Concepts in Arithmetic (1)

dice2

ARITHMETIC = Arithmos (number) + Techne (Skill) = Number skill

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(1)    Arithmetic starts with COUNTING.

Counting starts with one. The next count is one more.

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(2)    What is counted is called a UNIT.

UNIT = what is counted one at a time.

If one is counting houses, then each house is a unit.

If one is counting inches in a length, then each inch is a unit.

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(3)    Each count is represented by symbols called NUMBERS.

The numbers are 1, 2, 3, etc. They are read as ONE, TWO, THREE, and so on.

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(4)    The counting numbers start from ‘1’. The next number is always one more.

ONE         = 1

TWO        = 1 + 1      =   2

THREE     = 2 + 1     =   3

FOUR       = 3 + 1     =   4,     and so on.

These numbers go on forever.

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(5)    Numbers are patterns (see the picture of the dice above).

Numbers are visualized as ‘patterns’, such as, a pattern of five dots for five.

If we visualize five as the symbol ‘5’, then it is like visualizing a cat as the symbol “CAT’.

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(6)    The concept of ‘no unit’ is represented by 0 (zero).

0 (zero) means “nothing”, and, therefore, it is not used in counting.

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(7)    The set of counting numbers is called NATURAL NUMBERS.

The smallest natural number is 1 (one) and not 0 (zero).

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(8)    When zero is included, the number set is called WHOLE NUMBERS.

0 (zero) is a whole number, but not a natural number.

1 (one) is a whole number as well as a natural number .

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(9)    Numbers are written with DIGITS.

This is similar to writing words with letters.

The word CAT is written with three letters: C – A – T.

The number 105 is written with three digits: 1 – 0 – 5.

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(10)   There are ten different digits:  0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

This is just like having 26 different letters in English, which are used to write all the words in English.

The word ‘I’ is written with a single letter;  ‘ME’with two letters; and ‘YOU’ with three letters.

The number ‘7’ is written with a single digit; ‘15’ with two digits; and ‘164’ with three digits.

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(11)   Numbers are grouped as ONES, TENS and HUNDREDS.

There are 1 to 9 ONES.

There are 1 to 9 TENS.

There are 1 to 9 HUNDREDS, and so on.

Thus, ONES, TENS and HUNDREDS, may be used as “units.” See (2) above.

For example:   369   =   3 HUNDREDS + 6 TENS + 9 ONES

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(12)   The 1 to 9 ONES are single-digit numbers.

(1, 2, 3, 4, 5, 6, 7, 8, 9)

The 1 to 9 TENS are double-digit numbers.

(10, 20, 30, 40, 50, 60, 70, 80, 90)

The 1 to 9 HUNDREDS are three-digit numbers.

(100, 200, 300, 400, 500, 600, 700, 800, 900)

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(13)  10 ONES become 1 TEN, like 10 pennies become 1 dime.

10 TENS become 1 HUNDRED, like 10 dimes become 1 dollar.

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(14)   The ONES may be counted between two consecutive TENS, such as.

Between 10 and 20: 11, 12, 13, 14, 15, 16, 17, 18, and 19 (1 more gives 20)

Between 20 and 30: 21, 22, 23, 24, 25, 26, 27, 28, and 29 (1 more gives 30)

And so on…

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(15)   The TENS may be counted between two consecutive HUNDREDS, such as,

Between 100 and 200: 110, 120, 130, 140, 150, 160, 170, 180, and 190 (10 more gives 200)

And, the ONES may be counted between two consecutive TENS within HUNDREDS.

And so on…

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(16)   The single-digit numbers are from:   1 to 9

Note: There are nine single-digit numbers.

Zero (0) is a whole number but not a counting number.

The double-digit numbers are from:   10 to 99

Note: There are ninety double-digit numbers.

The three-digit numbers are from: 100 to 999

Note: There are nine hundred three-digit numbers.

And so on…

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(17)   The “units” of ONES, TENS HUNDREDS make a BASIC group.

The next group of ONES, TENS HUNDREDS is called the THOUSANDS.

The next group of ONES, TENS HUNDREDS is called the MILLIONS.

The next group of ONES, TENS HUNDREDS is called the BILLIONS.

The next group of ONES, TENS HUNDREDS is called the TRILLIONS, and so on.

Number System

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For further details, please refer to: MATH MILESTONE #1: NUMBERS & PLACE VALUES

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Comments

  • vinaire  On February 18, 2013 at 7:51 AM

    Numbers give us the concept of digital as precipitating from an analog space. We count what has precipitated. The first item precipitated is ONE. Such precipitation increases by one each time. It can also increase by TENS or HUNDREDS at a time too. And this can go on for ever. Here we have COUNTING or NATURAL numbers. Here we also get our first intimation of INFINITY.

