This page is a tutorial on how iconic container numbers work. *Addition is putting outside, while multiplication is putting inside*. The container numbers on this page are expressed as unit-ensembles with depth-value. The Unit-ensembles page describes how to represent numbers as collections of identical tokens. The Depth-value Notation page shows how enclosures can be used to group units. Container numbers represent magnitude by enclosing units in ** containers**, that is, by drawing circles, boxes, or other enclosures around units. Each container multiplies the value of its contents by the base of the number system. The base is usually 10 (the decimal system) or 2 (the binary system). Here we will show

*binary container numbers in order to focus on how iconic arithmetic works.*. The

*shows the binary container number*

**Image**Three different container notations are used on this page. *Parens notation* is a textual representation that uses parentheses as delimiting tokens to stand in place of surrounding containers. *Circle notation* is a graphic two-dimensional form that represents containers by surrounding circles and ovals. For example, the image shows how to read the binary container number ** dynamic circle notation**, which shows the process of computing with container numbers.

*Dynamic circle notation shows how arithmetic works.*The symbolic numbers that are used universally in our culture do not permit dynamic interaction, they cannot illustrate the mechanism of addition or of multiplication. Instead, symbolic numbers rely on memorization of number facts and the abrupt substitution of results in place of operations. We write

In the following, doing arithmetic is illustrated by video animations of counting, adding, subtracting, multiplying and dividing using a two-dimensional version of container numbers. The animations not only show how container arithmetic works, they show how arithmetic itself works. The intent of the tutorial is to provide models of *activity*, not passive viewing experiences. The animations show *what to do* in order to do arithmetic. The Parens Notation page includes the parens form of each of the dynamic container arithmetic examples shown in the videos. The parens form shows the skeleton outline of the animated transformations, bridging the gap between conventional symbolic representation and the dynamic interaction of the iconic representation. The Parens Notation page also describes the alignment between one-dimensional sequential and two-dimensional parallel computation.

Four 2006 videos that compile the animations on this page are available for download. All four are VGA (640×480) size in .MP4 format.

• Add/multiply Container Numbers (2 minutes, 2Mb) shows condensed animations of reading, adding, and multiplying container numbers.

• Container Number Operations (3 1/2 minutes, 3Mb) shows condensed animations of reading and the four basic operations (+, –, x, ÷).

• Container Numbers (8 1/2 minutes, 6Mb) shows most of the animations on this page.

• Spatial Arithmetic (11 minutes, 26Mb) is a *narrated* tour through the animations on this page.

## Containers

Containers are a natural iconic form, *they look like what they do*. Iconic containers represent the single concept of containment, distinguishing what is inside and what is outside. Although physical containers have properties such as shape and size, iconic containers are abstract in that we do not associate any properties with them other than containment. Without an underlying size or shape, any iconic container can contain any other container. But a container cannot contain itself, it is not possible to put a box inside itself for example. A container can contain any collection of other containers. It’s possible for a container to contain nothing, or to contain one other object, or to contain many other objects.

Although containers are two and three dimensional forms, they can also be represented typographically by delimiters, such as parentheses ** parens**.

Container numbers represent a **unit** as *a container that is full*. To make this distinction, a unit is represented by a solid circle

The depth-value notation represented by nested containment does not have the same limitations as place-value notation. In particular, each container is independent of the others. There is no reliance on the position of a digit in a form, nested containers take care of magnitude by *structure* rather than by an arbitrary placement convention. When addition is represented as *putting things into the same container*, both the Additive Principle and the natural parallelism of unit-ensembles is maintained.

Any particular number has many different iconic representations depending upon how much it has been condensed from unit-ensemble notation to container notation. For example, some of the ways we can write the number

There are *two canonical ways* to represent iconic numbers, as units only (hard to work with) or as maximally nested (easy to work with). The effort in using container numbers is in simplifying them to a standard nested form. The operations of addition and multiplication of container numbers then require very little effort, both are achieved by solely putting numerals in particular containers.

These techniques were first published in 1995 for a one-dimensional sequential version of base-2 container numbers by Louis Kauffman of the University of Illinois at Chicago. Spencer Brown Numbers presage the iconic representation and manipulation of whole numbers.

## Reading Container Numbers

This video shows the binary container numbers from

Counting from 0 to 16 with binary container numbers

The content of this animation is laid out in a strip here, so that the entire sequence can be seen at the same time.

To read a binary container number, begin with the deepest nested unit. Crossing outward over a boundary doubles the accumulated value. This video shows the doubling effect of container boundaries.

Enclosures multiply their contents by two

When crossing into a new space, if there is a unit in that space, then add one. This is the Additive Principle at work, objects in the same space add together. Here are three videos that illustrate reading the container numbers

Reading binary three

Reading binary twelve

Reading binary thirty-five

## Standardizing Container Numbers

The purpose of standardization is *to make numbers easier to work with*. Of course there are many ways to define what it means to be *easier to work with*. We’ll initially adopt a criterion of *minimal structural complexity*, based on the original motivation to make it easier to read the cardinality, or exact size, of large ensembles.

