Hexadecimal Words

Certain hexadecimal numbers contain only letters between A (10) and F (16) and so can form English words. The commonly accepted words are:

a, aba, abaca, abed, accede, acceded, ace, aced, ad, add, added, baa, baad, babe, bad, bade, baff, baffed, be, bead, beaded, bed, bedded, bee, beef, beefed, cab, cad, cade, cafe, cede, ceded, cee, dab, dabbed, dace, dad, daff, dead, deaf, decade, dee, deed, deeded, deface, defaced, ebb, ebbed, efface, effaced, fa, facade, face, faced, fad, fade, faded, fed, fee, feed

These hexadecimal "words" with their decimal equivalents are shown below (arranged alphabetically):

10 --> a

2746 --> aba

703178 --> abaca

44013 --> abed

11325150 --> accede

181202413 --> acceded

2766 --> ace

44269 --> aced

173 --> ad

2781 --> add

712173 --> added

2986 --> baa

47789 --> baad

47806 --> babe

2989 --> bad

47838 --> bade

47871 --> baff

12255213 --> baffed

190 --> be

48813 --> bead

12496365 --> beaded

3053 --> bed

12508653 --> bedded

3054 --> bee

48879 --> beef

12513261 --> beefed

3243 --> cab

3245 --> cad

51934 --> cade

51966 --> cafe

52958 --> cede

847341 --> ceded

3310 --> cee

3499 --> dab

14334957 --> dabbed

56014 --> dace

3501 --> dad

56063 --> daff

57005 --> dead

57007 --> deaf

14600926 --> decade

3566 --> dee

57069 --> deed

14609901 --> deeded

14613198 --> deface

233811181 --> defaced

3771 --> ebb

965613 --> ebbed

15727310 --> efface

251636973 --> effaced

250 --> fa

16435934 --> facade

64206 --> face

1027309 --> faced

4013 --> fad

64222 --> fade

1027565 --> faded

4077 --> fed

4078 --> fee

65261 --> feed

The decimal numbers in ascending order are:

10, 173, 190, 250, 2746, 2766, 2781, 2986, 2989, 3053, 3054, 3243, 3245, 3310, 3499, 3501, 3566, 3771, 4013, 4077, 4078, 44013, 44269, 47789, 47806, 47838, 47871, 48813, 48879, 51934, 51966, 52958, 56014, 56063, 57005, 57007, 57069, 64206, 64222, 65261, 703178, 712173, 847341, 965613, 1027309, 1027565, 11325150, 12255213, 12496365, 12508653, 12513261, 14334957, 14600926, 14609901, 14613198, 15727310, 16435934, 181202413, 233811181, 251636973

OEIS A132676 shows these same numbers. 

If we are only interested in decimal numbers whose hexadecimal equivalents contain only letters and no digits from 0 to 9 then these are listed in OEIS A228774. Up to 40000, there are 258 of them and they can be generated via this permalink.

