We’ve already seen a bit of Modular Arithmetic in the previous lesson (Number Bases and Conversion). In this lesson we will delve a little deeper and explore Modular Arithmetic further.
When dividing two integers we can break down the parts of the division equation into several components. Those components are the Dividend, Divisor, Quotient and Remainder. If expressed as a formula they look like the following:
Dividend/Divisor = Quotient and Remainder
Modular Arithmetic in Number Theory focuses on the Remainder component or what is referred to as the Modulo Operation, Modulus or Mod for short.
Dividend mod Divisor = Remainder
Example: What is 23 mod 7 ?
23/7 = 3 remainder 2
Since the mod operation only cares about the remainder
23 mod 7 = 2
Modular Arithmetic and Clocks
What does Modular Arithmetic have to do with Clocks? A lot! In fact, one very useful way to understand Modular Arithmetic is by thinking of clocks and how we use them. A clock has the numbers 1 – 12 on its face. Once we pass 12 o’clock we start again at 1 o’clock and continue in a circular manner. Another way we tell time is with a 24-hour system commonly used by the military. In this system we have the usual hours 1 – 12, but instead of starting at hour 1 again after we pass hour 12, we count hours 13, 14, 15 and so on until 24. In this 24-hour system the 13th hour or 13 o’clock is like 1 o’clock, 14 is like 2 o’clock, 15 like 3 o’clock and so forth. Notice a pattern?
If we extend this idea past 24 hours, we can say that 25 o’clock is also like 1 o’clock or 13 o’clock because they both wrap around the clock and land on the 1 o’clock position. In this clock system we can represent where we are in relation to the numbers 1 – 12 (or 0 – 11 as illustrated). That means that 1, 13, 25, 37, 49 … fall on the same position on the clock and are thought of as the same. 2, 14, 26, 38, 50 … are also thought of as the same.
Image Source: https://mathstats.uncg.edu/
In other words, 13 = 1 + a multiple of 12. 26 = 2 + a multiple of 12. Another way to express this is by saying that 1 is the remainder when we divide 13 by 12. 2 is the remainder when we divide 26 by 12. Mathematically, we write this as 13 ≡ 1 mod 12, 26 ≡ 2 mod 12, etc… This is read as 13 is congruentHaving the difference divisible by a given modulus to 1 mod (or modulo) 12. 26 is congruent to 2 mod 12.
There are many different mods we can work in. Above, we worked in Mod-12. Whatever mod we work with, it is always helpful to think of a clock face with that many numbers. Mod-8 for example can have a clock face numbered 1 – 8 (or in a similar fashion 0 – 7, like our illustration above). Remember that when we pass our highest number, we reset to 1 again.
If we put together what we’ve learned so far, we can formulate a general concept regarding mod. Namely, if we are working in a system of Mod-N (N being a whole number), we write a ≡ b mod n if aand b result in the same remainder when you divide them by n. In other words, a ≡ b mod nif ndivides a−b.
Example: What is 17 mod 5?
We can divide 17 by 5 and whatever the remainder is, is our answer.
17/5 = 3 remainder 2.
17 mod 5 = 2
Alternatively, we can continually subtract 5 from 17 until we are left with a number less than 5 (1, 2, 3, or 4).
17 – 5 – 5 – 5 = 2
Example: What is -3 mod 4?
Negative or positive numbers, mods work the same way.
-3 + 4 = 1
A helpful thing to remember is that if we have amod b and we increase a by a multiple of b, we get the same result.
For example, the following are all equal to the same number.
3 mod 7 = 3
(3 + 7) mod 7 = 10 mod 7 = 3
(3 + 14) mod 7 = 17 mod 7 = 3
(3 + 21) mod 7 = 24 mod 7 = 3
(3 + 28) mod 7 = 31 mod 7 = 3
You may come across notation that states: a ≡ b mod c. This simply means that a is the same as b when taken mod c.
For example, 3, 10, 17, 24, and 31 are all equal to 3 when taking mod 7. See the examples above. Therefore, we can say:
Human beings are most comfortable or familiar with counting using the numbers 0 – 9. This is also referred to as the Base-10 system. We call it the Base-10 system because there are exactly 10 symbols or numbers/digits represented.
