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Dials for Counting
Dials for counting, three rows, be able to mount them, change them around, and of course, spin them - stitching together the number system
- Distinguish between amounts (circular order, as in drumming, repetitive) and units (linear order, as with multiple units, what is most important, has priority). Zero is part of every amount, it is the reference for circular counting, whereas there is no zero in the units, but is a blank instead, a nothing.
Note: A dial has "labels", whereas a circle is before the labels
- A number system - the units don't matter, or are internal - everyone's number system is private - but then is shared, thus public, thus explicitly defined abstract units of counting (such as "ones", so that we have eight "ones", we can have "eight" anything, eight "whatevers")
Dials (names become amounts) Circular Counting: Ordered list: Memory Loop
- ABC's are a list, as in the ABC song.
- rhythmic chants help for memory, for example, Dr.Seuss's mother's names of pies
- What to do for memory, to memorize math facts? Circular counts leverage the way autoassociative memory works
- The counting dial makes numbers work as actions. We are counting the counting.
- Is there a 0 dial, a dial with just 0 on it, that represents an item abstractly, as a unit?
Formality: Answers, Amounts and Units
- The counting dial allows us to formally distinguish between amount and unit, thus have a formal answer. In this way a system is elaborated, formally. There is no longer any confusion as to whether an item corresponds to a unit. This is because what is now counted is the counting. Thus there must be a maximum to the dial. The unit is taken as given and equated to the dial.
- A system defines what we can think about and in terms of.
- System is definition.
- We can think in terms of units, which are the atoms, nodes, thoughts, concepts, encapsulations of our thinking.
- We can't think of what's particular to the units as those distinctions have been discarded; nor can we think in terms of whatever no units.
- In physics, the basic atoms (protons, electrons) are indeed indistinguishable as such like such units.
- We can think up to the equivalence of the units, so that if the unit is "apples", then all apples are the same.
- We can think about what can be expressed in the units, the relationships between the units.
- A system keeps track of "answers" and thus minimizes the need for recounting (or adding).
- An "answer" is an amount and a unit
- amount => what doesn't change, unit is what changes
- amount is what you count with your fingers, unit is what you don't have to count, it's fixed
- what is your finger worth? unit. 5 one dollar bills = 1 five dollar bill
- A system is inherently ambiguous
- Amount and Unit divide up the context into two parts, allowing for simplification
- The counting dial shows the difference between amounts and units.
- running out of fingers, thus decimal system, or simply, a dial
- Why don't we use a system of elevens? for we have ten digits
- We can't count past 10 - or whatever is our maximum.
- Thus the Y2K problem with 2 digits
- Computers define integers and other numeric types up to some maximum, after which, they won't work as desired.
Repeating - Setting an external rhythm
- Rhythm is set by a recurring activity
- Setting a rhythm - converting an internal rhythm to an external rhythm that others can tune to
- Example: stroke for a crew of rowers - obedience
- Example: drummer for a band
- Example: metronome
- Recurring activity stretches from negative infinity to positive infinity
- Christopher Alexander's Timeless Way of Building, and Pattern Languages
- Make a list of every day activities - Make dials for them - How does recurring activity relate to the structure it evokes?
- Creating marks, instances, events in time.
- Following one event with another
- Rhythm is driven by recursion, for example, plus one
- Laying down "tracks" and superimposing them yields patterns upon patterns which then appear as groups of strokes, beats
- Beats within beats
- Internal verbalizations
- Each is naturally distinct in time.
- Different dials for different musical beats - study the beats (polka, waltz, etc.)
- Processing - write out commas after every three units, starting from the right and going left - a helpful process for making numbers usable - and can be done without comprehending the numbers
Canonizing - number system
Time: Internal counting: Counting out an obligation
- Counting the amount of work to be done - such as forty lashes
- Counting the amount of coins to be paid out - such as thirty pieces of silver
- Use your fingers to calculate 2**9 (keep track as you double 9 times).
- Counting out the number of pull ups that you can do. (Arbitrary counting? Note the sequential integral review - each pull up is checked independently - and this check occurs in time)(Counting depends on a deep structure of "checks" of the count.)
- Label on a dial lets you tally.
