What if arithmetic were simple? Really simple, so that addition and multiplication took no thought at all. The earliest arithmetic was based on identical units: fingers, pebbles, shells, marks, notches, strokes or tallies. This arithmetic is sometimes called stroke or tally arithmetic. Tally sticks were in use 30,000 years ago. Tally arithmetic is really easy. To add, just put units together. Putting together takes no effort, no thinking. Tally arithmetic uses the Additive Principle, that a sum looks just like its parts. In the Image the objects being added are collections of indistinguishable units called unit-ensembles. Unit-ensembles are added simply by removing the boundary between them. This boundary can be physical or it can be conceptual. All that is necessary to add unit-ensembles conceptually is to look at them differently, to stop seeing them as different collections and to start to see them as one single collection.
A unit-ensemble is a grouping of identical units without specific names. Unit-ensemble arithmetic
• is base-1, there are no special size groups and no positional notation
• uses one-to-one correspondence, there is no counting , just matching
• adds by putting units together via the Additive Principle
• is often considered to be the definition of whole numbers
We are very familiar with unit-ensembles. Consider the American flag. The fifty stars form a unit-ensemble. They are placed artistically, and where they are placed has no particular meaning. They are not in any particular groups and no particular ordering. The stars are in one-to-one correspondence with the States, but no particular star corresponds to a particular state. We can match stars with states and account for all of them without counting up how many stars or States there are. The stars are added together by being put together on the blue field. Similarly, the thirteen stripes are a unit-ensemble that corresponds to the thirteen original colonies. No particular stripe corresponds to a particular colony. The six white stripes and the seven red stripes do form different ensembles, but these colors have no meaning, they are artistic choices, presumably to be able to tell them apart. The flag adds these two red and white ensembles, what we see is their sum.
We can assign names to each type of unit-ensemble. These are the usual whole numbers. Here we’ll use a dot,
The dotted lines highlight the location of the ensemble. They are informal containers and not part of the iconic numeral. Notice that there is no zero, the absence of units indicates that there is nothing.
Addition is placing ensembles into the same container. We do not need an addition sign, changing the informal boundaries indicates addition.
Unit-ensemble addition is not limited to only two ensembles. In general any number of ensembles can be added together at the same time. Nor is there any first or second ensemble being added, there is no ordering when ensembles are added at the same time.
It is often convenient to be able to type iconic numbers, so there’s a textual notation. A bar between objects shows the intention to add, while deleting the bar shows the result of the addition. In general, to add, merge containers containing ensembles. Merging containers can also be described as putting ensembles into the same container.
Unit-ensemble arithmetic does have one significant drawback, it is difficult to know how many units are in large ensembles. Roman numerals label specific ensembles of
To support subtraction, it’s necessary to add a new type of unit, the negative unit. We’ll represent it by a hollow diamond,
Subtracting a positive number is the same as adding a negative number. To subtract, replace the units of the number being subtracted by negative units and add. When a positive and a negative unit turn up in the same space, they cancel one another. The only new rule needed to handle negative numbers and subtraction is to Cancel opposites. Positive units are wholes while negative units are holes. When they come together, they both cease to exist.
This rule can be stated in general, for any matching ensembles. There are many ways to indicate a general negative number, the conventional method is to prefix a horizontal bar, the minus sign. Here, we’ll use typography to draw a hollow letter.
Multiplication is replacing every unit of one group with a copy of another group. Instead of putting together, to multiply is to put inside. Conventional symbolic notation does not provide an inside, so the Multiplicative Principle was also lost when we moved to symbolic numerals. Substitution has always been difficult to show in mathematical notation, because the number or form being substituted into changes when the substitution is made. Since a specific form changes, there is no convenient notation to show the result of an arbitrary substitution into an arbitrary form. Below, square brackets are a compromise notation for the action “Substitute a for b in c“:
Unlike addition, multiplication by substitution does have an inherent ordering, which is expressed as the commutativity of substitution. Substituting
Finally, iconic arithmetic has a distribution rule, distribute addition over substitution. Distribution says that ensembles can be merged first and then substituted into, or they can be substituted into first and then merged.
Like multiplication, division of unit-ensembles occurs by substitution. Just like addition and subtraction are accomplished by the same action of putting together, multiplication and division are accomplished by the same action of putting inside. To divide, replace each ensemble that matches the divisor by a single unit. The division of
As an illustration, the division of
The axioms of unit-ensemble arithmetic and the unit-ensemble conceptualization of reciprocals, fractions, and exponents are discussed briefly on the Container Algebra page.