From an early age we are taught to use these numbers and after some time it becomes second nature. This is why Base-10 feels so natural to us. Base-10 systems have existed for a very long time too. Many civilizations throughout history have used a Base-10 numbering system. In fact, it is widely accepted that the reason we use Base-10 and not some other arbitrary number system is because humans have 10 fingers!
What is a Number Base (Radix)?
A number base, radix, or “base” for short is a system of counting that describes numbers using symbols (such as numbers and letters) to represent values. The base of a number system also tells us how many unique number notations exist.
For example, Base-2 or the binary numbering system has exactly 2 unique numbers used to express different values (0 and 1). Binary is used in programming and this makes sense because computers “think” in binary. This isn’t the only number base that we often see in programming.
Another example of a programming-friendly number base is Hexadecimal or Base-16. As you may have already figured out, in this number base we have 16 unique symbols to represent different values. They are the digits 0 – 9 and the letters A – F.
Example: How many symbols/digits exist in Base-8 (Octal)?
8 – The numbers 0 – 7
When working with numbers in different bases we need to be clear about which base we are representing. One way we do this is by using a simple subscript next to the number we are working with. For example, 1002 is in Base-2 or the number 410 in decimal. 10010 is in Base-10 or the number “One Hundred”. 10016 is in Base-16 or the number 25610 in decimal. Each value very different from the others. If no base is specified, we generally assume Base-10 as the default.
Counting in Different Number Bases
Counting in another number base is very similar to counting in our more familiar Base-10. In decimal or Base-10 we can count 0, 1, 2, … 7, 8, 9, and then 10. Notice how in Base-10 there is no unique symbol/digit for the value “10”. Instead, ten is made up of the digits 1 and 0. This is also true for all other bases.
For example, Base-2 doesn’t contain the symbol “2” but rather the idea of “2” is represented by “10” (1 and 0. Not the number 10, or ten).
Another thing to note is how when we reach the last digit in our number system, we reset, and shift/carry. For example, in decimal 9 + 1 = 10. Adding 1 to the last symbol/digit (9) in our number system (Base-10) resets the “One’s” place to 0. We also shift or carry a 1 over to the left in what we call the “Ten’s” place. This gives us 1 – Ten and 0 – Ones, or Ten. If we continue to shift left we get to the “Hundred’s” place, then the “Thousand’s” place and so on.
Every time we shift to the left the value of our number is effectively multiplied by the base we are operating in. In Computer Science, this operation actually has a name. We call it an Arithmetic Shift (to the left). Arithmetic Left Shifts are equivalent to multiplication by the radix or base we are operating in.
Following this logic what happens when we add or count in Base-2 (binary)? Since binary only has the symbols 0 and 1, we will need to reset, shift/carry after “1”. In binary, 1 + 1 = 10 (One Zero. Not to be confused with the number Ten in decimal). As we do in decimal-land, adding 1 to the last symbol/digit (1) in our number system (Base-2) resets the “One’s” place to a 0 and we shift or carry a 1 over to the left which we call the “Two’s” place. Why the “Two’s” place? Well, because we are counting in Base-2. Every shift or carry over to the left is a power of that base. So in binary, we have the 1’s (20), 2’s (21), 4’s (22), 8’s (23) place and so on. Remember that every time we shift to the left we are effectively multiplying by the base or radix which we are operating in.
So far we’ve seen what counting in a base lower than 10 looks like. What does counting in a base greater than 10 look like? Something like Hexadecimal or Base-16?
Let’s start and see where it takes us. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, … What comes after “F”? If counting in hexadecimal is anything like counting in binary or decimal then F + 1 should be similar to 1 + 1 in binary or 9 + 1 in decimal. Indeed it is! Again, adding 1 to the last symbol/digit (F) in our number system (Base-16) resets the “One’s” place to a 0 and we shift or carry a 1 over to the left which we call the “16’s” place.
Converting between Number Bases
We already have some good clues that will help us in converting between number bases. Namely, we now know that whenever we reach the highest single digit/value in a column or place, we reset and shift or carry to the left, which we determined was equivalent to a multiplication by that base. Depending on the base we are operating in, that number column or place will be some power of that base. This little nugget of information will help us convert a number in ANY base to its equivalent value in Base-10. What follows are a few examples.
