Most systems for representing numbers are a combination of the visible and invisible. In the familiar base-ten system, we represent numbers by linking visible digits to an invisible scaffold of powers of ten. For example, 534.08 implies the respective multiplication of 5, 3, 4, 0, and 8 with the unseen place values 100, 10, 1, 0.1, and 0.01. Also invisible are the leading and trailing zeros: there is no need to write …0000000534.080000000….
Many other number systems involve multiplication by an invisible set of powers. For instance, the binary number system multiplies 1s and 0s by invisible powers of two.
A different way to represent numbers occurred to me recently when I was thinking about my daughter’s math studies, which in the last few years has ranged from putting fractions into simplest form to understanding the difference between rational and irrational numbers. I thought about how every positive whole number has a unique factorization in primes. For example, 20 is the product of two 2s and one 5. So 20 = 22 x 5, but there is no other way to factor 20 into primes. This property of numbers is so important that it has been enshrined as the Fundamental Theorem of Arithmetic. It is the key principle underlying some of the my daughter’s math topics. It is also connected to need for well-tempered tunings in music.
I imagined a different way to represent numbers that might clarify some of the concepts that involve prime factorization. The prime power representation is based on prime factorization. Let’s take 63 = 32 x 7. If I don’t want to skip any primes on the way to 7, I could write 63 = 20 x 32 x 50 x 71 (recall that numbers raised to the 0th power are equal to one and have no effect on the multiplication). Now remove everything but the exponents, so that 63 is represented by 0 2 0 1. In this system, the primes are the invisible scaffold. As with other number representations, we don’t have to add any more zeros (which would only go off to the right). All of the subsequent primes are assumed to be raised to the 0th power, with no impact on the grand multiplication.
Let’s count to ten with prime powers:
|0||1 = 20|
|1||2 = 21|
|0 1||3 = 20 x 31|
|2||4 = 22|
|0 0 1||5 = 20 x 30 x 51|
|1 1||6 = 21 x 31|
|0 0 0 1||7 = 20 x 30 x 50 x 71|
|3||8 = 23|
|0 2||9 = 20 x 32|
|1 0 1||10 = 21 x 30 x 51|
As you can see, the prime power representation is not very useful for counting. Adding or subtracting numbers in this form would also be a nightmare. But perhaps the representation could help students deepen their understanding of some of the fundamental properties of numbers.
Let’s look at how the power representation can be used. In the following examples, we will include some trailing zeros in an attempt for more clarity.
It’s easy to multiply two numbers in the prime power representation. Let’s try 18 x 20. Since 18 = 21 x 32 and 20 = 22 x 30 x 51, all we have to do is add the corresponding exponent values:
1 2 0
2 0 1
3 2 1
The answer is 360 = 23 x 32 x 51. We can easily generalize from multiplication to exponentiation. To raise a number to a higher power, simply multiply all its exponents by that power. For example 15 is 0 1 1 and 152 is 0 2 2, or 225.
To divide one number by another, simply subtract the corresponding values. For 99÷33, since 99=20 x 32 x 50 x 70 x 111 and 33=20 x 31 x 50 x 70 x 111, we have:
0 2 0 0 1
0 1 0 0 1
0 1 0 0 0
The resulting quotient is 3, a nice whole number. What about something like 54÷60? Then, with 54=21 x 33 and 60=22 x 31 x 51, we have
1 3 0
2 1 1
-1 2 -1
What are we to make of these negative numbers? This is just a fraction, where the numerator is represented by the positive exponents and the denominator by the negative exponents. The fraction is 9/10, since 9= 32 and 10= 21 x 51.
The fraction 9/10 is in simplest form. Using the power representation for division will always give you a fraction in simplest form. This is a very nice property that I hadn’t anticipated.
Rational and Irrational Numbers
The power representation also got me thinking about rational and irrational numbers. A rational number is equal to a fraction where both the numerator and denominator are integers. An irrational is any real number, such as pi, that is not equal to such a fraction. In the power representation, every rational number corresponds to a unique combination of positive and/or negative integers in the list of exponents.
Pythagoras famously believed that the world was based on whole numbers, and that therefore it should be possible to construct a square where the lengths of the sides and of the diagonals are multiples of the same unit. In other words, the square root of two must be a rational number. However, one of his disciples discovered that this was impossible, undercutting Pythagorean dogma. He was supposedly drowned for his trouble.
The power representation sheds some light on the issue. A good approximation for the square root of two is 99/70, which we can represent as -1 2 -1 -1 1. If we square 99/70, we get 2.0002 (to four digits), which is pretty close. However, the power representation of the square is -2 4 -2 -2 2. We need a rational number that squares to 2, which is 1 0 0 0… in the power representation. There is no way to ever make this work. No integer can be doubled to give us 1 in the 2s place, and no non-zero integers can be doubled to give us 0s in the other places.
If we go beyond the integers and allow rational exponents, we now represent the square root of two by .5. This is the only way to represent the irrational square root of two with rational powers of prime numbers. Some, but not all, irrational numbers can be represented with rational exponents in the power representation. These are unique representations. If we permit irrational exponents, we can represent any irrational number, but these representations are not unique.
I created a spreadsheet to explore several properties of the prime power representation. Click here for a free copy.
By now I hope you can see that the title of this post refers to the prime power representation of 2015, which factors as 5 x 13 x 31. At least next year will have a more compact representation: 5 2 0 1.
Happy New Year!
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 Recall that a number raised to a negative power gives the reciprocal of that number raised to the positive power (e.g. 3-4=1/34).