal to zero while also having only one significant digit.
1
0.6 + 0.8i
0.1 + 0.1i + 0.7j + 0.7k
0.1 + 0.3i + 0.3j + 0.9k
0.1 + 0.5i + 0.5j + 0.7k
0.2 + 0.4i + 0.4j + 0.8k
0.5 + 0.5i + 0.5j + 0.5k
0.28 + 0.96i
0 + 0.36i + 0.48j + 0.8k
0 + 0.48i + 0.6j + 0.64k
0.02 + 0.02i + 0.34j + 0.94k
0.02 + 0.1i + 0.5j + 0.86k
0.02 + 0.14i + 0.14j + 0.98k
0.02 + 0.14i + 0.7j + 0.7k
0.02 + 0.22i + 0.26j + 0.94k
0.02 + 0.22i + 0.46j + 0.86k
0.02 + 0.26i + 0.62j + 0.74k
0.02 + 0.34i + 0.38j + 0.86k
0.02 + 0.34i + 0.46j + 0.82k
0.02 + 0.34i + 0.58j + 0.74k
0.06 + 0.06i + 0.18j + 0.98k
0.06 + 0.06i + 0.62j + 0.78k
0.06 + 0.1i + 0.42j + 0.9k
0.06 + 0.18i + 0.54j + 0.82k
0.06 + 0.42i + 0.46j + 0.78k
0.06 + 0.42i + 0.62j + 0.66k
0.08 + 0.08i + 0.64j + 0.76k
0.08 + 0.12i + 0.24j + 0.96k
0.08 + 0.16i + 0.44j + 0.88k
0.08 + 0.2i + 0.56j + 0.8k
0.08 + 0.24i + 0.48j + 0.84k
0.08 + 0.32i + 0.56j + 0.76k
0.08 + 0.4i + 0.44j + 0.8k
0.08 + 0.52i + 0.56j + 0.64k
0.1 + 0.1i + 0.14j + 0.98k
0.1 + 0.26i + 0.5j + 0.82k
0.1 + 0.3i + 0.54j + 0.78k
0.1 + 0.34i + 0.62j + 0.7k
0.1 + 0.5i + 0.5j + 0.7k
0.12 + 0.24i + 0.64j + 0.72k
0.14 + 0.22i + 0.22j + 0.94k
0.14 + 0.22i + 0.62j + 0.74k
0.14 + 0.46i + 0.62j + 0.62k
0.16 + 0.16i + 0.32j + 0.92k
0.16 + 0.2i + 0.4j + 0.88k
0.16 + 0.32i + 0.64j + 0.68k
0.18 + 0.26i + 0.3j + 0.9k
0.18 + 0.26i + 0.54j + 0.78k
0.18 + 0.54i + 0.54j + 0.62k
0.22 + 0.26i + 0.38j + 0.86k
0.22 + 0.26i + 0.46j + 0.82k
0.22 + 0.26i + 0.58j + 0.74k
0.22 + 0.46i + 0.5j + 0.7k
0.24 + 0.44i + 0.48j + 0.72k
0.26 + 0.34i + 0.38j + 0.82k
0.26 + 0.46i + 0.58j + 0.62k
0.28 + 0.32i + 0.64j + 0.64k
0.3 + 0.3i + 0.46j + 0.78k
0.3 + 0.3i + 0.62j + 0.66k
0.32 + 0.4i + 0.4j + 0.76k
0.32 + 0.52i + 0.56j + 0.56k
0.34 + 0.34i + 0.62j + 0.62k
0.34 + 0.38i + 0.5j + 0.7k
0.34 + 0.46i + 0.58j + 0.58k
0.34 + 0.5i + 0.5j + 0.62k
0.36 + 0.48i + 0.48j + 0.64k
0.4 + 0.4i + 0.52j + 0.64k
0.42 + 0.42i + 0.46j + 0.66k
This file contains all unit quaternions that do not have more than three significant figures in each coefficient.
These are quaternions that contain only one significant digit in each component of the number. Other multiples of these may also have only one digit. Those marked with an asterisk can also be made into nice unit quaternions listed above. Of course, you can change the signs and rearrange the components, as you wish. You can multiply or divide these numbers by 2, 2.5, 4 or 5 and still have "nice" quaternions, as these are "nice" real numbers.