10, 11, 12, 13, 14, 15, 170, 171, 172, 173, 174, 175, 186, 187, 188, 189, 190, 191, 202, 203, 204, 205, 206, 207, 218, 219, 220, 221, 222, 223, 234, 235, 236, 237, 238, 239, 250, 251, 252, 253, 254, 255, 2730, 2731, 2732, 2733, 2734, 2735, 2746, 2747, 2748, 2749, 2750, 2751, 2762, 2763, 2764, 2765, 2766, 2767, 2778, 2779, 2780, 2781, 2782, 2783, 2794, 2795, 2796, 2797, 2798, 2799, 2810, 2811, 2812, 2813, 2814, 2815, 2986, 2987, 2988, 2989, 2990, 2991, 3002, 3003, 3004, 3005, 3006, 3007, 3018, 3019, 3020, 3021, 3022, 3023, 3034, 3035, 3036, 3037, 3038, 3039, 3050, 3051, 3052, 3053, 3054, 3055, 3066, 3067, 3068, 3069, 3070, 3071, 3242, 3243, 3244, 3245, 3246, 3247, 3258, 3259, 3260, 3261, 3262, 3263, 3274, 3275, 3276, 3277, 3278, 3279, 3290, 3291, 3292, 3293, 3294, 3295, 3306, 3307, 3308, 3309, 3310, 3311, 3322, 3323, 3324, 3325, 3326, 3327, 3498, 3499, 3500, 3501, 3502, 3503, 3514, 3515, 3516, 3517, 3518, 3519, 3530, 3531, 3532, 3533, 3534, 3535, 3546, 3547, 3548, 3549, 3550, 3551, 3562, 3563, 3564, 3565, 3566, 3567, 3578, 3579, 3580, 3581, 3582, 3583, 3754, 3755, 3756, 3757, 3758, 3759, 3770, 3771, 3772, 3773, 3774, 3775, 3786, 3787, 3788, 3789, 3790, 3791, 3802, 3803, 3804, 3805, 3806, 3807, 3818, 3819, 3820, 3821, 3822, 3823, 3834, 3835, 3836, 3837, 3838, 3839, 4010, 4011, 4012, 4013, 4014, 4015, 4026, 4027, 4028, 4029, 4030, 4031, 4042, 4043, 4044, 4045, 4046, 4047, 4058, 4059, 4060, 4061, 4062, 4063, 4074, 4075, 4076, 4077, 4078, 4079, 4090, 4091, 4092, 4093, 4094, 4095

The final number (4095) in the list above is equivalent to "f f f" in hexadecimal. All the English words listed at the start of this post will be included in this list of course. Up to one million, there are 8034 such decimal numbers with letter-only hexadecimal equivalents and there are significant gaps between groups of numbers. See Figure 1.

Figure 1: permalink
On vertical scale 1.0 = One Million

Radix Economy

In a coffee shop this afternoon, I was reading an interesting article in Quanta Magazine titled How Base 3 Computing Beats Binary. I like the graphic also that began the article and which I've reproduced in Figure 1.

Figure 1: source

The article introduces the notion of "radix economy" that is explained as follows:

The hallmark feature of ternary notation is that it’s ruthlessly efficient. With two binary bits, you can represent four numbers. Two “trits” — each with three different states — allow you to represent nine different numbers. A number that requires 42 bits would need only 27 trits.

If a three-state system is so efficient, you might imagine that a four-state or five-state system would be even more so. But the more digits you require, the more space you’ll need. It turns out that ternary is the most economical of all possible integer bases for representing big numbers.

To see why, consider an important metric that tallies up how much room a system will need to store data. You start with the base of the number system, which is called the radix, and multiply it by the number of digits needed to represent some large number in that radix. For example, the number 100,000 in base 10 requires six digits. Its “radix economy” is therefore 10 × 6 = 60. In base 2, the same number requires 17 digits, so its radix economy is 2 × 17 = 34. And in base 3, it requires 11 digits, so its radix economy is 3 × 11 = 33. For large numbers, base 3 has a lower radix economy than any other integer base.

This Wikipedia article explains it in more formal terms for a number \(N\):$$ \text{radix economy of } N =b \lfloor \log_b(N)+1 \rfloor $$For large \(N\) we can thus write:$$ \begin{align} \text{radix economy of } N &\approx b \log_b(N) \\ &= \frac{b}{ln \,(b)} ln \,(N) \end{align}$$Using the number 123456789 as an example, the radix efficiency for integer bases from 3 to 16 is shown in Figure 2.

Figure 2: base 3 is best

The Quanta article goes on to say that:

In addition to its numerical efficiency, base 3 offers computational advantages. It suggests a way to reduce the number of queries needed to answer questions with more than two possible answers. A binary logic system can only answer “yes” or “no.” So if you’re comparing two numbers, x and y, to find out which is larger, you might first ask the computer “Is x less than y?” If the answer is no, you need a second query: “Is x equal to y?” If the answer is yes, then they’re equal; if the answer is no, then y is less than x. A system using ternary logic can give one of three answers. Because of this, it requires only one query: “Is x less than, equal to, or greater than y?”

