I showed how the positive powers of ten and two are interleaved, and said that the interleaving of the negative powers of ten and two is its mirror image. In this article, I will show you why, and prove that the same properties hold.
The Negative Powers Mirror the Positive Powers
Based on the above diagram, it appears that the interleaving of the negative powers is the mirror image of the interleaving of the positive powers. If you fold the number line over at 100 = 20, you’ll see that the placement of the powers line up, and the absolute value of their exponents match.
This is no coincidence. The logarithm of a negative power is the negative of the logarithm of the corresponding positive power. For example, log2(10-a) = log2((10a)-1) = -1 · log2(10a) = -log2(10a).
The two sides are mirror images because they both start from the same point: 1. Going one power of ten unit to the right of 1 brings you to 101; going that same distance to the left of 1 brings you to 10-1 . Similarly for the powers of two, going one power of two unit to the right of 1 brings you to 21, and going that same distance to the left of 1 brings you to 2-1 .
Here is a diagram showing just (the start of) the nonpositive powers:
As I showed for the positive powers, a fixed linear distance represents a fixed factor: going right multiplies by that factor; going left divides by it. This same relationship holds for the negative powers. The same distance anywhere on either side of the number line represents the same factor. Let’s look at my “tick mark” diagram in the context of negative powers:
You can think of the tick marks in two ways: going right one tick mark is multiplication by 101/10 ≈ 1.26; going left one tick mark is division by 101/10, or multiplication by 10-1/10 ≈ 0.79. Right and left are inverses of each other.
Three tick marks to the right is multiplication by 103/10 ≈ 2; three tick marks to the left is multiplication by 10-3/10 ≈ 0.5, or approximately 2-1 = 1/2. Nine tick marks to the left represents approximately three factors of one half, or 1/8 = 0.125.
The Same Properties Hold for the Negative Powers
Because of the mirror image, the same properties that hold for the positive powers hold for the negative powers:
- There are three or four powers of two between powers of ten.
- Four powers of two never occur twice in a row.
- Three powers of two can occur three times in a row.
The proofs of these properties for the positive powers and the negative powers are themselves “mirror images”: “greater than” and “less than” are swapped; multiplication and division are swapped; factors and their inverses are swapped; negative exponents and positive exponents are swapped; increasing exponents and decreasing exponents are swapped; the qualifiers “above” and “below” are swapped.
I will show the details of the proofs for the negative powers with those transformations applied. Some of the terms I use are the same as for the positive powers, but will have the “opposite” meaning. When I refer to one value as being “after” another, I’ll mean to its left — less than it. When I say the “first” power of two after a power of ten, I’ll mean the power of two to its immediate left; in other words, the biggest power of two that’s less than it. This is because my orientation will be from right to left; that is, in terms of increasing absolute value of the exponents.
There are Three or Four Powers of Two Between Powers of Ten
There are three or four powers of two between every consecutive pair of powers of ten. Whether it’s three or four depends on where the first power of two in the interval falls.
In my diagram above, you can see there are four powers of two between 10-3 and 10-4. (There are four powers of two in the interval (10-1, 100], but since 100 = 20, I won’t count that as “between”.) A span of four powers of two represents a factor of 1/8 = 0.125 (the first one is not a factor; it’s just the starting point). Under what conditions do four powers of two occur between powers of ten?
Let p10n and p10n-1 be consecutive powers of ten, and let p2m be the greatest power of two less than p10n. Four powers of two appear between p10n and p10n-1 when p2m > 0.8 · p10n; three powers of two appear when p2m < 0.8 · p10n (p2m can never be equal to 0.8 · p10n ; multiplying it by 0.125 would make it a power of two and power of ten).
For example, there are four powers of two between 10-3 and 10-4 because 2-10 = 1/1024 = 0.0009765625 is greater than 0.8 · 10-3 = 0.0008; there are three powers of two between 10-2 and 10-3 because 2-7 = 0.0078125 is less than 0.8 · 10-2 = 0.008.
On the Factor 0.8
I got the factor 0.8 by recognizing that 0.8 · 0.125 = 0.1. In terms of my logarithmic scale diagram, 0.8 is just slightly greater than one tick mark between powers of ten.
Four Powers of Two Never Occur Twice in a Row
There are never two consecutive power of ten intervals that contain four powers of two each. The constraint on where the first power of two must occur cannot be met in back-to-back intervals.
Assume there are four powers of two between p10n and p10n-1; call them p2m , p2m-1 , p2m-2 , and p2m-3. Here’s why there can’t be four powers of two between p10n-1 and p10n-2:
- The first power of two, p2m, satisfies the inequality p10n > p2m > 0.8 · p10n
- The first power of two in the next power of ten interval, p2m-4, satisfies the inequality 0.0625 · p10n > p2m-4 > 0.05 · p10n (I multiplied everything by 2-4 = 1/16 = 0.0625)
- That inequality can be rewritten as 0.625 · p10n-1 > p2m-4 > 0.5 · p10n-1 (I pulled a factor of one-tenth out of each constant to reflect the crossing of a power of ten boundary)
- That inequality says that p2m-4 starts after 0.625 · p10n-1, which means that there can’t be four powers of two between p10n-1 and p10n-2
Three Powers of Two Can Occur Three Times in a Row
Let’s prove this, but less formally:
- Assume the worst case start for the first power of two in a three power of two range, just above 0.5 · p10n
- The worst case start for the first power of two in the next power of ten range will be just above 0.625 · p10n-1
- The worst case start for the first power of two in the next power of ten range will be just above 0.78125 · p10n-2, just barely meeting the criteria for a three power of two range.
- The worst case start for the first power of two in the next power of ten range will be just above 0.9765625 · p10n-3, which means four powers of two will occur in that range.
All told, three powers of two occurred in three consecutive power of ten ranges.
Here’s an example where this occurs
- Between 10-28 and 10-29, since 2-94 ≈ 0.505 · 10-28
- Between 10-29 and 10-30, since 2-97 ≈ 0.63 · 10-29
- Between 10-30 and 10-31, since 2-100 ≈ 0.79 · 10-30
The three powers of two run stops there because 2-103 ≈ 0.99 · 10-31.
Summary for All Powers
Three or four powers of two appear between every power of ten. Three powers of two can occur in three consecutive power of ten ranges; four powers of two cannot occur consecutively. The intervals (10-1, 100] and [100, 101) are slightly different than the rest: they contain four powers of two each, but only all or part of four power of two ranges. This is because 100 and 20 are equal, the only powers of ten and two that ever are.
When you look at the powers logarithmically, a linear pattern emerges, making the above properties more apparent. Logarithmic scale shrinks values greater than 1 and magnifies values less than 1. As you go in either direction from 1, the true scale is increasingly distorted.