- Dividing up is backwards counting on a dial. It is repeated subtraction and can yield a remainder. Whereas dividing out is the inverse of multiplying and can be seen on a grid, in two dimensions.
- Division (100 divided by 5) can be counting (if you are dividing the amount, dividing by a clustered unit, thus "dividing up", yielding and counting 20 instances of 5, counting them each out, dividing evenly, bottom up, if you don't know the answer yet) or can be clustering, "dividing out" (if you are dividing 100 among 5, and you know the answer already, and so you give 5 instances of 20).
- Dividing up can yield remainders.
- 4 thousand divided by 4 = 1 thousand. Why? Because 4 divided by 4 is 1, and the thousand is just a unit.
- Dividing is "counting by": 60 miles can be counted by hours, making a chart
Dials that do not start from zero
- counting by odds is counting by 2's but starting from 1
- division with remainder = dial that does not start from zero
- Division is sharing - remainder is what you can't share
Two dials: Counting by
Counting by ...
- we can count the dial turns if we have a second dial
- in order to have a unique expression, we have the second dial's value be one more than the max of the first dial, thus we introduce 0
- we use 0-9 instead of 1-10 so we are forced to clump - because we are using a second dial, a dial to count the counting, to count the number of times the dials went around
- Counting by 1's, 10's, 2's, 5's, 3's, 1/2's, fractions, decimals, percents, X's
- multiplying by -1
- multiplying by i
- roots of unity
- Counting is the paradigm for linear growth. But such counting is stitched together from circular counting, and it is the stitching that makes it linear.
- Tally - skip counting, repeated addition
- "Tallying" is a two dial system, counting by 1s and 5s.
- Tally marks used by shepherds
- count by 1/2, in general, count by numeric units
- 10 divided by 1/2 = 20, counting by 1/2s, grouping by 1/2s, We do that rarely! so it seems strange.
- Exercises: counting game
- Arithmetic sequence: counting by X
- counting up to a number by a number = dividing
- Exercise: Counting by 3's - double counting in your mind
- multiplication by 9s patterns is 10 - 1
- Use your fingers to keep track of counting by 12s. How many times do they go into 120?
- 10 x 3 vs. 9 x 3 which is bigger? and why?
Multiplication turns counting up into counting
- counting what (multiplication) by what (division)
- we count up "what" = units = nouns, things, natural numbers
- multiplication is "counting up" "by what" = an "answer", an amount and a unit, a grouping. This allows us to treat the amount as our unit and treat the unit as an "abstraction". We can then "count" (by numbers) rather than "count up" (actual things, units). We then relate back to the original unit. Thus it is accelerated counting, speed counting. Multiplication relates "counting up" (items in space) with "counting" (activity in time).
- This works because multiplication is associative.
- We can thus memorize and reuse multiplication problems.
- multiply = distribute while you can and wait, lump if you can't, thus multiplication = distribution
- multiplication is organized counting
- multiplication is repetitive counting
- Multiplication is super counting
- multiplication makes sense when counting is organized, "correct", well defined
- Multiply disks by simply placing one after the other - this works if you set each multiplied dial to zero because you are treating it as a unit - you can do this for multiple dials - and you can commute the dials - and you can combine dials - but how do you divide by a dial? you remove it
- Illustrate multiplication: Label - splitting, sets, per each, symmetry - multiplying dials. Note that there is no inverse! no way to divide, except to remove dials.
- multiplication patterns
- Multiplication can be defined as one dial counting another dial.
- Multiplication is the direct product.
Division using dials
- 3 divided by 5 can be done by adding a dial that is unit "1/5". Then count by 5's. This turns the next dial around to 1. Do this three times and you get 3. So it's a very straightforward way of getting "3/5". And it looks like sharing a pie 3 times, which it is. To get 3 pies divided you can also relate to the circle (for the fifths) and to tokens (for the 3 wholes).
- 3 divided by 5 can be discussed in terms of two objects (3 apples divided by 5 people) whereas 3 out of 5 discusses in terms of one object, a whole (3 out of 5 slices). Note that 3 divided by 5 can lead to 3 out of 5 as its solution. Thus 3 divided by 5 builds on 3 out of 5.
- Do we need to have 3 objects that we divide in 5? Or can we simply take the circle, divide it into 3, then divide it into 5? How do we then interpret 3/5?