Base 2:
From right to left we have the 1’s column, the 2’s column, the 4’s column, the 8’s column, the 16’s column, etc… Notice how each column is 1 power greater than the column before it. The “X’s” represent the symbol/digit of the number.
Similarly, in the decimal system from right to left we have the 1’s column, the 10’s column, the 100’s column, the 1000’s column, the 10000’s column, etc… Each column 10 times greater than the column before it.
Following the same formula in Base-16; starting from the right and going to the left, we have the 1’s column, the 16’s column, the 256’s column, the 4096’s column and the 65536’s column and so on … ever increasing 16 times whenever we shift left.
We can more generally express this as the following:
Converting a Number from Base-X to Base-10
Let’s say we have a number – 123XYZB (Base-B)
Generally, we take the symbol/digit and multiply by the base (B) raised to the power of the column or place that symbol/digit is in. The columns/places start from 0 and the rightmost symbol/digit of the number. Finally, we add all of our products together and get our decimal equivalent number.
What about going the other way around? That is, taking a number we have in Base-10 and converting it to another base? Let’s take the number 35 and convert it into Binary (Base-2), Octal (Base-8) and Hexadecimal (Base-16).
As is true with many things in math, there are multiple ways to approach this task. One common and simple method involves using Modulo Division. Unlike regular division, Modulo Division or Modulus gives us just the remainder of a division operation rather than the quotient and remainder. For example, 10/3 = 3r1 or 3 remainder 1. However, 10%3 = 1. Also expressed as 10 mod 3 = 1. Using this method of Modulo Division we can convert any number from Base-10 to whatever other Base-X we choose. The idea behind this method is somewhat the reverse of what we did when converting a number of Base-X to Base-10. We are going to find how many times our original number can be divided into by our desired Base (X) and use that result as the number we divide into (dividend) in the following step. We also determine what the remainder of that division (Modulus) is while keeping track of our results. We repeat these steps until we reach a dividend of 0 at which point we calculate the Modulus one last time. We then piece together the results of our Modulo Division operations in reverse order (starting from our last result to our first result) to get our converted number.
Example: What is 35 in Binary?
Binary is Base-2 so we will be dividing by 2.
35/2 = 17 and 35%2 = 1
Repeat until the dividend is 0.
17/2 = 8 and 17%2 = 1
8/2 = 4 and 8%2 = 0
4/2 = 2 and 4%2 = 0
2/2 = 1 and 2%2 = 0
1/2 = 0 and 1%2 = 1
In the following step our dividend is 0 so we stop.
Putting the results of our Modulus Division operations all together in reverse order we get:
1000112 = 3510
Example: What is 35 in Octal?
Octal is Base-8 so we will be dividing by 8.
35/8 = 4 and 35%8 = 3
Repeat until the dividend is 0.
4/8 = 0 and 4%8 = 4
In the following step our dividend is 0 so we stop.
Putting the results of our Modulus Division operations all together in reverse order we get:
438 = 3510
Example: What is 35 in Hexadecimal?
Hexadecimal is Base-16 so our divisor will be 16.
35/16 = 2 and 35%16 = 3
Repeat until the dividend is 0.
2/16 = 0 and 2%16 = 2
In the following step our dividend is 0 so we stop.
Putting the results of our Modulus Division operations all together in reverse order we get:
2316 = 3510
Example: What is 672 in Hexadecimal?
Hexadecimal is Base-16 so our divisor will be 16.
672/16 = 42 and 672%16 = 0
Repeat until the dividend is 0.
42/16 = 2 and 42%16 = 1010 or A16 (Remember that in Base-16 we use the symbols A – F to represent the numbers 10 – 15)
2/16 = 0 and 2%16 = 2
In the following step our dividend is 0 so we stop.
Putting the results of our Modulus Division operations all together in reverse order we get:
2A016 = 67210
Converting a Number from Base-X to Base-Y
Now that we’re comfortable converting a number to and from Base-10 it will be easy for us to convert a number from any base to any other base. All that is required is for us to convert our original number into Base-10 and then from Base-10 to the desired base.
Example: What is 7358 (Base-8) in Base-5?