    Then we put our attention back at the background of this analog space, What is there before anything digital has precipitated from it. There is nothing “digital” and so we call it ZERO. This gives us a “stable datum” or a reference point to look from at anything digital. The reference point itself is neither digital nor analog. Once we have this reference point included, we get the set of WHOLE numbers.

    What is next?

    • Chris Thompson  On February 18, 2013 at 12:36 PM

      Vin: Numbers give us the concept of digital as precipitating from an analog space.

      Chris: I think we should be careful about beginning this thread with that assumption — analog space.

      • Nia Simone  On February 18, 2013 at 1:17 PM

        Good point, Chris.

        I like analog, though.

        It would be good to define analog. For this purpose, I’d like to say analog is the integrated whole. Digital is the breakdown of the whole into slices. Counting numbers are the the breakdown of all numbers into a countable, infinite set. The whole set is analog, then.

        If you take the slices of a digital graph of a manifestation to the limit, as with integration, the graph becomes analog (whole) again.

        Now, given all this, I agree, Chris, I don’t think zero fits the definition of analog. It’s not the whole. It’s the absence of the whole.

        You’ve been luring me into this contemplation of nothing, Chris, try though I might to resist it. 😉

        Nia

        • Chris Thompson  On February 18, 2013 at 2:42 PM

          Thank you Simone. As a fledgling contemplator of these mysteries and while I live out the fantastic adventure of life with my family and friends like Vinaire; I have found my heart. This is who I was born to be. What could be better or more fun than knowing what is my purpose and following it?

        • Chris Thompson  On February 21, 2013 at 10:18 PM

          Nia, there’s nothing (no pun) in my experience that prepares me for the zero. I wonder is the zero point a freeze frame with something in it or is zero the in-between frames without anything (much?) in them… don’t pay too much attention to me. I wheeled bricks and dirt in a wheelbarrow today.

        • vinaire  On February 22, 2013 at 6:08 AM

          I like the description of analog space as the integrated whole and digital is the breakdown of this whole into slices. It is two different ways at looking at the same thing..

          Zero seems to be something else. It is the unknowable. It is Brahma. It is fathomless. It is the reference point from which to assess both the analog space as well as the digital breakdown of it.

          Zero is the perception-point with no filters. It adds nothing to what is observed.

          But zero may also be the absence of all of above.

          .

        • Nia Simone  On February 22, 2013 at 8:15 AM

          I like what you say about zero, Vinaire. How would you relate it or compare it to infinity? Using the idea of limits, I guess you hit infinity before you ever get to zero. That’s interesting.

        • Chris Thompson  On February 22, 2013 at 10:35 AM

          Vin: But zero may also be the absence of all of above.

          Chris: Philosophically, yes. But practically, we can use it to good result. As in a location marker on a line or zero peanuts, etc.,. Zero has contextual usefulness and truth. And it can be unknowable.

          You know Vin, I have changed my mind. Your harping on unknowable has moved me to look until I’ve gotten new understandings.

        • vinaire  On February 22, 2013 at 2:28 PM

          Yes, practically, we can use zero as

          (1) Reference point for all numbers, quantities, things, you name it.

          (2) The unknowable Brahma is simply an extension of (1) above.

          (3) Origin on a graph.

          (4) The height or depth at sea level.

          (5) No relative motion, etc.

          Now, watch out. If you accept unknowable the universe may go poof! on you. LOL!

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        • vinaire  On February 22, 2013 at 8:56 AM

          Infinity seems to be the analog space and its breakdown into digital. It is the manifestation that is being perceived. It is the sum total of all considerations. It is everything and more. It is the existence.

          Absence of infinity is zero. It is through infinity we approach zero. It is neti-neti of Vedas.

          Limit is the digital elimination of the analog space to take a shot at what is beyond. Limit depends on the path you take. Each function denotes a path.

          The Limit seems to define the basic construction of the path rather than its starting point of zero.

          Any limit, any starting consideration, thus, is of infinite dimension at its manifestation. The limit seems to define the fractal structure of the path it takes to develop further.

          That’s enough of rambling on my part. 🙂

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        • Nia Simone  On February 22, 2013 at 11:09 AM

          I was with you up to the last two paragraphs:

          VINAIRE: The Limit seems to define the basic construction of the path rather than its starting point of zero.

