The apparent ease of divisibility of numbers using different bases
[warning: yet another number theory laden post. It's not heavy mathematics, but more like a philosphical discussion of numbers]
In everyday usage, the most commonly used language to describe numbers is the decimal system, or the base-10. In base-10, numbers (objects) are represented using 10 symbols, namely:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Of course, there are more numbers (objects) than there are symbols, so larger numbers will have to be represented by a combination of symbols. An easy way to think of this is to say that there are 26 alphabets in the English language, but there are definitely more than 26 objects that need to be named. Hence we combine several alphabets to form words like “prototype”, “hippopotamus”, “coffee” and “cheebye”.
Similarly, several symbols might be combined to represent large numbers (objects) like “55”, “252452787”, “100” and “47”.
How are the symbolic representations of numbers (objects) constructed?
Start from the first symbol, 0. 0 will be used to represent zero. The next symbol, 1, will represent one. The next symbol, 2, will represent two. This goes on until we reach 9 (the last symbol available in base-10).
We then add one more character to the name, and it becomes a two-symbol long name: 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21 … 98, 99, 100, 101, 102 … 998, 999, 1000, 1001, 1002…
Sometimes, it appears as if it is easier to divide by some number than divide by other numbers. Sticking to the base-10 notation, it dead easy to see that 4322 can be divided by 2, but it is not as easy to know that 4322 cannot be divided by 3. Is it some inherent property of numbers (objects), or is it purely due to our way of expressing numbers?
Lets write out a list of numbers in base-10, and highlight those that can be divided by 2:
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…
It happens that all those that divide by 2 are numbers that have their last digits that divide by 2 (even numbers).
Similarly, below is a list of numbers with those that divide by 3 highlighted:
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…
It’s slightly messier with this case.
In truth, numbers written in base-10 are easy to divide by 2 (and 5) precisely because 10 can easily be divided by 2 (and 5). As such, as long as the last digit can be divided by 2 (or 5), the number can be divided by 2 (or 5).
To prove the point, we will now work with base-12, using the following twelve symbols:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b
Writing the same numbers (from one to thirty-six) in base-12:
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2a, 2b, 30…
With the numbers that can be divided by 2 highlighted. Again, it is dead easy to see that if a number can be divided by 2- the last digit needs to be an even number (in base-12, a is also an even number, since a is ten)
What if we highlight those that divide by 3?
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2a, 2b, 30…
The pattern should be easy to spot. All numbers that have the last digit of 0, 3, 6 or 9 will divide by 3.
This is because we are working with base-12, and 12 divides by 3 (and 2).
Seven is often a difficult number to divide by if working with base-10. What if we use base-14 instead? The 14 symbols used in base-14 are as follows:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d
The numbers one to thirty-six, in base-14:
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 1c, 1d, 20, 21, 22, 23, 24, 25, 26, 27, 28…
Highlighting numbers that divide by 3:
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 1c, 1d, 20, 21, 22, 23, 24, 25, 26, 27, 28…
Its trickier isn’t it? That’s because 14 does not divide by 3.
Highlighting numbers that divide by 7:
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 1c, 1d, 20, 21, 22, 23, 24, 25, 26, 27, 28…
Easy stuff. All numbers ending with the digit 7 or 0 can be divided by 7.
Here is something worth thinking about while you eat your nasi lemak or cheese naan:
How messy will be base-P systems, where (P is a prime number)?
In everyday usage, the most commonly used language to describe numbers is the decimal system, or the base-10. In base-10, numbers (objects) are represented using 10 symbols, namely:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Of course, there are more numbers (objects) than there are symbols, so larger numbers will have to be represented by a combination of symbols. An easy way to think of this is to say that there are 26 alphabets in the English language, but there are definitely more than 26 objects that need to be named. Hence we combine several alphabets to form words like “prototype”, “hippopotamus”, “coffee” and “cheebye”.
Similarly, several symbols might be combined to represent large numbers (objects) like “55”, “252452787”, “100” and “47”.
How are the symbolic representations of numbers (objects) constructed?
Start from the first symbol, 0. 0 will be used to represent zero. The next symbol, 1, will represent one. The next symbol, 2, will represent two. This goes on until we reach 9 (the last symbol available in base-10).
We then add one more character to the name, and it becomes a two-symbol long name: 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21 … 98, 99, 100, 101, 102 … 998, 999, 1000, 1001, 1002…
Sometimes, it appears as if it is easier to divide by some number than divide by other numbers. Sticking to the base-10 notation, it dead easy to see that 4322 can be divided by 2, but it is not as easy to know that 4322 cannot be divided by 3. Is it some inherent property of numbers (objects), or is it purely due to our way of expressing numbers?
Lets write out a list of numbers in base-10, and highlight those that can be divided by 2:
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…
It happens that all those that divide by 2 are numbers that have their last digits that divide by 2 (even numbers).
Similarly, below is a list of numbers with those that divide by 3 highlighted:
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…
It’s slightly messier with this case.
In truth, numbers written in base-10 are easy to divide by 2 (and 5) precisely because 10 can easily be divided by 2 (and 5). As such, as long as the last digit can be divided by 2 (or 5), the number can be divided by 2 (or 5).
To prove the point, we will now work with base-12, using the following twelve symbols:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b
Writing the same numbers (from one to thirty-six) in base-12:
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2a, 2b, 30…
With the numbers that can be divided by 2 highlighted. Again, it is dead easy to see that if a number can be divided by 2- the last digit needs to be an even number (in base-12, a is also an even number, since a is ten)
What if we highlight those that divide by 3?
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2a, 2b, 30…
The pattern should be easy to spot. All numbers that have the last digit of 0, 3, 6 or 9 will divide by 3.
This is because we are working with base-12, and 12 divides by 3 (and 2).
Seven is often a difficult number to divide by if working with base-10. What if we use base-14 instead? The 14 symbols used in base-14 are as follows:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d
The numbers one to thirty-six, in base-14:
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 1c, 1d, 20, 21, 22, 23, 24, 25, 26, 27, 28…
Highlighting numbers that divide by 3:
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 1c, 1d, 20, 21, 22, 23, 24, 25, 26, 27, 28…
Its trickier isn’t it? That’s because 14 does not divide by 3.
Highlighting numbers that divide by 7:
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 1c, 1d, 20, 21, 22, 23, 24, 25, 26, 27, 28…
Easy stuff. All numbers ending with the digit 7 or 0 can be divided by 7.
Here is something worth thinking about while you eat your nasi lemak or cheese naan:
How messy will be base-P systems, where (P is a prime number)?
posted by Tan Yee Wei at 7/25/2005 08:23:00 pm
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