- 3 divided by 5 can be discussed as "dividing out" 3 apples to 5 people. You divide each apple to 5 people.
Memory management - Copying - Dials as a number recording system
- Memorize math facts, don't overburden the mind.
- Math facts are meant to be like old friends
- Levels of memory - human brain
- Levels of memory - computer, artifacts
- Reliability of memory
- Obsolescence of memory artifacts, software, hardware, systems - example Windows 95 backup QIC
- Exercise: Memorize math facts. Motivate yourself with math fact racing.
- Writing (a system) is an aid to memory (but creates a new system, the context of which must be tracked and remembered)
- Memory relates to the quality of signs, what is memorable - symbol=index
- Clock, telling time
- 10 + 4 = 2
- Giving change doesn't affect the total quantity, just rebalances the amount and the unit.
- long division, 3 dimes = 30 pennies renaming
Division and remainder
- division with remainder = mixed fraction
- introduce fractions as remainders
- In division, we can drop the remainder, or not.
- long division, divide out, relate to money, get change, remainder
- Decimals are remainders 16/10 = 1 6/10 where the latter is 6 divided by 10, in other words, .6
- division by 10 (decimal) relates remainder and quotient
Single units or multiple units
- switch between single units and multiple units
Single units and multiple units
- Present your answer with multiple units.
- Group by the largest units to present. Multiple units help people who lack math intuition, they allow for various degrees of mathematical intuition.
- Single units help keep the answer exact, intact, we don't lose the minor units.
- Convert to the same unit to calculate.
- List in order the directions how to get somewhere, so that one can at least remember the first few steps.
- Being thoughtful in keeping smaller amounts: paying so that it's convenient to give change - making subtraction easy
- Exercises: give money using fewest number of bills and coins, give money so that the fewest number of bills and coins are returned as change
- Time is written in a variety of formats around the world. (Nonordered multiple units)
- Time formats can be read as a whole, or as in pieces, regardless of how they are disbursed.
- Fractions of time: hours, minutes, seconds, months.
- Subtraction - if you don't have to carry, then you can do it in your head, and you can do it from the left or the right or the middle.
- "Irregular" counting systems, note the variety - Roman numerals - French numbers like 80+ - linguistic evidence
- Suppose the sun went backward for a bit. How would that affect what we mean by day, hour, etc.?
Stitching together (Multiple units)
Ordered multiple units
- Multiple units like 24hours, 60 minutes, 60 seconds or cups, tablespoons
- build a tower from ones, tens, hundreds, thousands
- Decimal numbers are multiple units read right to left, and single units read left to right.
- decimals - well formed, easy comparison - regular (digits) all less than 10
- dollars, dimes, pennies, staples, confetti
- Multiple units are the basis for: Rounding, addition with regrouping, subtraction with regrouping.
- Multiple units allow for smaller amounts.
- decimal numbers - multiple units, right to left, single units (ones) left to right
- decimal system multiple units allows for approximation
- Because we carry, you have to start from the right, if you are a poor man, nobody will borrow from you. Otherwise, you could start from the left or the right or the middle.
- Abacus and Counting devices
- Compare slots (units) with pigeon hole principle.
Different names for the same thing (see also ambiguity)
- Different ways of writing the same number: 37.50 cents, 37.5 dimes, 37 1/2, 37 r 1, 75/2 All mean the same but used in different ways.
Max number on the dial
- Decimal system: 9 is the most allowed for any unit, therefore the largest units dictate the comparisons
- Truncating is discarding units when there are multiple units.
Avoiding large units
- 99 cents - the largest values for the units, without using a larger unit
- Multiple units allow for counterexamples - remainder shows that not evenly divisible.
- Rounding is expressing in terms of the largest unit when there are multiple units.
- Round 79.99 to 80 etc. then add prices
- Why choose decimals, why fractions? Decimal style vs. fraction style for problems like 10 / 20 / 30
- rounding = getting rid of units (when there are multiple units)* multiple units (show answer) single units (calculate)
Integrating multiple units
- Counting stacks of bills - need to be integrated
Dealing with a problem from one direction in a way so that you don't have to go back
- Start adding (multiple units) with the loose carrots because you can bag them, crate the bags
- Systemic counting, organized counting
- Reading from left or from right?