First we convert 7358 (Base-8) to Base-10:
7358 is expressed:
(7 * 82) + (3 * 81) + (5 * 80)
(7 * 64) + (3 * 8) + (5 * 1)
7358 = 44810 + 2410 + 510 = 47710
Next, we convert our result from step 1 (47710) into our desired base (Base-5).
Our desired base is Base-5 so we will be dividing by 5.
477/5 = 95 and 477%5 = 2
Repeat until the dividend is 0.
95/5 = 19 and 95%5 = 0
19/5 = 3 and 19%5 = 4
3/5 = 0 and 3%5 = 3
In the following step our dividend is 0 so we stop.
Putting the results of our Modulus Division operations all together in reverse order we get:
* Example: What is the smallest positive integer that has 10 factors?
We can observe that 10 = 2 * 5, meaning that the number in question has just two prime factors in its decomposition – one with the exponent of α = 1, the other of β = 4: N = (p1)(q4). To make N as small as possible, we have to choose the smallest available primes, 2 and 3. The answer then must be N = (31)(24) = 48
The smallest positive integer that has 10 factors is the number 48
The greatest common factor of a set of numbers such as a and b is the largest positive integer that divides both a and b. The GCF can be thought of like the inverse of LCM.
GCF(a, b) * LCM(a, b) = a * b
To find the GCF of a set of numbers:
List the prime factors of each number
Multiply the factors that all numbers have in common
If there are no common prime factors, then the GCF is 1.
Example: What is the Greatest Common Factor (GCF) of 42, 56 and 84?
42 = 2 * 3 * 7, 56 = 23 * 7, 84 = 22 * 3 * 7
The common prime factors among the set of numbers are: 2 and 7
The least common multiple of a set of numbers such as a and b is the smallest positive integer that is divisible by both a and b. Least Common Multiple is often referred to as LCM.
Example: What are the common multiples of 3 and 5?
Both 3 and 5 have an infinite number of multiples, therefore they also have an infinite number of common multiples.
The first 5 multiples of 3 are: 3, 6, 9, 12, 15
The first 5 multiples of 5 are: 5, 10, 15, 20, 25
The following are the first 5 common multiples of 3 and 5. In other words, the multiples of each that they have in common: 15, 30, 45, 60, 75
To find the next common multiple we add 15 to the previous multiple in the list. Therefore, we can say that the common multiples of both 3 and 5 can be expressed as 15m, where m is an integer.
So what is the Least Common Multiple? Well, it’s just the smallest common multiple between the set of numbers. In the case of 3 and 5, the LCM is 15.
The method described above for finding the LCM of a set of numbers works but can quickly become very tedious and inefficient.
There is a better way of finding the LCM of any set of numbers.
First, we list the prime factors of each number
Then we multiply the prime factor that occurs the greatest amount of times in any of the numbers
Finally, we check that all numbers evenly divide our result.
Example: What is the Least Common Multiple (LCM) of 8, 105, and 72?
8 = 23, 105 = 3 * 5 * 7, 72 = 23 * 32
2 occurs the most 3 times so we take 23. 3 occurs the most 2 times so we take 32. 5 and 7 occur the most 1 time so we take 5, and 7. All together we end up with 23, 32, 5, and 7. Multiplying these altogether we get 23 * 32 * 5 * 7 = 2520. LCM(8, 105, 72) = 2520
Perhaps one of the most important and fundamental concepts when learning Number Theory is Prime Factorization. Prime numbers are the building blocks of Number Theory because all of the Natural numbers can be expressed as a unique product of prime factors. This is what is called Prime Factorization. Knowing the prime factors of a number also gives us insight into other properties of that number as we shall see when working with Least Common Multiples (LCM) and Greatest Common Factors (GCF) in the following chapters.
A formal definition of Prime Factorization is the decomposition of a composite number into a product of prime numbers. In other words, we can ask ourselves, “which prime numbers can we multiply together to get our original number?”
Example: What is the Prime Factorization of 72?
We can start finding the prime factors by dividing our original number (72) by the first prime number (2). 72/2 = 36.
We continue dividing by 2 until we no longer can and move on to the next prime (3) if necessary repeating this process until our dividend is a prime number. 36/2 = 18, 18/2 = 9, 9/3 = 3.