And finally, also quoting from the article:

Surprisingly, if you allow a base to be any real number, and not just an integer, then the most efficient computational base is the irrational number e.

The table in Figure 2 looks as shown in Figure 3 when we add "e" to the list of bases.

Figure 3: e is best

Representing numbers using "e" as the number base is the stuff of a future post perhaps but here is a link to how to go about it. 

Quaternary Number System

The number associated with my diurnal age today, 27368, has the property that its prime factors contain only the digits 1, 2 and 3. Specifically:$$27368=2 \times 2 \times 2 \times 11 \times 311$$This got me thinking about what numbers in the range up to 333,321 contain only the digits 1, 2, 3 and have prime factors that also only contain the digits 1, 2, 3. It turns out that there are only seven such numbers and they are (permalink):$$ \begin{align} 132 &= 2^2 \times 3 \times 11\\312 &= 2^3 \times 3 \times 13\\11132 &= 2^2 \times 11^2 \times 23\\11232 &= 2^5 \times 3^3 \times 13\\13312 &= 2^{10} \times 13\\123132 &= 2^2 \times 3 \times 31 \times 331\\132122 &= 2 \times 31 \times 2131 \end{align}$$For brevity, I've written the factorisation in base-index format and so digits greater than 1, 2, 3 appear but these of course disappear when the factorisation is not written in base-index format.

The appearance of the digits 1, 2, 3 only is reminiscent of the quaternary number system even though what is written previously is in base 10. The quaternary equivalents of the previous decimal numbers are:$$ \begin{align} 132 &= 2010_{4}\\312 &= 10320_4\\11132 &= 2231330_4\\11232 &= 2233200_4\\13312 &= 3100000\\123132 &= 132003330_4\\132122 &= 200100122_4 \end{align} $$The factorisation of the quaternary numbers will contain the same number of factors and single digit factors will remain the same but multidigit factors will appear as their quaternary equivalents. Thus:$$ \begin{align} 13312 &= 2^{10} \times 13 \text{ and}\\3100000_4 &=2_4^{22_4} \times 31_4 \end{align} $$Of course, the quaternary number system also contains the digit 0 and so it's fair to ask what numbers up to 333,210 contain all the digits 0, 1, 2 and 3 (multiplicity allowed) and have prime factors that also contain only these digits (again with multiplicity allowed). It turns out that there is only one such number in the range up to one million. It is:$$133012 = 2^2 \times 11 \times 3023$$I thought I'd ask Google's Gemini about the quaternary number system.

Can you give me some interesting facts about and uses of the quaternary or base 4 number system?

The quaternary (base-4) system, while not as common as binary or decimal, holds some interesting properties and potential uses. Here are some fun facts and applications:

Interesting Facts:

• Fewer Digits: Compared to binary, quaternary uses fewer digits to represent the same numerical value. For example, 10 in binary is 1010, while in quaternary it's simply 22.
• DNA Connection: The four building blocks of DNA (Adenine, Guanine, Cytosine, and Thymine) can be seen as a quaternary system, with each combination representing a specific unit of genetic information.
• Error Detection: Quaternary finds some niche applications in fault-tolerant computer systems. By using 4 possible states (0, 1, 2, 3) instead of 2 in binary, it allows for additional error detection and correction mechanisms.

Potential Uses:

• Educational Tool: Quaternary can be a stepping stone for learning binary. Due to its simplicity (using only 4 digits), it can be easier to grasp the concept of positional number systems before moving to binary's base-2 system.
• Space Exploration: Some proposals for future spacefaring computers suggest using quaternary systems due to their potential radiation hardness. Quaternary systems might be more resistant to the corrupting effects of cosmic radiation compared to traditional binary systems. (This is an area of ongoing research)
• Signal Processing: Quaternary can be used in certain signal processing applications where reducing the number of states while maintaining some level of complexity compared to binary might be beneficial. However, extensive binary infrastructure makes widespread adoption less likely.