          NIA: I don’t think it starts at zero. I think it starts to the right of zero. Plus, I think the limit describes the manifestation as well as the path. I like that you came up with the description of the path because that is true. Putting this in Calculus terms, if you look at a function, then the limit gives you both the the path to the curve (the way you solve for the function) and it gives you graph. It’s just math so we don’t have the actual manifestation, but I find it helps me because the language is less ambiguous and I think there are 2 things: the path and the manifestation.

          VINAIRE: Any limit, any starting consideration, thus, is of infinite dimension at its manifestation. The limit seems to define the fractal structure of the path it takes to develop further.

          NIA: Not sure. Fractal Geometry is a different area of Math from Calculus, I thought?

        • Chris Thompson  On February 22, 2013 at 1:56 PM

          VINAIRE: Any limit, any starting consideration, thus, is of infinite dimension at its manifestation. The limit seems to define the fractal structure of the path it takes to develop further. NIA: Not sure. Fractal Geometry is a different area of Math from Calculus, I thought?

          Chris: Is not all mathematics language used to describe the relationship between considerations?

        • Nia Simone  On February 22, 2013 at 2:28 PM

          Yes. Good point. I just don’t understand that branch at all so it’s like not knowing the language, for me.

      • vinaire  On February 18, 2013 at 1:44 PM

        OK then, let’s look at analog (continuous) and digital as a dichotomy that precipitates from God or Unknowable.

        I prefer unknowable to God because there is less baggage associated with that word.

        So, as we look at discrete objects to count, we also look at a continuous background against which to count. I am sure mathematics gets back to looking at that continuous background too.

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        • Chris Thompson  On February 18, 2013 at 3:38 PM

          Vin: OK then, let’s look at analog (continuous) and digital as a dichotomy that precipitates from God or Unknowable.

          Chris: Thank you for that. And while we are being careful not to bog in terms like God and Unknowable, lets try to understand the implications of Schrodinger and see if there is a way to bring forward their terminology, based on sound mathematics, in a meaningful way for our philosophy. If we stand on the shoulders of these giants maybe we too can see a bit farther.

        • vinaire  On February 18, 2013 at 9:11 PM

          Yes, I want to examine Schrodinger, once I get through these basics.

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      • vinaire  On February 18, 2013 at 1:45 PM

        Don’t we call the continuous background a ‘dimension’?

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        • Chris Thompson  On February 18, 2013 at 3:50 PM

          Vin: Don’t we call the continuous background a ‘dimension’?

          Chris: Yes, I see what you mean. Agreed.

          Therefore lets drop — at least for a little while — the term unknowable. That word has a built in arrogance which points to the idea that we already know what is manifested. Our little scratches into the surface of “what is known” coupled with our predictions of “what is there” but yet to be known, should humble us away from declarations that smack of absolutism. I think that may hinder our discussion… If not between you and me, then maybe between others with newer and fresher looks at these conundrums. Understand that I don’t particularly care anymore. You and I understand each other. This is my opinion about the discussion itself and how to make it productive.

        • vinaire  On February 18, 2013 at 3:56 PM

          OK then let’s use the traditional term BRAHMA instead of God or unknowable.

          BRAHMA has no attributes, not even that of ‘being’.

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        • Chris Thompson  On February 18, 2013 at 6:05 PM

          Wow. Well ok, I’ll try to get used to that.

        • Nia Simone  On February 18, 2013 at 6:07 PM

          I always like a good, solid definition for the totally ethereal.

        • Chris Thompson  On February 18, 2013 at 6:15 PM

          LOL! I needed that.

        • Chris Thompson  On February 21, 2013 at 10:14 PM

          Vinaire: Don’t we call the continuous background a ‘dimension’?

          Chris: I missed how that’s relevant to my comment. Is it?

  • vinaire  On February 18, 2013 at 7:59 AM

    Last week, I was working with a 7th Grade student, who was having a terrible time with math. So, I started to work with on the basic concepts as laid out above.

    The penny dropped at point (5). The student never looked at numbers as certain patterns. He was looking at them as symbols. So, to him 3 was a symbol, and 2 was a symbol, and when you added these two symbols, a symbol 5 appeared. He was having a terrible time keeping all these relationships in his memory.

    The moment he realized that numbers were these patterns, the mystery of the transformation of 3 and 2 into 5 became clear. Yesterday, in one hour, he learned to add up to two 3-digit numbers in his head.