2 -1
|z| z z
10 1 + i + 2j + 2k 0.1 - 0.1i - 0.2j - 0.2k
50 3i + 4j + 5k -0.06i - 0.08j - 0.1k
50 3 + 3i + 4j + 4k 0.06 - 0.06i - 0.08j - 0.08k
100 5 + 5i + 5j + 5k 0.05 - 0.05i - 0.05j - 0.05k *
100 1 + i + 7j + 7k 0.01 - 0.01i - 0.07j - 0.07k *
100 2 + 4i + 4j + 8k 0.02 - 0.04i - 0.04j - 0.08k *
100 1 + 3i + 3j + 9k 0.01 - 0.03i - 0.03j - 0.09k *
100 1 + 5i + 5j + 7k 0.01 - 0.05i - 0.05j - 0.07k *
1000 6i + 8j + 30k -0.006i - 0.008j - 0.03k
Here, we present quaterions where either the quaternion or its reciprocal have no more than two significant digits. Those marked with an asterisk can also be made into nice unit quaternions listed above. Of course, you can change the signs and rearrange the components, as you wish. You can multiply or divide these numbers by 2, 2.5, 4 or 5 and still have "nice" quaternions, as these are "nice" real numbers.
2 -1 |z| z z 4 1 + i + j + k 0.25 - 0.25i - 0.25j - 0.25k * 20 1 + i + 3j + 3k 0.05 - 0.05i - 0.15j - 0.15k 25 1 + 2i + 2j + 4k 0.04 - 0.08i - 0.08j - 0.16k * 50 1 + 2i + 3j + 6k 0.02 - 0.04i - 0.06j - 0.12k 125 3i + 4j + 10k 0.024i - 0.032j - 0.08k 125 5i + 6j + 8k 0.04i - 0.048j - 0.064k 125 2 + 2i + 6j + 9k 0.016 - 0.016i - 0.048j - 0.072k 125 2 + 6i + 6j + 7k 0.016 - 0.048i - 0.048j - 0.056k 125 3 + 4i + 6j + 8k 0.024 - 0.032i - 0.048j - 0.064k 250 3i + 4j + 15k 0.012i - 0.016j - 0.06k 250 5i + 9j + 12k 0.02i - 0.036j - 0.048k 250 1 + 2i + 7j + 14k 0.004 - 0.008i - 0.028j - 0.056k 250 1 + 4i + 8j + 13k 0.004 - 0.016i - 0.032j - 0.052k 250 1 + 7i + 10j + 10k 0.004 - 0.028i - 0.04j - 0.04k 250 1 + 8i + 8j + 11k 0.004 - 0.032i - 0.032j - 0.044k 250 2 + 2i + 11j + 11k 0.008 - 0.008i - 0.044j - 0.044k 250 2 + 5i + 5j + 14k 0.008 - 0.02i - 0.02j - 0.056k 250 2 + 5i + 10j + 11k 0.008 - 0.02i - 0.04j - 0.044k 250 3 + 3i + 6j + 14k 0.012 - 0.012i - 0.024j - 0.056k 250 3 + 4i + 9j + 12k 0.012 - 0.016i - 0.036j - 0.048k 250 3 + 6i + 6j + 13k 0.012 - 0.024i - 0.024j - 0.052k 250 4 + 4i + 7j + 13k 0.016 - 0.016i - 0.028j - 0.052k 250 4 + 7i + 8j + 11k 0.016 - 0.028i - 0.032j - 0.044k 500 1 + 3i + 7j + 21k 0.002 - 0.006i - 0.014j - 0.042k 500 1 + 7i + 15j + 15k 0.002 - 0.014i - 0.03j - 0.03k 500 3 + 7i + 9j + 19k 0.006 - 0.014i - 0.018j - 0.038k 500 3 + 9i + 11j + 17k 0.006 - 0.018i - 0.022j - 0.034k 500 5 + 9i + 13j + 15k 0.01 - 0.018i - 0.026j - 0.03k 500 7 + 9i + 9j + 17k 0.014 - 0.018i - 0.018j - 0.034k 500 9 + 9i + 13j + 13k 0.018 - 0.018i - 0.026j - 0.026k 500 3 + 3i + 11j + 19k 0.006 - 0.006i - 0.022j - 0.038k 500 3 + 5i + 5j + 21k 0.006 - 0.01i - 0.01j - 0.042k 1250 3i + 4j + 35k 0.0024i - 0.0032j - 0.028k 1250 2 + 11i + 15j + 30k 0.0016 - 0.0088i - 0.012j - 0.024k 1250 9 + 12i + 20j + 25k 0.0072 - 0.0096i - 0.016j - 0.02k 2500 1 + 7i + 35j + 35k 0.0004 - 0.0028i - 0.014j - 0.014k * 2500 3 + 5i + 21j + 45k 0.0012 - 0.002i - 0.0084j - 0.018k * 2500 9 + 13i + 15j + 45k 0.0036 - 0.0052i - 0.006j - 0.018k * 2500 11 + 23i + 25j + 35k 0.0044 - 0.0092i - 0.01j - 0.014k * 2500 17 + 19i + 25j + 35k 0.0068 - 0.0076i - 0.01j - 0.014k *