Overall, the quaternary number system offers a unique perspective on representing numbers. While not as widely used as binary or decimal, it has interesting properties and potential applications in specific areas like education, fault-tolerant systems, and possibly even future space exploration.

Here is some further information about the connection with DNA taken from this source:

Parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in alphabetical order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0, 1, 2, and 3. With this encoding, the complementary digit pairs 0↔3, and 1↔2 (binary 00↔11 and 01↔10) match the complementation of the base pairs: A↔T and C↔G and can be stored as data in DNA sequence. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010 (= decimal 9156 or binary 10 00 11 11 00 01 00).

Gray Code

I'd never heard of this before. I came across it when looking at the properties associated with my diurnal age of 27104. One of its properties is that it's a member of OEIS A265385:

A265385

Sequence defined by a(1) = a(2) = 1 and a(\(n\)) = gray(a(\(n\)-1) + a(\(n\)-2)), with gray(\(m\)) = A003188(\(m\)).

Here's what Wikipedia has to say on the topic:

The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).

For example, the representation of the decimal value "1" in binary would normally be "001" and "2" would be "010". In Gray code, these values are represented as "001" and "011". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.

Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems.

Table 1 shows the relationships between binary, gray code and decimal.

Table 1: source

To understand how this OEIS arises, I needed a way to convert binary to gray code and this site provided some Python code to achieve this. I modified it slightly to accept decimal input (permalink). Armed with this, I began to investigate the sequence. 

The initial values are 1 and 1 and these sum to 2. The binary code for 2, as can be seen from Table 1, is 0010 which the Gray code converts to 0011 and this has a decimal equivalent of 3. Proceeding like this we have:

• 1 first seed value
• 1 second seed value
• 1 + 1 = 2 --> 3
• 1 + 3 = 4 --> 6
• 3 + 6 = 9 --> 13
• 6 + 13 =19 --> 26
• 13 + 26 = 39 --> 52
• 26 + 52 = 78 --> 105
• 52 + 105 = 157 --> 211
• 105 + 211 = 316 --> 418
• 211 + 418 = 629 --> 847
• 418 + 847 = 1265 --> 1673
• 847 + 1673 = 2520 --> 3380
• 1673 + 3380 = 5053 --> 6755
• 3380 + 6755 = 10135 --> 13404
• 6755 + 13404 = 20159 --> 27104

The OEIS comments note that:

This recurrence is reminiscent of Fibonacci's, except that the result of each step is passed through the binary-reflected Gray code mapping, which introduces a degree of pseudo-randomness. 

However, the ratio of successive terms doesn't approach the Golden Ratio but instead fluctuates around 2 with a variation of about 1% above and below. Here are the successive ratios of \( \frac{(n+1)^{th} \text{ term}}{n^{th} \text{ term}}\):

3.0000000, 2.0000000, 2.1666667, 2.0000000, 2.0000000, 2.0192308, 2.0095238, 1.9810427, 2.0263158, 1.9752066, 2.0203228, 1.9985207, 1.9843079, 2.0220830, 1.9752804, 2.0203033, 1.9986779, 1.9841292, 2.0220523, 1.9752920, 2.0202956, 1.9986750, 1.9841298, 2.0220479, 1.9752930, 2.0202950, 1.9986749, 1.9841299, 2.0220478, 1.9752930, 2.0202950, 1.9986725

There's lots of YouTube videos explaining about Gray code and I'm happy to have finally stumbled upon this clever manipulation of binary code. The OEIS A003188 referred to above lists the decimal equivalents of the Gray code for integers \(n\). The result is a permutation of the order of the cardinal numbers as shown for \(n\) up to 70:

• 0 --> 0
• 1 --> 1
• 2 --> 3
• 3 --> 2
• 4 --> 6
• 5 --> 7
• 6 --> 5
• 7 --> 4
• 8 --> 12
• 9 --> 13
• 10 --> 15
• 11 --> 14
• 12 --> 10
• 13 --> 11
• 14 --> 9
• 15 --> 8
• 16 --> 24
• 17 --> 25
• 18 --> 27
• 19 --> 26
• 20 --> 30
• 21 --> 31
• 22 --> 29
• 23 --> 28
• 24 --> 20
• 25 --> 21
• 26 --> 23
• 27 --> 22
• 28 --> 18
• 29 --> 19
• 30 --> 17
• 31 --> 16
• 32 --> 48
• 33 --> 49
• 34 --> 51
• 35 --> 50
• 36 --> 54
• 37 --> 55
• 38 --> 53
• 39 --> 52
• 40 --> 60
• 41 --> 61
• 42 --> 63
• 43 --> 62
• 44 --> 58
• 45 --> 59
• 46 --> 57
• 47 --> 56
• 48 --> 40
• 49 --> 41
• 50 --> 43
• 51 --> 42
• 52 --> 46
• 53 --> 47
• 54 --> 45
• 55 --> 44
• 56 --> 36
• 57 --> 37
• 58 --> 39
• 59 --> 38
• 60 --> 34
• 61 --> 35
• 62 --> 33
• 63 --> 32
• 64 --> 96
• 65 --> 97
• 66 --> 99
• 67 --> 98
• 68 --> 102
• 69 --> 103
• 70 --> 101

The March of Time

I've written about what I term AD and BC numbers in a post titled, quite sensibly, AD and BC Numbers. I was reminded of them because my diurnal age today, 27068, converts to 69BC in hexadecimal. For some time now, these BC numbers have been occurring every 256 days.

24764 --> 60bc

25020 --> 61bc

25276 --> 62bc

25532 --> 63bc

25788 --> 64bc

26044 --> 65bc

26300 --> 66bc

26556 --> 67bc

26812 --> 68bc

27068 --> 69bc

However, this regular march of time is now at an end because if 256 is added to 27068, the resultant number (27324) is 6ABC. The decimal equivalent of 70AD is 28845, representing a jump of 1792 or 7 x 256 days.

Similarly, I recently turned 27053 days old which is 69AD in hexadecimal. Notice the difference of 14 days between 69AD and 69BC. This number was also the end of a run of numbers differing by 256 days.

 

24749 --> 60ad

25005 --> 61ad

25261 --> 62ad

25517 --> 63ad

25773 --> 64ad

26029 --> 65ad

26285 --> 66ad

26541 --> 67ad

26797 --> 68ad

27053 --> 69ad

The decimal equivalent of 70AD is 28845, again a jump of 1792 or 7 x 256 days. Taken over a long enough time period, this represents an average advance of a little over 395.6 days, about a month longer than the solar year. Both AD and BC dates are advancing at this average rate.

However, it will almost five years before I encounter another hexadecimal AD and BC number so I thought it important to mark the fact in this post. While exercises like this may seem frivolous, they nonetheless provide an opportunity to work with hexadecimal numbers and convert from decimal to hexadecimal and vice versa. I'm always thinking in terms of the former mathematics teacher who I once was.

How can hexadecimal numbers be made interesting for students? These AD and BC numbers are a way of doing this. The question could be asked of students:

Find out your diurnal age and determine when you will next have a connection to AD or BC year (via decimal to hexadecimal conversion). What was significant about that year.

See my post titled 69BC for details on what was significant about this year in history. Students could be shown how to determine their diurnal age using Wolfram Alpha and they could also use it to convert between decimal and hexadecimal. Overall, a useful and interesting exercise.