Three simple transformation rules are used to convert container numbers into their standard nested minimal form, technically called a *normal* form. The ** Group Rule** collects units into groups. For binary numbers, each group consists of two units. For decimal numbers, each group consists of ten units. The

**joins containers that share the same space. The**

*Merge Rule**deals with negative numbers.*

**Cancel Rule**Units are collected into groups by the **Group Rule**, which converts a pair of units into one unit inside an enclosure. The enclosing container multiplies its contents by two. In standard notation, this rule is

The Group units action

Iconic grouping collects units into ensembles of a specific size (the *base*), and represents each group by a boundary around one unit. Boundaries in the same space merge by combining their contents. In standard notation, the **Merge Rule** is the distributive property,

The Merge boundaries action

*Positive units* are represented by a full circular container. *Negative units* are represented by the arbitrary choice of a pentagon. The **Cancel Rule** deletes positive and negative units in pairs. In standard notation, this rule is

The Cancel opposite units action

### The Additive Principle

Putting-into-the-same-space, grouping units, merging containers, and canceling opposites are all manifestations of the Additive Principle. Putting-into-the-same-space is, in fact, the *definition* of the Principle. Standardization is motivated by our criterion to minimize structure, so that all three standardization transforms remove structure. Grouping and Merging both reduce two separate forms into one. Grouping defines a specific collection of units and puts them together in their own space as an ensemble, while at the same time transferring the cardinality or multiplicity of the collection onto a single unit within the newly bounded space.

Technically, as an aside, space cannot support any type of relation, since we have given it no structure. It is always the container and not the contained space that defines or implements transformations. This very important principle assures us that only existant forms have properties. It can be made notationally explicit by writing a *localization container* around each transformation. We’ll use explicit brackets here but render them implicit and invisible throughout the other sections. That turns the phrase “put into the same space” into “put into the same container”. A localization container is an imaginary rather than an actual boundary — it doesn’t multiply, rather its purpose is to draw attention and to assure that all forms have an outermost container.

Here are the three standardization rules with the localization boundary made explicit. I’ll use a white lenticular bracket because it draws attention.

## Addition

To add, put numbers together. Group and Merge then take over to simplify the numbers into the nested form of the sum. To illustrate Group and Merge operating in concert, the next animation shows four single units being converted into the nested form of the number

Standardizing four units

It is sometimes convenient to record the progress of the dynamic simplification process using text. ** Parens notation** provides a typographical representation that shows each transformation step of an iconic number. The parens sequence on the right is read vertically, each line replicates a step taken in the animation going from the unit-ensemble form to the fully nested form. Parens notation provides a static outline of the transformation sequence.

Here is another dynamic addition, the parens form is in the Supplement.

Adding 7 and 5

Here is the content of this animation laid out as a strip, so that the entire sequence can be seen at one time.

## Subtraction

To subtract, *add* positive and negative numbers and then cancel pairs of opposites. Often a nested unit must be brought to the same level of nesting as a canceling unit, the container analog to *borrowing*. To do this, the Group Rule is run backward, as ** Ungroup**. This animation shows an enclosed unit ungrouping.

The Ungroup action

To facilitate the cancellation of deeply nested units, it is sometimes also necessary for the Merge Rule to run backwards as ** Unmerge**. Here is an animation illustrating unmerging, the deconstruction of an enclosured unit,

The Unmerge action

To put Ungroup in context, here is the video of

Adding 2 and –1

The Group and Merge operations of iconic arithmetic are the same for both positive and negative numbers, they are independent of type of unit. This video shows

Adding 1 and –2

As illustrated in the next animation of

Adding 2 and –2

The addition *backwards* is avoided, as is the difficulty of trying to explain (often with banking metaphors) that negative seven minus five yields negative twelve,

Adding –7 and –5

The next two videos show

Adding 7 and –5

Adding –7 and 5

## Multiplication

Multiplication is the replacement of the units of one number by another entire number. This process is stated as ** substitute number-A for each unit in number-E**. Square brackets serve as an abbreviation for this action.

*Substitute A for*• in Eis written as

*into*is being multiplied by the number being substituted, so the above substitution is read as

*, or*E multiplied by A

Once substitution has occurred, the two numbers have been multiplied. Then Merge and Group take over to simplify the form of the result. In the simple case of replacing *Substitute • for • in E*), nothing changes. The animation shows

Multiplying 3 by 1

In contrast, the next video shows

Multiplying 1 by 3

The commutativity of multiplication, that

When

Multiplying 2 by 3

When

Multiplying 3 by 2

The next four animations show

Verifying that substituting 7 into 5 is 35

Verifying that substituting 5 into 7 is 35

When Group and Merge are applied to each initial multiplication, the two forms quickly converge to a single common form that then continues to simplify until the final nested form of

Multiplying 5 by 7

Below, the content of this animation is laid out in a strip so the entire animation can be seen at one time.

For comparison here is the animation of

Multiplying 7 by 5

Multiplication by negative numbers is discussed on the Container Algebra page.

## Division

Iconic division is the inverse of multiplication, ** Unsubstitution** so to speak. However, the general operation of substitution itself is sufficient to incorporate the concept of division. This animation shows

Division of 2 by 2 using substitution

This elementary case illustrates that division is also a type of substitution. The iconic form of two is first identified within the form to be divided and then replaced by a unit. This can be written as *Substitute a unit for every 2 in 2*. In general,

*Substitute a unit for every*A in E .

To divide *Substitute • for 2 in 6*, which can be abbreviated as

Dividing 6 by 2

The final two animations show

Dividing 35 by 5

Dividing 35 by 7

The Parens Notation page shows an alternative textual notation for the dynamic containers in each of the videos.