    So, what is 389 + 143? Can you quickly add them in your head? Well he can do that now. 🙂

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    • Nia Simone  On February 18, 2013 at 1:06 PM

      No, I can’t. I have to think about this patterns thing more.

    • vinaire  On February 18, 2013 at 1:53 PM

      See point (11): A three-digit number is made up of HUNDREDS, TENS, and ONES.

      See point (10): There are only ten different digits.

      389 = 3H + 8T + 9O
      143 = 1H + 4T + 3O

      Sum = 4H + 12T + 12O = (see point 13) = 5H + 2T + 12O = 5H + 3T + 2O = 532

      Mental Math works better from left to right.

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      • Nia Simone  On February 19, 2013 at 12:08 PM

        Thank you, Vinaire. You are an amazing teacher.

      • vinaire  On February 19, 2013 at 1:23 PM

        Mathematics is all about patterns. Symbols are necessary distractions.

        But if you understand this, you won’t be distracted. 🙂

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        • Nia Simone  On February 19, 2013 at 6:28 PM

          I understand the distraction of symbols and that got me through school, but I still don’t see the patterns you describe. I understand the logic but I don’t see the pattern. I will keep looking. I am not primary visual at all, so it will take me time.

          By the way, please don’t go down a rabbit hole with the Laplace transform. Wikipedia made a total mess of it. I happen to have had a very good teacher, who led us through the proofs and derivations of the knowledge area of differential equations, so that when she presented the leap that Laplace took with the transform, we could all appreciate it. She told it as a mathematical story and it was the class in which I did best in college. (I nominated her for outstanding teacher and she won. She was a grad student, a minority as a female in the math department, so it was a good thing and she deserved it. I used my writing skills to sway them. There had to be some benefit of being a former English major in the math department.)

          Only the idea, the intuitive leap that Laplace made, is of interest here. It is the leap that he took to transform a difficult problem into a space in which it became an easy problem. That’s the only thing I’m borrowing here, the details of transforming problematic differential equations into another space and back, are not relevant.

        • vinaire  On February 19, 2013 at 7:15 PM

          Here is the pattern for number base:

          Take an abacus. Have as many beads on each wire as is the number base. One counts from left to right. Have the rule that when all the beads are to the right they are regrouped as counting one bead on the next wire. If you do this then the number would appear as the pattern of beads appearing on the right on the wires.

          The above operation with base 10 on the regular abacus is described here: MATH MILESTONE #1: NUMBERS & PLACE VALUES .

          The base of the number works as a pattern for higher and higher place values.

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        • Nia Simone  On February 19, 2013 at 7:30 PM

          Thank you. I never had an abacus. That might be the root of the problem I’m having. Will check it out. You are generous. Thank you, again.

        • vinaire  On February 19, 2013 at 8:38 PM

          Can you give me some idea of the problem you are referring to.

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        • vinaire  On February 19, 2013 at 7:28 PM

          I programmed a base 10 abacus for computer where one could move the beads with the mouse cursor. Whenever there were 10 beads to the right they automatically regrouped as one bead to the right on the next wire.

          I closed the web site where it was after transferring all the my math materials to this blog. I have yet to figure out how to put such a program on a blog.

          I can improve this program to have the option for different number bases.

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        • vinaire  On February 19, 2013 at 7:29 PM

          Nia, I wish I could follow the same path that you followed with Laplace transform.

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        • Nia Simone  On February 19, 2013 at 8:18 PM

          Oh wow, Vinaire. I wish you could, too. If I can find my notes, I’ll try to recreate it for you, but it will take time to refresh this stuff. In the meantime, you have already used the essential leap in your rewrite of perception, by deliberately eliminating time.

  • vinaire  On February 18, 2013 at 1:56 PM

    The next concept after WHOLE numbers is that of INTEGERS. I shall start adding that to the above.

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  • vinaire  On February 18, 2013 at 9:58 PM

    I have added points (18) to (22) on Integers.

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  • vinaire  On February 18, 2013 at 10:04 PM

    So. from a reference point, the numbers would be infinite in any direction.

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  • vinaire  On February 19, 2013 at 7:59 AM

    Anything countable is digital. This provides us the dimension of analog or continuous. We won’t be aware of continuous unless we are also aware of digital. The continuous provides us with the background called ‘dimension’ in which the digital may be spotted.

    We can count discrete objects like cats, dogs, bodies, etc. because their boundaries are defined.

    We also attempt to count continuous attributes like length, time, mass and energy etc. by devising units, such as, inches, seconds, grams, ergs, etc.

