We simplify math by generalizing its many verbs (*, /, powers, log, root) into a single concept called the blead, allowing the early introduction of logs & roots at the same time as we introduce multiplication and division.
Self-plus *, self-times ^, and their opposites /,, , are much alike. In this section, we will study their likeness. They share a common relationship call the BLeaD:
L / \ / \ B D
The blead diagram shows the construction of the verbs *,^,/,, by repeating a base. The term is an initialism of its components: Base, Leaf/Lattice, and Depth. The table below shows how the verbs * and ^ act by taking a base amount and repeat it to form lattices and leaves. The repetition count is called "Depth". Tracing the B-L-D diagram bottom-up gives the "forward" verbs *,^ and tracing it top-down gives the "reverse" verbs /,, .
|generic BLeaD||self-plus blead||self-times blead|
Leaf /\ / \ Base Depth
lattice /\ / \ base depth lattice for 5 * 3 = 15 | | | | | - 0 - 0 - 0 - 0 - 0 - | | | | | - 0 - 0 - 0 - 0 - 0 - | | | | | - 0 - 0 - 0 - 0 - 0 - | | | | |
To build a picture of the self-plus 5 * 3, we simply start with a base
leaf /\ / \ base depth leaves of a "snowflake" for 2 ^ 3 = 8 1 2 = 0------------0 0 0 2 | | 2 = +------------+ | | 0 0 0--+--0 0--+--0 3 | | 2 = +------------+ | | 0--+--0 0--+--0
To build pictures of self-times, say 2^3, we start with a base of 2, pictured as 0-----0. We call this the 1st repeat. At the 2nd repeat, we overwrite each leaf with another base. At the 3rd repeat, again, we overwrite each leaf with another base. We continue this repeat if we want higher powers. The "depth" of these repeats is the repetition count. When repeating with other bases, such as 3 and 4, the picture begins to resemble a snowflake.
Role Reversal (ror), aka Commutivity
In self-plus, the base and depth can reverse their roles and still give the same lattice count. For example 5 * 3 = 15 = 3 * 5, that is,
0 - 0 - 0 | | | 0 - 0 - 0 - 0 - 0 0 - 0 - 0 | | | | | rotate | | | 0 - 0 - 0 - 0 - 0 ---------> 0 - 0 - 0 base 3, depth 5 | | | | | | | | 0 - 0 - 0 - 0 - 0 0 - 0 - 0 | | | base 5, depth 3 0 - 0 - 0
With self-times, this does not hold true. For example, while .
In other words, "snowflakes don't ror" (can you imagine a dainty little snowflake making any load noises like a "roar"? No). That's why self-times has 2 divides, right-divide & left-divide, and self-plus has only one.
Traveling downward in the blead diagram, we form the verbs /,, . Because of the symmetry in self-plus between base and depth, that is "lattices do ror", traversing down its blead diagram towards either base or depth yields only one algorithm -- the divide /. Since snowflakes don't ror, traversing downwards yields 2 algorithms, the left & right divides and . The left & right divides are also commonly written as roots and logs. For example:
Here's a mnemonic to help remember the sided-ness of the verbs:
Log = Left divide =
leaf leaf o---------------- = o------ = depth L_eft of blead base
left divide by left of blead
Root = Right divide =
leaf leaf -----------------o = -------o = base R_ight of blead depth
right divide by right of blead
|left & right "denominator" algebra||
leaf log (leaf) = o---- = _ d b base |_| | ^ | | +--------+ depth leaf = (base)
denominator moves to the Left of rhs (right hand side)
_ d _____ Leaf |_| <-+ root = V leaf = -----o = base | depth | | | +-----------------+ depth leaf = (base)
denominator moves to the Right of rhs (right hand side)
All the sided-ness of the above table is self-consistent, that is left sided traits stay together. Similarly for right sided traits. There is one inconsistency that occurs when the above table is put into blead diagram. Note that the right-divide appears on the left side, and visa-versa.
L / \ L / \ L left = root = --o = B D = o-- = log = right side D B
We could resolve this by making a D-L-B diagram instead of B-L-D. However this would ruin the fact that the denominator of left-divide should be the left side of blead, a mnemonic that is convenient during algebraic manipulation and one that we prize.
The Evert section will show further similarities between these self-ish verbs.
Powers, Logs & Roots
Logarithms are a confusing concept for many, but they are conceptually simple when viewed as a type of fraction. As a matter of fact, they are SO similar to fractions that we propose a fraction-like notation for them: the right & left fractions.
If we visualize multiplication as "shaping a quantity into boxes" with length & width, we can visualize powers as "shaping a quantity into snowflakes". The length and width of the snowflake is the length & width of its shape. Here are a few thoughts on numerical representation of quantities as bases, that will help solidify the ideal of length & width in powers.
- A quantity 9, 27, 81, 16, 64, ... can be thought of as a bunch of dots, independent of it's numerical representation in base-2, ... base-10, ... or any base. Children are so used to base 10 that they think that base-10 is the quantity and cannot conceive of a quantity in any other bases. In terms of representations of an object, a person's name or social security number does not define the person -- the person thinks, feels, and exists even without the identifier. Similarly, a quantity exists regardless of its base-n representation. It's also useful to bring back the analogy of language for numbers.
- There are many foreign languages, each with its own alphabet -- the English alphabet has 26 letters, while the Russian alphabet has 33. Numbers can have many alphabets. The base-4 alphabet is (0,1,2,3), base-16 (hex) alphabet is (0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f). Because of the limited alphabet and the endless concepts to be written, we have to string letters together into words to represent the concept. Similarly, because of the limited characters in a base, and the endless quantities to be written, we string the characters together as numbers. In terms of the alphabet, the log is the number of digits in a number (its length), and the base is the size (width) of that alphabet. As a matter of fact, the "D" in BLeaD can also be thought of as an abbreviation for "digits".
- Even as symbols, the log & root has a very natural interpretation as length & width when writing a quantity in symbols: the length is the digit count, and the width is the alphabet size. The alphabet of a written number is the characters in its base. For example, 1023 (in base 10) = 33333 (in base 4) has length of 5 digits & width of 4 alphabet characters. Even fractions don't have this interpretation of length & width! Hence, we argue that log & roots are even more natural as fractions.
Some numbers do not fit perfectly as a lattice or a snowflake. For example, 1234 is "3.1 digits" long in an alphabet that's 10-characters wide; as a lattice, it can be 12.34 long and 100 wide.
The term blead borrows from Warren Ellis' Bleed in his Planetary story, where the term is defined as the space between universes that make up the multiverse. Similarly, our blead is a core concept that pervades mathematics by connecting all the algebraic verbs. However, for children we intentionally avoided complete likeness to Ellis' Bleed to avoid the thought that math is as painful as blood letting. Ellis' fictional story also uses a snowflake construct which we also use, but the similarity here is in word only and not any further reaching metaphor.