- A dial can be thought of as a unit but also as an action. As an action it becomes the set of possible values, thus a unit of multiplication.
- Multiplication can be thought of as adding dials between the unit dial on the left and the decimal point on the right, thus separating them. If the dial is 0,...,N-1 then we set the amount to 0 and we are multiplying by N.
- Division can likewise be thought of as adding dials between the unit dial on the right and the decimal point on the left.
- We can have two kinds of dial colors to distinguish multiplication and division.
- Multiplication and division dials can be rearranged and can be cancelled.
- Adding or removing such a dial can also be thought of as multiplying or dividing.
- The dials can be in different amounts.
- One dial can be broken up into two dials, or two can be joined as one.
- If the dial has a non-zero number, then there is also addition.
A scale of dials
Keep adding a dial
- Do the numbers ever end? Is there a largest number? 999,999,999,999. Then add one. So long as you can add a dial.
- Parking spot - every family gets one of them
- Reserved parking is given by zero, a place maker
Need for commas
- But this only helpful up to seven or eight commas. After that we need line breaks, then page breaks, then books, then bookshelves, then rooms, then floors, etc.
- Perhaps this is why in large libraries you're not allowed to reshelve books. Also, they aren't able to check, so a book misshelved is a book lost.
- Large numbers: Say forward, read it backward, by 3s.
- two ways of looking at it, long division dividing 350 million, may have intermediate remainder 20, which means 20 millions and stands for 20,000,000 ones.
- numbers have first and last names. At home they call themselves by the same first name 1-2-3, baby-mommy-daddy
- When you multiply you can reorganize, as in 1/3 x 3,000,000
- Think of multiplication variously in terms of units, actions, as in 4 x 5 millions
- Wallet for money (maximum of 9 for each slot), up to 9 ones, 9 tens, 9 hundreds, 9 thousands and so on.
Remainders - Division algorithm
- Justification for powers of 10: why is that how we can write down any number? division algorithm?
- Multiply by 10 to shift units
- dividing by 10 implies multiplying by 1/10 so division and multiplication can be combined
- Show doubling of generations - shift by binary units
- Fractal - shift unit - addition rule a**m a**n = a**(m+n)
- zeros = scoot overs, pushies
- We use zero so we don't confuse ourselves.
- .8 x 80 decimal points move vs. zeroes added
- folding paper = multiplying
- Such a system of units is uniform in its conversions.
- Such a system limits the size of any amount, for example, to less than 10.
- binary numbers: switches, or: cups, pints, quarts, half-gallons, gallons
- bits and bytes
- binary multiplication: cups to gallons, whole notes to sixteenth notes
- Addition formulas - extending the domain: 2**(x+y) = (2**x)(2**y)
- Multiplication/division and addition/subtraction are groups and linked by the addition formula (N**X)(N**Y) = N**(X+Y). Adding the number of dials/units makes for multiplying the amounts/values of the dials.
- "hundreds" family marries "millions" family equals "hundred millions" family
- You can choose which dial you want to be the unit - for example, the millions - and you can ignore the other dials
- Coins are spaced logarithmically, and in EU evenly .01,.02,.05,.10,.20,.50,1,2,5,10,20,50,100,200,500 whereas in the US unevenly .25 quarter and $20 but no $2 bill, no 50 cent.
- Logarithmic scales are based on multiplying evenly
- Numbers to memorize: such as the doubling sequence: 1, 2, 4, 8, 16, 32, ... , 1024
Name system for units
- Roman numerals is like currency, bills, banknotes, each has a name.
- Need to come up with a system of names - and even the decimal number system has limitations - we run out of names, at least, in theory
Orders of magnitude
- Show up to 45 orders of magnitude in three rows, such as size of electron/quark 10**-18 meter to ant 10**-3 to distance to Saturn 10**12 to diameter of known universe 10**27.
- Model all different systems in life. How many are there? Make use of the different units, dimensions. Compare the units and have rate problems. Use the three rows for the dials.
