Picking Arbitrary Values and Numberline Subtraction

Society has a plethora of arbitrary values. I was inspired to see that kids are learning subtraction using the concepts of distance and the number line (LearnZillion: Solve subtraction problems using a number line). I’ll give an example if you didn’t check out the link.

  1. Pick two numbers x, y (given: 273, 834).
  2. Construct a number line with the given numbers as points:
    <--273---------------------------------834-->
  3. Step up or down from the numbers in hops:
    <--273-(1)-274-(60)-334------(500)-----834-->
  4. Add up the hop distances:
    500 + 60 + 1 = 561

So 834 - 273 = 561.

But the neat thing is that it gives children the chance to learn about making choices of where to hop. Given 105 - 35 they might hop to 55 and then to 105 or they might hop (bidirectionally) to 40 and 100.

The best choices for this sort of subtraction is, as far as I can tell, the following:

  1. For each common column, from smallest to largest, hop from the low number until that column is normalized.
  2. If the last hop increased the column count, include that column as a common column.
  3. Add the remaining uncommon amount of the larger number.

So for 123,456 - 789:

<--789-(7)-796-(60)-856-(600)-1,456-(2,000)-3,456---(120,000)---123,456-->
120,000 + 2,000 + 600 + 60 + 7 = 122,667

At each step we only focus on matching the single column. If we overrun the default number of columns for the smaller number (eg, 856 + 600 = 1,456), we may take more steps. But we will never add more than nine of whatever unit size we’re focused on.

So in this case we probably have a good algorithm for picking what would otherwise be arbitrary hops. Kids can still learn the method without this algorithm, and they can play around with finding their own hops.

But we find arbitrary numbers throughout the law and in our daily lives. They have economic, social, and psychological ramifications. Tax brackets tend to be based on arbitrary income levels, for example. Inflation and other factors (such as the number of people with incomes previously considered outliers increase) may invalidate those existing, arbitrary values.

We should tend to avoid arbitrary values in the law. Fines should be based on an individual’s income, as is done in some European countries. Fining a poor person the same as a rich person for a minor infraction such as speeding makes zero sense. Either the fine will be excessive for the poor person, or it will be meaningless to the rich person.

We may again face problems with arbitrary values as new technologies come along. Autonomous vehicles may be able to safely exceed speed limits, but will undoubtedly be forced to comply with limits that make little sense for years beyond the widespread adoption of such vehicles.

And yes, the federal budget. What amount should the budget be? We have endless scoring of laws and regulations from the Congressional Budget Office and Office of Management and Budget. But when we actually formulate the budget, it is entirely arbitrary. It is based on whims and beggings of special interests, or on emotional appeals for the social welfare.

I favor social welfare, but just as prison sentences should be chosen for their effect and not out of emotional reflexes, so should social welfare programs budgets.

But at least young kids will become acquainted with picking arbitrary values using the number line. Maybe they will find algorithms for picking other values that have profound repercussions on society. Or maybe they’ll just get better at picking the arbitrary ones.