Definition: Two integers n and b that are both perfect integer powers of a single, common underlying integer root d.
Chapter 1: The Same Family Bricks (Elementary School Understanding)
Imagine you have a giant collection of LEGO bricks. You notice that some of the big, special "power bricks" seem to be related.
You look at the 8-brick. You realize it's made of three 2-bricks snapped together (2³).
You look at the 16-brick. You realize it's made of four 2-bricks snapped together (2⁴).
The numbers 8 and 16 are Commensurable Powers. "Commensurable" is a fancy word that means "they can be measured by the same thing." In this case, they can both be perfectly built using only 2-bricks. They come from the same root, which is 2.
Now look at the number 27. It's made of three 3-bricks (3³).
Are 8 and 27 Commensurable Powers? No. You can't build 8 using 3-bricks, and you can't build 27 using 2-bricks. They come from different root families.
Two numbers are Commensurable Powers if they are like cousins, both born from the same single "root" number.
Chapter 2: The Common Root (Middle School Understanding)
Commensurable Powers are two integers that share a "common root."
Formal Definition: Two integers n and b are Commensurable Powers if there exists an integer d (the root) and two integers k and j such that:
b = d^k
n = d^j
How to Check if Two Numbers are Commensurable Powers:
Find the prime factorization of both numbers.
Express each factorization as a single root raised to a power.
Check if the roots are the same.
Example 1: Are 64 and 512 Commensurable Powers?
64 = 2⁶
512 = 2⁹
The root is the same: d=2. The powers are k=6 and j=9.
Yes, they are Commensurable Powers.
Example 2: Are 27 and 81 Commensurable Powers?
27 = 3³
81 = 3⁴ (or 9², but we need the same root).
The root is the same: d=3.
Yes, they are Commensurable Powers.
Example 3: Are 16 and 36 Commensurable Powers?
16 = 2⁴ (or 4²).
36 = 6² = (2×3)² = 2² × 3².
The roots are different. 16 is a pure power of 2. 36 is a power of 6 (or has prime roots of 2 and 3). They do not share a single, common integer root.
No, they are not Commensurable Powers.
Chapter 3: The Key to Rational Logarithms (High School Understanding)
The concept of Commensurable Powers is most important for understanding when a logarithm is a rational number.
The Law of Logarithmic Irrationality states:
The logarithm log_b(n) is a rational number if and only if n and b are Commensurable Powers.
Proof:
"If" part: Assume n and b are Commensurable Powers.
This means b = d^k and n = d^j for some integers d, k, j.
Let's evaluate log_b(n) = log_{d^k}(d^j).
Using the change of base formula for logarithms: log_x(y) = log(y)/log(x).
log_{d^k}(d^j) = log(d^j) / log(d^k)
Using the power rule of logarithms, log(a^b) = b*log(a):
= (j × log(d)) / (k × log(d))
The log(d) terms cancel out, leaving j/k.
Since j and k are integers, j/k is a rational number.
"Only If" part: Assume log_b(n) is rational.
This means log_b(n) = j/k for some integers j, k.
By the definition of a logarithm, this means b^(j/k) = n.
Raising both sides to the power of k: (b^(j/k))^k = n^k, which simplifies to b^j = n^k.
This final equation shows that the prime factors of b and n must be the same, which means they must both be powers of a common root d.
This provides a perfect, airtight test. To know if log_b(n) is rational, you don't need to calculate it. You just need to check if n and b have a common root.
Chapter 4: A Statement of Multiplicative Linear Dependence (College Level)
The concept of Commensurable Powers is a statement about the linear dependence of the exponents in the prime factorizations of two numbers.
Let the unique prime factorization of an integer N be represented by a vector of its p-adic valuations:
V(N) = < v₂(N), v₃(N), v₅(N), ... >
V(64) = <6, 0, 0, ...>
V(512) = <9, 0, 0, ...>
V(27) = <0, 3, 0, ...>
V(36) = <2, 2, 0, ...>
Two integers n and b are Commensurable Powers if and only if their valuation vectors V(n) and V(b) are linearly dependent. This means one vector is a scalar multiple of the other.
V(n) = c × V(b) for some rational scalar c.
Example: 64 and 512
V(512) = <9, 0, ...>
V(64) = <6, 0, ...>
Is <9, 0, ...> = c × <6, 0, ...>? Yes, for c = 9/6 = 3/2.
Since the vectors are linearly dependent, the numbers are Commensurable Powers.
Connection to Logarithms:
The logarithm can be seen as the scalar c that connects these two vectors.
log_b(n) = j/k.
V(n) = V(d^j) = j × V(d)
V(b) = V(d^k) = k × V(d)
Therefore, V(n) = (j/k) × V(b).
c = log_b(n).
The Law of Logarithmic Irrationality is therefore a statement that the logarithm log_b(n) is rational if and only if the "prime factor exponent vectors" of n and b point in the exact same direction in an infinite-dimensional vector space. If they point in different directions at all (are linearly independent), the logarithm must be irrational.
Chapter 5: Worksheet - The Common Root
Part 1: The Same Family Bricks (Elementary Level)
Are the numbers 9 (3²) and 81 (3⁴) Commensurable Powers? What is their common root?
Are the numbers 4 (2²) and 27 (3³) Commensurable Powers? Why or why not?
Part 2: The Common Root (Middle School Understanding)
Using prime factorization, determine if 32 and 128 are Commensurable Powers.
Determine if 100 (10²) and 1000 (10³) are Commensurable Powers.
Determine if 25 (5²) and 100 (10²) are Commensurable Powers.
Part 3: Rational Logarithms (High School Understanding)
Without using a calculator, is log₃₂(128) a rational or irrational number? Justify your answer using the concept of Commensurable Powers.
Is log₁₀(20) rational or irrational? Why?
If log_b(n) is rational, what does that prove about the prime factors of b and n?
Part 4: Linearly Dependent Vectors (College Level)
Write down the "valuation vectors" V(n) for n=100 and n=1000.
Show that these two vectors are linearly dependent by finding the scalar c such that V(1000) = c × V(100).
What is the value of log₁₀₀(1000)? How does it relate to your scalar c?
The vectors for n=25 and b=125 are V(25) = <0,0,2,0...> and V(125) = <0,0,3,0...>. Are they linearly dependent? What is log₂₅(125)?