Orders of magnitude - a row of units, objects as units (sun, earth, dime, dollar) as labels for the dials - listed with pertinent facts- a personal database of facts
- Degrees of imperfection - and multiple reasons - Earth is not perfect sphere; also Earth has Mt.Everest, sea gorges; even water surface is not even; tidal effects
- Fortunately, forces of nature are almost intentionally spaced out to be of different orders of magnitude.
- Orders of magnitude: "Penny wise and pound foolish"
- Inflating errors: Get a calculator to create errors in various ways.
- Racial caste system - and ultimately must be visible
- Orders of Magnitude - Nikos Salingaros's Theory of Architecture
- Orders of magnitude in various dimensions (relate with grid)
- Scientific notation
Compare binary and decimal
- 2**10 is roughly 10**3
- kilobytes, megabytes, gigabytes, terabytes
- Square roots, geometric mean (2**5 = 32 is halfway between 1 and 1,000)
Two rows of dials
Comparing two rows
- Show how 2**10 is roughly 10**3, and then 2**20 is roughly 10**6, and 2**30 is roughly 10**9, and so on, and the story of the inventors of checkers and chess. Can use two rows to compare binary and decimal.
- Show how exponential growth of 1.06 becomes doubling. Compare also with Pascal's triangle (1+x)**N.
- Illustrate doubling problems - Adam and Eve, Manhattan - and how they break down.
Lining up rows - setting the main unit of reference
- When all in line (column of dogs) makes it easy to imagine, thus the value of units.
- lining up the decimal points lines up all the units
- If you don't line up the numbers, then you can get confused.
Units can be actions
- how do you change units? redeciding which dials is the unit
- word problems convert fractions (units) what we're counting by
- Fraction is a grouping instruction (translation) 600 miles / hour. For 4 hours "per" means substitution?
Three rows of dials
Computation on three rows
- Three rows so that we could do problems and get answers.
- You can multiply by rebalancing the first and second rows. (What is the natural way to do canceling? actions on the number line?)
- Subtraction involves two or three different roles: brought, ate, how many left
- Prove that .999999.... = 1.000000 (equivalence classes?)(subtract?)
Choosing a sequence of steps
- Choose how to do an 5,000 x 50,000 can be broken down in different ways such as 5 x 50 x 1,000 x 1,000
- Adding 256 + 256 in your head is easier to do from left to right than from right to left because the intermediate response (500) is simpler because there is nothing to carry
- Show each step, write it down!
- In math there are typically several ways to get the right answer, and there is an advantage to solving in the way that keeps the intermediate answer simplest - though other times there is an advantage to doing things in a general way that doesn't require case-by-case thinking
Word problem rates and proportions
- Speed = distance / time. A way of counting distance by time. I'll be there in half an hour.
Synchronizing: Use the three rows of dials, like audio tracks
- Consciously tracking a recurring activity
- Joining in: clapping hands, tapping feet, marking a beat
- Synchronizing, thus amplifying, channeling
- synchronizing and compatible patterns
- Synchronized counting = "per" means "counting by" 600 miles per hour
Counting fosters intuition about systems and spirit
- How do we foster intuition about spirit? We look inward by reflecting on our counting. We do this by division, which is "counting by", for example, dividing by 1/2 is counting by halves.
- How do we foster intuition about systems? We use single units to calculate, but we present answers in multiple units (such as running a marathon in 2 hours 10 minutes 20 seconds). This lets us communicate with people to allow for different intuitions, and to foster our intuitions. We can truncate, round, approximate. We stitch together our intuition by addition. When the units are different, addition means list, and when units are the same, addition means combine. This leads to counting, which is a very sophisticated behavior, as it relates qualitatively different types of numbers into one system. Multiplying is a form of counting that makes this evident. We foster intuition about systems by counting, by getting practice in counting. Perhaps: Counting is extending the domain for the addition formula. Extending the domain is stitching together our various faculties for sizing things up. Counting stitches them together by way of the addition formula, which is the heart of counting. Perhaps the initial domain is the divisions and the operation +0, +1, +2, +3, thus whether, what, how, why. This is transformed from circular (mod 8) to linear counting by way of the addition formulas?
- System has us present our thoughts to multiple audiences - to others, externally - whereas spirit has us present our thoughts to ourselves, internally. Thus system manages "breaking down" of answers, models, uses multiple units. Whereas spirit holds models, answers true, keeps true to them, internally, sees them through.