Putting it all together we see the prime factors of 72 are 2, 2, 2, 3, 3. Another way to express this is: 23 * 32.
Example: What are the prime factors of 31?
31 is prime so it does not have any prime factors other than itself.
A factor of an integer n, also known as a divisor of n is an integer m that when multiplied by another integer produces n. This can also be worded that n is a multiple of m. An integer n is (evenly) divisible by another integer m if m is a divisor of n. This means that dividing n by m results in no remainder.
A multiple is the product of any quantity of an integer. For example, 35 is a multiple of 7 because 35 = 7 * 5.
There are many ways one can describe the relationship between factors, divisors and multiples. For example, the following three statements are the same:
n/m is an integer
m divides n
m is a factor of n
Example: What are the factors of 8?
1, 2, 4, 8
Is this all? No. We must remember the negative factors as well.
-8, -4, -2, -1
For a total of 8 factors:
-8, -4, -2, -1, 1, 2, 4, 8
Knowing that we have both negative and positive factors, it is important to remember that the minimum number of factors a number can have is 2!
Example: What are the factors of 1?
-1, 1
Example: How many multiples does the number 7 have?
Numbers can be categorized into different sets or groups. Some sets of numbers are subsets of others.
By HB – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=34938003
Natural Numbers
Natural numbers are your common counting numbers. These are all the positive numbers from 1 and on.
{1, 2, 3, 4, 5, … n}
Among the Natural numbers are the Prime and Composite numbers.
Prime Numbers: A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A prime number has exactly 2 factors.
Composite Numbers: A composite number is any number that is not a prime number. Composite numbers contain more than 2 positive factors.
Whole Numbers
Whole numbers are all of the Natural numbers and Zero (0).
{0, 1, 2, 3, 4, 5, … n}
Integer Numbers
Integers are the positive and negative Natural numbers and Zero (0).
{-n, … -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … n}
Rational Numbers
Rational numbers are all numbers that can be expressed as the ratio of two Integers.
Examples of Rationals are:
{-5/3, 3/7, 0, 2}
Irrational Numbers
Irrational numbers on the other hand, cannot be expressed as the ratio of two Integers.
Examples of Irrationals are:
{pi, e, phi, sqrt(2)}
Real Numbers
Real numbers are all numbers that can be found on a number line, positive, negative or Zero.
Back in my early to mid-twenties, I worked as a night-auditor for a popular ocean-front resort in Montauk – the small town I grew up in. My main responsibility was to balance the entire days worth of business services against the money/credits/debits/etc… submitted by all revenue centers (Spa, Beach, Bar, Cafe, Hotel, Dining room and so on). This was one of the many “odd” jobs I found myself working and it was because of this job that I discovered a few pretty fascinating properties between the relationships certain types of numbers hold with one another. I’ve always had a passion for numbers and a natural proclivity for deeply understanding the things that interested me.
When working with numbers as a night-auditor your primary goal is to “check and balance”; to reconcile. When going about your job, it’s not uncommon to find transposition errors. A transposition error in accounting is when two or more adjacent digits or figures are reversed. These errors are very easy to spot because you’ll usually end up with a difference such that the amount out of balance is always evenly divisible by 9. For example, you tally up a stack of receipts knowing that the total amount has to equal, let’s say $320.89, but you end up with a total of $320.98 (89 cents to 98 cents), or your total needs to equal $181 but you somehow end up with $190. If you’ve crunched numbers long enough, you’re bound to run into this phenomena.
It wasn’t after probably a couple of years that one (very boring) night I decided to purposely make transposition errors and start jotting down how my intentional errors correlated to a deviation in my balance. In other words, I wanted to see exactly how by making these errors, if there was a way to understand more deeply the error itself and how, if at all, it related to anything else. I simply wanted more insight into this basic pattern I began to recognize. I did not know about this common accounting phenomena at the time nor any accounting “tricks” used to spot these types of errors. It fascinated me however, so I thought to investigate some more.
I began by making a simple chart (table 1) with random 2-digit numbers. I wrote down the intended number – meaning the correct number. The reverse of that number or what we’re calling the transposition error, and finally what the difference between these two numbers were.