    Each unit provides us with a ‘dimension’ for that unit. So there are different dimensions. There would definitely be a deeper background against which different dimensions may be spotted. Hence there is a deeper dimension.

    For example, the dimensions of cats, dogs, elephants, whales, etc. may be considered to have the background dimension of animals. Here we are looking at the categories of Aristotle.

    So what is the background of ‘units’ in general. It has to be ‘no units’ or nothing (zero). Thus, zero acts as the reference point for all units, all numbers, and all ‘somethings’.

    From this reference point, we may now look in different directions. The primary directions are negative and positive.

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    • Chris Thompson  On February 19, 2013 at 8:20 AM

      That is written very clearly and succintly.

      Keeping in mind a sense of layers, I try to remember that the background of all things is just the next background but not all backgrounds.

      • Nia Simone  On February 19, 2013 at 12:09 PM

        Not sure about that.

        • Chris Thompson  On February 19, 2013 at 12:13 PM

          I write it that way because I have never seen any reason to think we are next to or near the final layer of anything.

        • Nia Simone  On February 19, 2013 at 12:22 PM

          I think we are very near, between the introduction of “is thought” (in SIX through TEN) and “Thus, zero acts as the reference point for all units, all numbers, and all ‘somethings’.
          From this reference point, we may now look in different directions.”

    • vinaire  On February 19, 2013 at 10:17 AM

      A dimension may be represented by a Number Line.

      On this Number Line, the reference point of zero may be assigned

      This reference point of zero may be assigned arbitrarily.

      ..

  • vinaire  On February 19, 2013 at 8:37 PM

    I have added concepts (23) to (33).

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  • vinaire  On February 22, 2013 at 3:05 PM

    This is in response to NIA:

    VINAIRE: The Limit seems to define the basic construction of the path rather than its starting point of zero.

    NIA: I don’t think it starts at zero. I think it starts to the right of zero. Plus, I think the limit describes the manifestation as well as the path. I like that you came up with the description of the path because that is true. Putting this in Calculus terms, if you look at a function, then the limit gives you both the the path to the curve (the way you solve for the function) and it gives you graph. It’s just math so we don’t have the actual manifestation, but I find it helps me because the language is less ambiguous and I think there are 2 things: the path and the manifestation.

    .

    You are right. The limit is off center. It does not start from zero. It is a ‘seed’ that is closer to zero than any other consideration. The path is like the indefinite integral that proceeds from the ‘seed’ or limit. Manifestations of considerations follow this path.

    There can be infinite number of different ‘seeds’ from which come infnity of paths for considerations to travel upon. I like the the word ‘seed’ which I got from The Creation Hymn of Rig Veda. Now associating this seed with limits in calculus and the unfolding of it as in integration seems to be making sense.

    .

    • Chris Thompson  On February 22, 2013 at 3:36 PM

      Vin: So, when I think of a fractal, I think of a fixed pattern, iterated again and again but with some harmonic upped by one in each iteration. Hope this clarifies the language I am using.

      Chris: . . . and the result of the earlier computation used as the “seed” for the next iteration, repeating the process thusly:

    • Nia Simone  On February 22, 2013 at 8:15 PM

      VINAIRE:Now associting this seed with limits in calculus and the unfolding of it as in integration seems to be making sense.
      NIA: It sure does make sense! That’s great. Thanks.

      VINAIRE: So, when I think of a fractal, I think of a fixed pattern, iterated again and again but with some harmonic upped by one in each iteration. Hope this clarifies the language I am using.
      NIA: Yes, I understand now, thank you.

  • vinaire  On February 22, 2013 at 3:23 PM

    This is in response to NIA:

    VINAIRE: Any limit, any starting consideration, thus, is of infinite dimension at its manifestation. The limit seems to define the fractal structure of the path it takes to develop further.

    NIA: Not sure. Fractal Geometry is a different area of Math from Calculus, I thought?

    .

    Maybe you’ll understand the sense in which I am using the word ‘fractal’ from the following example from computer language.

    I first came across programming language in 1964 as an engineering student. One little computer instruction was really interesting. It was the following statement from FORTRAN.

    N = N+1


    My first reaction was to find it nonsensical from my algebraic thinking. But when I got over it, I realized that the same math symbols were being used to communicate something entirely different. It was simply a command to reiterate a computation after replacing “N” with “N+1”.

    The pattern of computation was exactly the same, but the variable used was upped in value by 1. So, when I think of a fractal, I think of a fixed pattern, iterated again and again but with some harmonic upped by one in each iteration. Hope this clarifies the language I am using.

    .

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