Numbers Within Numbers

As sometimes happens, the number associated with my diurnal age seems to contain nothing much of interest. Today the number was 26930 and I was temporarily stuck. However, I noticed something in Numbers Aplenty. See Figure 1.

Figure 1

The hexadecimal representation of 26930 is 6932. All the digits in the hexadecimal representation are contained within the decimal representation, each in the same proportion. I thought that this would be an interesting criterion to apply to the numbers from 10 to 40000. The numbers from 0 to 9 trivially satisfy the criterion. 

It turns out that there are only 90 decimal numbers that satisfy. This is the list (Permalink):

Hex     Decimal

35 --- 53

173 --- 371

391 --- 913

411 --- 1041

412 --- 1042

413 --- 1043

414 --- 1044

415 --- 1045

416 --- 1046

417 --- 1047

418 --- 1048

419 --- 1049

541 --- 1345

791 --- 1937

821 --- 2081

822 --- 2082

823 --- 2083

824 --- 2084

825 --- 2085

826 --- 2086

827 --- 2087

828 --- 2088

829 --- 2089

1004 --- 4100

1415 --- 5141

1524 --- 5412

1826 --- 6182

2008 --- 8200

2419 --- 9241

3012 --- 12306

4010 --- 16400

4011 --- 16401

4012 --- 16402

4013 --- 16403

4014 --- 16404

4015 --- 16405

4017 --- 16407

4018 --- 16408

4019 --- 16409

4167 --- 16743

4815 --- 18453

5021 --- 20513

5042 --- 20546

5221 --- 21025

6251 --- 25169

6528 --- 25896

6582 --- 25986

6702 --- 26370

6872 --- 26738

6921 --- 26913

6932 --- 26930

7221 --- 29217

7412 --- 29714

7830 --- 30768

7840 --- 30784

8020 --- 32800

8021 --- 32801

8022 --- 32802

8024 --- 32804

8025 --- 32805

8026 --- 32806

8027 --- 32807

8028 --- 32808

8029 --- 32809

8052 --- 32850

8335 --- 33589

8834 --- 34868

8840 --- 34880

8841 --- 34881

8842 --- 34882

8844 --- 34884

8845 --- 34885

8846 --- 34886

8847 --- 34887

8848 --- 34888

8849 --- 34889

9026 --- 36902

9056 --- 36950

9274 --- 37492

9830 --- 38960

9831 --- 38961

9832 --- 38962

9833 --- 38963

9834 --- 38964

9835 --- 38965

9837 --- 38967

9838 --- 38968

9839 --- 38969

9848 --- 38984

9982 --- 39298

The hexadecimal numbers will have the same or fewer digits than the decimal ones. Here is the list of decimal numbers on their own:

53, 371, 913, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1048, 1049, 1345, 1937, 2081, 2082, 2083, 2084, 2085, 2086, 2087, 2088, 2089, 4100, 5141, 5412, 6182, 8200, 9241, 12306, 16400, 16401, 16402, 16403, 16404, 16405, 16407, 16408, 16409, 16743, 18453, 20513, 20546, 21025, 25169, 25896, 25986, 26370, 26738, 26913, 26930, 29217, 29714, 30768, 30784, 32800, 32801, 32802, 32804, 32805, 32806, 32807, 32808, 32809, 32850, 33589, 34868, 34880, 34881, 34882, 34884, 34885, 34886, 34887, 34888, 34889, 36902, 36950, 37492, 38960, 38961, 38962, 38963, 38964, 38965, 38967, 38968, 38969, 38984, 39298

The algorithm can  be easily modified to accommodate bases up to 36 and the range can be extended, let's say to 100,000. Here are the results for base 36 in the range up to 100,000 (Permalink):