Intended number
Transposition Error
Difference
19
91
72
34
43
9
73
37
36
41
14
27
82
28
54
74
47
27
57
75
18
95
59
36
29
92
63
Table 1
As we already know or what I later found out, is that the difference will always be evenly divisible by 9. This means that the difference will be a multiple of 9. I thought this was pretty neat but was there more? How do these numbers relate to each other besides the obvious in that their difference is always a multiple of 9? “Is there some deeper pattern?”, I asked myself. Well, there was. So I thought about how the difference between the intended number and the reverse of that number (transposition error) relate. Was there some information in the numbers themselves that could shine some light on the matter? I believed there was and so I looked a little closer.
If I take the number 91 by itself, what can that tell me about all the other information in my chart? Well, let’s see – 91 is made up of the digits 9 and 1. Okay, so what? Hmm, what does 9 and 1 tell me about 19 and the difference 72? What’s the pattern? Well, let’s play a little. Why not?
9 + 1 = 10
How does 10 relate to 72 ? … Nah, doesn’t really lead me anywhere obvious.
What about 9 – 1 = 8. How does 8 relate to 72 and 9 ? …
8 * 9 = 72. Okay, this seems like an interesting path to follow, let’s continue.
Looking back at my chart 91 – 19 = 72, or to get the reverse of 91 (19) we are exactly 72 units away. Another way to look at this is that we are 72/9 or 8, 9‘s apart.
I put the few things I found together and came up with my first steps towards an algorithm for 2-digit numbers.
91 is made up of the digits "9" and "1". If I take their difference I get "8".How does "8" relate to the difference between "91" and "19" or "72"? "8 x 9 = 72". This means that if I take the difference between the digits that make up either number "91" or "19" and ignore the sign (take absolute value), and multiply by the magical number "9", I'll have the difference between the original 2-digit number and its reverse!
At this point a general sense of euphoria rushes through my body as if I’ve discovered a little secret. As insignificant as it may be, a secret nonetheless! I felt as if I was somehow in direct contact with GOD. It was a great feeling to say the least. The beauty of what I had just discovered was that a lot of the information I found was inherent in the original number itself. It was as if the original number was packed with this secret information that just needed a little unraveling. Somewhat recursive if you will. This will make more sense later (I hope) when I try to generalize this discovery to 3-digits.
So now I see that from any single 2-digit number I can quickly figure out the difference between it and its reverse. Not just that it’s divisible by 9 but beyond that.
A common accounting mistake turns into a quick math “trick” you can use to impress your friends with Mental-Mathe-magic!
Algorithm for reversing any 2-digit number using the number 9 or how to quickly calculate the difference between any 2-digit number and its reverse:
1.) Pick any 2-digit number (ex: 38)
2.) Take the difference of the digits that make up the 2-digit number
and take its absolute value. (3 - 8 = -5), |-5| = 5
3.) Multiply the result from step 2 by 9 (5 * 9 = 45)
4.) If the first digit in the original 2-digit number is less than the
second digit, ADD the result of step 3, otherwise SUBTRACT.
(3 is less than 8 so, 38 + 45 = 83).
5.) Result of step 4 is always the reverse of original number from step 1.
I couldn’t contain myself. I shared this neat little trick with anyone that would listen and one night while I was explaining this to my future wife, I thought how I could do something similar with 3-digit numbers and what I had already learned.
This time around I didn’t have a chart so I just used my memory. For the purpose of making this easier to follow I’ll make a chart.
The chart would’ve looked something like the following:
Number
Reverse of Number
Difference
387
783
396
979
979
000
123
321
198
489
984
495
100
001
099
111
111
000
Table 2
It may not be so obvious, but there is a neat little pattern in the chart above. More on this pattern later. This time around I started with any 3-digit number and simply reversed it. I then took the difference and asked myself similar questions to those when I was investigating the 2-digit numbers and their reverse. The main question this time was how does the difference relate to the number 9? Why 9? because that was the magic number that made the last algorithm work. Why not this time?
I began trying to apply the same simple math rules and operations to the original 3-digit number and after some trial and error I discovered another pattern. My thought process went something like this:
Let’s start with any 3-digit number.
387
What happens if I take 3 – 8? Or 8 – 7? Or 3 – 7? Notice that I’m simply breaking apart the original 3-digit number 3 8 7 and running it through step 2 and on of my 2-digit algorithm.
