From: 3blue1brown
The concept of “distance” in mathematics can be re-evaluated and redefined, leading to new and counter-intuitive results, such as the infinite sum 1 + 2 + 4 + 8… equaling -1 [00:00:11]. Understanding this requires exploring alternative ways of defining how “close” numbers are [00:09:01].
Convergent Infinite Sums (Standard Definition)
Historically, mathematicians wrestled with defining infinite sums [00:00:57]. A classic example is the sum ½ + ¼ + ⅛ + ₁⁄₁₆ + … up to infinity, which equals 1 [00:00:48]. This can be visualized by imagining an object moving from 0 towards 1 on a number line, halving the remaining distance with each step [00:01:38]. The numbers the object touches form a list of partial sums: ½, ½ + ¼, ½ + ¼ + ⅛, and so on [00:01:48].
Defining “Approach” and “Equal”
For infinite sums, the definition of “approach” is crucial [00:03:14]. It’s not just that the distance between each number in the partial sum list and the limit (e.g., 1) gets smaller [00:03:20]. The key insight is that the numbers can get arbitrarily close to the limit [00:03:33]. This means no matter how small a desired distance (e.g., 1/100, 1/1,000,000), if you go far enough down the list of partial sums, all subsequent numbers will fall within that tiny distance of the limit [00:03:36]. This formal definition of an infinite sum “equaling” a number x means that the list of finite partial sums approaches x in this arbitrarily close sense [00:04:09].
Generalizing Infinite Sums
This idea can be generalized. If an interval is repeatedly cut into pieces of size p and 1-p (where p is between 0 and 1), the sum of the 1-p portions will equal 1 [00:05:40]. This leads to the formula:
(1-p) + p(1-p) + p²(1-p) + ... = 1 [00:06:04]
Dividing by (1-p), we get the geometric series formula:
1 + p + p² + p³ + ... = 1 / (1-p) [00:06:16]
A popular example derived from this is 0.9 repeating equals 1 [00:05:24]. This arises from cutting an interval into 9/10 and 1/10 proportions repeatedly [00:05:00].
Divergent Sums and “Nonsense”
While the formula 1 + p + p² + p³ + ... = 1 / (1-p) makes sense for p values between 0 and 1, plugging in other values can yield “nonsensical” results under the traditional definition of infinite sums [00:06:28].
- Plugging in p = -1: The equation becomes
1 - 1 + 1 - 1 + ... = 1/2[00:06:40]. The partial sums (1, 0, 1, 0…) don’t approach a single value in the standard sense [00:07:17]. - Plugging in p = 2: The equation becomes
1 + 2 + 4 + 8 + ... = -1[00:06:59]. The partial sums (1, 3, 7, 15, 31…) clearly grow larger and larger, not approaching -1 [00:07:48].
Under the rigorous definition of convergent infinite sums, these results are ignored because their partial sums do not “approach” anything [00:07:14]. However, a mathematician, not a robot, might seek to make sense of this “nonsense” [00:07:30].
The Need for Alternative Distance Metrics
The limitation in making sense of sums like 1 + 2 + 4 + 8… = -1 lies in our traditional definition of distance between rational numbers [00:09:01]. Our familiar number line imposes a specific way of organizing numbers [00:09:05].
A “distance function” is essentially a function that takes two numbers and outputs a value indicating how far apart they are [00:09:15]. To be useful, a new distance function should share some properties with the familiar one [00:09:35]. One crucial property is shift invariance: the distance between two numbers shouldn’t change if the same amount is added to both [00:09:42]. For example, the distance between 0 and 4 should be the same as between 1 and 5 [00:09:48].
The p-adic Metrics
To make sense of 1 + 2 + 4 + 8… = -1, mathematicians needed to redefine distance in a way that would make powers of two approach zero [00:10:18]. This was a profound shift in thinking, taking until the 20th century to be discovered [00:08:55].
One such redefinition organizes numbers into a hierarchy of “rooms, subrooms, sub-subrooms, and so on” [00:10:34].
The 2-adic Metric
In the context of powers of two, the idea is that 0 is considered to be in the same “room” as all powers of two greater than one, in the same “sub-room” as powers of two greater than two, and so on [00:10:40]. This hierarchy extends infinitely [00:10:54].
Applying shift invariance, the structure of these “rooms” for other numbers can be deduced [00:11:13]. For example, since 0 is in smaller and smaller rooms with powers of two, -1 must be in smaller and smaller rooms with numbers that are one less than a power of two (1, 3, 7, 15…) [00:11:47].
This leads to the definition of the 2-adic metric, where the “distance” between two numbers is determined by the size of the smallest “room” they share [00:12:33].
- Numbers in different large yellow rooms are a distance of 1 apart [00:12:43].
- Numbers in the same large room but different orange sub-rooms are a distance of ½ apart [00:12:50].
- Numbers in the same orange sub-room but different sub-sub-rooms are a distance of ¼ apart [00:12:59]. This continues, using reciprocals of increasingly larger powers of 2 to indicate closeness [00:13:09].
The 2-adic metric is a perfectly legitimate notion of distance that satisfies properties like the triangle inequality [00:13:26]. It is part of a general family of distance functions called p-adic metrics, where p is any prime number [00:13:40]. These metrics give rise to completely new types of numbers, neither real nor complex, and are fundamental in modern number theory [00:13:48].
Resolution of the “Nonsense”
Using the 2-adic metric, the sum 1 + 2 + 4 + 8… = -1 actually makes sense [00:13:58]. The partial sums (1, 3, 7, 15, 31, etc. – which are all one less than a power of two [00:07:48]) genuinely approach -1 under this alternative distance measure [00:14:02]. This highlights how redefining fundamental concepts, such as distance, can profoundly change mathematical outcomes and reveal new, useful areas of study [00:14:27].
The process of mathematical discovery often involves nature presenting ill-defined or nonsensical observations, which then prompt the definition of new concepts to make them sensible, leading to the creation of genuinely useful math that broadens traditional notions [00:14:22].