Base 36 Decimal

41 --- 145

75 --- 257

185 --- 1589

213 --- 2631

247 --- 2743

268 --- 2816

291 --- 2917

340 --- 4032

573 --- 6735

659 --- 7965

680 --- 8064

814 --- 10408

850 --- 10548

896 --- 10698

1357 --- 50731

1504 --- 53140

1875 --- 57281

There are only 17 decimal numbers that satisfy in this range. These are listed below in decimal form only:

145, 257, 1589, 2631, 2743, 2816, 2917, 4032, 6735, 7965, 8064, 10408, 10548, 10698, 50731, 53140, 57281

If we wish to look at bases below 10 then the numbers will have the same or more digits and so the algorithm will need to be modified. We will require all the digits in base 10 format to be contained within the number in the lower base format and these digits need to be in the same proportion. Here are the results for base 8 in range from 8 to 40,000 (Permalink):

Base 8 Decimal

107 --- 71

1347 --- 743

1357 --- 751

2371 --- 1273

3165 --- 1653

4152 --- 2154

6130 --- 3160

6131 --- 3161

6132 --- 3162

6133 --- 3163

6134 --- 3164

6135 --- 3165

6136 --- 3166

6137 --- 3167

6232 --- 3226

12765 --- 5621

13620 --- 6032

14562 --- 6514

15713 --- 7115

15720 --- 7120

15721 --- 7121

15722 --- 7122

15723 --- 7123

15724 --- 7124

15726 --- 7126

15727 --- 7127

15764 --- 7156

17074 --- 7740

37621 --- 16273

52371 --- 21753

73560 --- 30576

75341 --- 31457

102345 --- 34021

102473 --- 34107

103462 --- 34610

104365 --- 35061

105436 --- 35614

106273 --- 36027

106347 --- 36071

There are only 39 numbers that satisfy. Here is the list in decimal format only:

71, 743, 751, 1273, 1653, 2154, 3160, 3161, 3162, 3163, 3164, 3165, 3166, 3167, 3226, 5621, 6032, 6514, 7115, 7120, 7121, 7122, 7123, 7124, 7126, 7127, 7156, 7740, 16273, 21753, 30576, 31457, 34021, 34107, 34610, 35061, 35614, 36027, 36071

Thus it can be seen that whenever a number seems boring, there is always something of interest that remains to be discovered and that property can often be generalised, as was done here, to find interesting properties of other numbers.

Getting back to comparing decimal numbers to their hexadecimal equivalents we can make conditions more stringent by requiring that the lengths of both numbers be the same so that the digits of both are simply permutations of the other. In the range up to 100,000, there are 24 numbers that qualify. These are (Permalink):

Hex Decimal

35 --- 53

173 --- 371

391 --- 913

1004 --- 4100

1415 --- 5141

1524 --- 5412

1826 --- 6182

2008 --- 8200

2419 --- 9241

12570 --- 75120

12571 --- 75121

12572 --- 75122

12573 --- 75123

12574 --- 75124

12575 --- 75125

12576 --- 75126

12577 --- 75127

12578 --- 75128

12579 --- 75129

12765 --- 75621

15086 --- 86150

16549 --- 91465

18197 --- 98711

18499 --- 99481

The numbers in decimal format only are:

53, 371, 913, 4100, 5141, 5412, 6182, 8200, 9241, 75120, 75121, 75122, 75123, 75124, 75125, 75126, 75127, 75128, 75129, 75621, 86150, 91465, 98711, 99481

Similarly in the range up to 100,000, there are only 16 numbers in base 8 format that have the same digits as their decimal equivalants (Permalink):

Base 8 Decimal

2371 --- 1273

3165 --- 1653

4152 --- 2154

6130 --- 3160

6131 --- 3161

6132 --- 3162

6133 --- 3163

6134 --- 3164

6135 --- 3165

6136 --- 3166

6137 --- 3167

6232 --- 3226

37621 --- 16273

52371 --- 21753

73560 --- 30576

75341 --- 31457

Here are the numbers in decimal format only:

1273, 1653, 2154, 3160, 3161, 3162, 3163, 3164, 3165, 3166, 3167, 3226, 16273, 21753, 30576, 31457

Lightweight mathematics of course but sometimes it's good just to have fun with numbers. The table below shows the number of decimal numbers whose value in another base is an anagram of its digits. The bases range from 4 to 28. For bases 2 and 3 and for 29 and beyond, there are no such numbers.