Okay, so 45… How’s this relate to the difference between 387 and 783 or 369? Not seeing anything that sticks out here… Let’s try the second set of numbers.
We come to 36. This is a bit interesting isn’t it? Looks like the difference (369) but it’s missing a “9” in between the “3” and the “6“. Perhaps there’s something to this.
I continue down this path trying this method with several numbers and to my surprise I stumble across another neat little pattern.
I put together my findings and begin formulating another algorithm. This time for 3-digit numbers and their reverse or rather, their difference.
If I take the first and last digits of any 3-digit number and subtract them ignoring the sign (absolute value), then multiply the result by my magical number “9“, I always end up with something that looks like their difference except it’s missing a “9” right smack in the middle… Interesting.
“Could that be it?” I ask myself. Just subtract, multiply by “9” and insert a “9” in the middle of my product? Can’t be… I’m a little skeptical so I try to fudge up the process by picking numbers that may not play nice. A little something I learned as a programmer. I decide to try this with the numbers like “000“, “111“, “393”
1.) 000 -> 000 with a difference of 0.
2.) 0 (first digit "000") - 0 (last digit of "000") = 0
3.) |0| = 0
4.) 0 * 9 = 0, or 00
5.) 0 9 0
The result of “090” being the difference didn’t make sense… perhaps it’s an exception. Let’s keep going.
1.) 111 -> 111 with a difference of 0.
2.) 1 (first digit "111") - 1 (last digit of "111") = 0
3.) |0| = 0
4.) 0 * 9 = 0, or 00
5.) 0 9 0
This also didn’t make sense. Not surprising considering “000” and “111” are very similar numbers. What about the next one?
1.) 393 -> 393 with a difference of 0.
2.) 3 (first digit "393") - 3 (last digit of "393") = 0
3.) |0| = 0
4.) 0 * 9 = 0, or 00
5.) 0 9 0
Well, this didn’t make sense either. What could be the problem? Turns out to be very simple. Just look for the pattern. Whenever the first and the last numbers in my original 3-digit number are the same, the difference is 0. The 3-digit number basically “pivots” off the middle number when reversed. So if the original 3-digit number’s first digit is the same as the last, the reverse of that number is also the original and there will not be a difference! Okay, all clear.
That takes care of all the 3-digit numbers that are palindromes 😉 What about the others?
Let’s try “102“, “225“, “907“
1.) 102-> 201 with a difference of 99.
2.) 1 (first digit "102") - 2 (last digit of "102") = -1
3.) |-1| = 1
4.) 1 * 9 = 1, or 09
5.) 0 9 9
Neat! it works. What about the other two 3-digit numbers?
1.) 225-> 522 with a difference of 297.
2.) 2 (first digit "225") - 5 (last digit of "225") = -3
3.) |-3| = 3
4.) 3 * 9 = 27
5.) 2 9 7
Oh cool! that works too. That last one?
1.) 907-> 709 with a difference of 198.
2.) 9 (first digit "907") - 7 (last digit of "907") = 2
3.) |2| = 2
4.) 2 * 9 = 18
5.) 1 9 8
Yup! it checks out. It seems I now have my second algorithm.
Algorithm for determining the difference between any 3-digit number and its reverse:
1.) Pick any 3-digit number whose first and last digits are not the same.
If they are, result is 0. (ex: 358)
2.) Take the difference of the first and last digits that make up the
3-digit number and take its absolute value. (3 - 8 = -5), |-5| = 5
3.) Multiply the result from step 2 by 9 (5 * 9 = 45)
4.) Take your result from step 3 and insert the number 9 between
the digits in the product. (495).
5.) Result of step 4 is always the difference between original number
from step 1 and its reverse.
And this is where I stopped. What started out as a simple curiosity became an exciting and educational experience with numbers but most importantly, a somewhat spiritual journey as well. I developed a deep connection to my discoveries. Few things in this world have given me such a pure spiritual/emotional feeling than this has. Nothing beats the sensation of discovering something new for yourself without prior knowledge of the matter. I plan to continue my journey deeper into this little rabbit hole I began exploring decades ago. Perhaps it will lead me somewhere pure again. I’m sure it will 🙂