Base 26 is interesting because the ten decimal numbers that satisfy are consecutive and the hexadecimal numbers formed from them are similarly consecutive and formed simply by moving the leading 9 of the decimal number one place to the right. Thus 910 in decimal becomes 190 in hexadecimal, 911 becomes 191 and so on until 919 that becomes 199.

Base 26 Decimal

190 --- 910

191 --- 911

192 --- 912

193 --- 913

194 --- 914

195 --- 915

196 --- 916

197 --- 917

198 --- 918

199 --- 919

All this can be explored via this permalink.

On the Randomness of Pi's Digits

Much has been written about the randomness of the \(\pi\)'s digits but usually in the context of the decimal number system. Looking at \(\pi\) in terms of the binary number system however, introduces a simplification that can be exploited to investigate its randomness.

The first ten decimal digits of \(\pi\) are 3.141592654 which is represented in binary as:

11.00100100001111110110101010001000100

SageMathCell enables us to examine the first one million decimal digits of \(\pi\), convert these to binary and then graph the cumulative balance between the number of 0's and 1's. The first two 1's are ignored and only the 0's and 1's following the decimal point are considered. Figure 1 shows the result:

Figure 1

Figure 2 shows the code that I used to generate this graph:

Figure 2: permalink

How does this graph compare to that of a sequence of randomly generated 0's and 1's? It's easy enough to undertake this comparison. To this end, I looked at a string of three million randomly generated 0's and 1's because this is approximately the number of binary digits resulting from the conversion of one million decimal digits. The result is shown in Figure 3. Of course, successive repetitions of this algorithm will produce different graphs whereas the graph for \(\pi\) will be always the same. However, it's the general form of the two graphs that I want to compare.

Figure 3: permalink

Figure 4 shows the code that was used to produce this result.

Figure 4: permalink

Not surprisingly, the form of the two graphs is very similar. The point of the exercise is that the conversion of \(\pi\) to binary digits provides an easy visual comparison of its digits with randomly generated 0's and 1's.

ADDENDUM: March 17th 2021

There are of course many interesting facts concerning \(\pi\). Here are some of them:

• The number pi is literally infinitely long. But the number 123456 doesn’t appear anywhere in the first million digits of \( \pi \). It is a bit shocking because if a million digits of \( \pi \) don’t have the sequence 124356, it definitely is the most unique number. Source.
  
  
• Did you know there is actually an entire language based entirely on \( \pi\)? Called Pilish, the numbers of letters in successive words match the digits in \(\pi\). Mike Keith, a devoted Pilish-lover, even wrote an entire book in Pilish called "Not a Wake." The rules of the language and its variants are described here.
  
  
• Rivers bend to \( \pi\): The way a river meanders is described by its sinuosity; the length of its winding path divided by the distance from the source to the ocean as measured in a straight line. Strange as it may be, the average river has a sinuosity of around 3.14, according to the journal Science. Source.

If you hold a mirror to a circle, it looks like a circle. If you hold a mirror up to 3.14, it spells PIE! See Figure 2.

Figure 2: source

Someone has developed a search engine that can be used to find digit sequences in \( \pi \) in the first \(2 \times 10^0 \) digits of \pi. Figure 3 shows the output if 123456789 is input.

Figure 3

Pressing the next link bring brings up what is shown in Figure 4:

Figure 4

      

The site can also be used to find positions of e, \(\sqrt{2} \) and \(\phi\) where SageMath for example struggles with one million decimal places.

Here is a GeoGebra link that will allows the user to use a slider to display decimal places of \(\pi\).