Additive recursion at dy/dan

The blog Dy/dan discusses an additive recursion formula.  It is apparent from the discussion that they are familiar with recursive and explicit as terms.

It is clear from the discussion that the Peano Axioms are not any harder.  Addition itself is defined recursively in terms of counting is a conceptual jump not present in this discussion.

The recursion is

z[0] = 9

z[n+1] = z[n] + 2

As an explicit formula we can write

z[n] = 9 + 2 * n


Addition is defined recursively as

x+0 = x

x+y’ = (x+y)’

These are the right additive identities.

Here ‘ means next or successor.  It can also be read as count one.  So 2′ means count one starting from 2.  That gets you to 3.

So the right additive identities state

Count 0 from a number keeps you at the number.


The sum of a number and zero is the number.

Given two addends, count 1 from the right addend gets you to one more than the sum of the two starting addends.


Given two addends with a sum, count 1 from the right addend equals counting one from said sum.

All of these statements are as easy to understand as the recursion of adding 2 at a time starting from 9 being discussed at Dy/dan.

Addition is defined recursively.   This means we can’t reduce addition to a formula that itself uses addition and multiplication.

The recursion discussed at Dy/dan

z[0] = 9

z[n’] = z[n] + 2

has the explicit solution

z[n] = 9 + 2 * n

The explicit form uses addition and multiplication.  So the recursion that defines addition has no explicit form in this sense.

This is an important point to point out to build an understanding of the difference between defining addition recursively in terms of counting by one and of solving this recursion after addition and multiplication have been defined.

This conceptual discussion is absent in the dy/dan thread.  However, it could be learned by these teachers and then taught to students in K-6.  However, this would require overcoming the apathy potential barrier.  That requires quantum mechanics and time.

Even though these teachers can understand and engage in this discussion of recursion given that addition and multiplication are already defined, the conceptual step to defining addition recursively in terms of counting and then defining multiplication recursively in terms of addition is nowhere present in their discussion.

This would take a huge step for the teaching profession and perhaps for individuals as well.  My books cover this huge step divided into baby steps.  More numerical examples of the Peano Axioms are in my books than likely exist in all other materials combined.

My homepage at book e-vendor

You can read them with a free computer ap without buying the e-reader.  They are well formatted and were never pdf or word docs at any stage in preparation.  They were uploaded to e-vendor as pure html.


The discussion at these two blogs illustrates the need to explicitly solve the problem and then explicitly discuss it in detail down to the final nit picks.  That the solution of the recursion problem at Dy/dan requires that addition and multiplication are already defined whereas the definition of addition by recursion obviously can not do so is something that has to be pointed out explicitly in writing.

This is why the anti-textbook tendency of progressive math is wrong.  This has to be spelled out in words and symbols and with examples for teachers, students, tutors, college math ed profs, and parents to get it.


About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
This entry was posted in Explicit solutions of recursion. Bookmark the permalink.

5 Responses to Additive recursion at dy/dan

  1. Bowen Kerins says:

    If this is intended for pre-algebra, why do you use variables? At what point do you plan to introduce students to negative numbers, fractions, irrationals? When Devlin says multiplication isn’t repeated addition, he means for real numbers. Or, explain how you do pi * sqrt(2) using repeated addition.

    Also if you plan to get serious attention from anyone, I recommend skipping lines like “Teachers and college professors as part of the anti-America coalition don’t care.”

    • I think that letters like x are now used at lower grade levels then most people realize. The use of ? is actually no easier than x. 2 + ? = 5 and ?+2 = 5 are as hard as 2+n = 5 and n+2 = 5. Possibly easier, see the discussion in one of Borovik’s books. I think ? as a number inside an equation is more difficult to peg as representing a number than n or x. Same with blank, an empty square, and so on.

      Students are taught numbers 11 to 20 in grade K. This is place value notation which depends logically on addition and multiplication of whole numbers. Multiplication of whole numbers starts in grade 2. So this is a cart before the horse problem already in common core. This is a bigger problem than when to introduce real numbers.

      My strategy on fractions is to teach the logic of whole numbers well so that students can understand 1/2 as a function on the even integers. I teach them functions by teaching them successor as (0,0′) (1,1′) etc. I teach 1 = 0′ starting in K in my proposal.

      Your advice on the last point is likely sound. However, I do have to follow my own path to move forward on it.

  2. Bowen Kerins says:

    Place value notation doesn’t depend on addition and multiplication of whole numbers. It depends on knowledge of units. 1 ten = 10 ones. 17 is one ten and seven ones. Why wouldn’t you say students shouldn’t be taught the number “10” — you started with “11” but “10” has the same issue. Two- and three-year-olds learn “10”.

    There is also a serious difference between ? + 2 = 5, which asks students to replace a missing number or recognize the relationship between addition and subtraction, versus an identity like (x+y)+1 = x+(y+1). Recognition of properties that apply to all numbers, then generalizing those properties using suitable notation, is a skill that takes a long time to develop fluency.

    How do you teach students to multiply 3 * 1/2? 3 * (-2)? (-3) * (-2)? 2 * sqrt(2)? It is very difficult for students to move from one model to another, and I don’t see how the Peano model could lead to an understanding of all these pieces.

    • Addition behaves like

      m+0 = m

      m+n’ = (m+n)’

      multiplication behaves like

      m’*m = m*n + 1

      and is another word for plus?

      let A = 10

      17 stands for 1*A+7

      A and 7 is the same as A+7?

      is 18 one more than 17?

      A+7′ = A+8 = 18

      Grouping by 10 behaves like multiplication?

      Multiplication is the same as grouping, it is a way to keep one’s place in the count using groups.

      The students don’t have to invent the generalization themselves.

      Sure in teaching place value notation you are not teaching all of these generalizations, just not giving them names?

      Fractions are a next stage. Fractions are functions on natural numbers is first step.

      Real numbers are later still. One mathematician says he did not realize that elementary math is the rational numbers not the real numbers. I think it was Wu.

    • I am inclined now to think you are right on place value notation. However, you do need some machinery. For 0 to 9 you need order. However, to do place value in terms of n places, you need initial segments it appears to me. Also you need to be able to loop over initial segments. You also need to prove some variant of the recursion theorem I think.

      n = maximum place.


      i[j] jth place

      Algorithm of increment by one.

      if i[0] is not 9, just increment it by one.

      If you do it this way, and then define addition by recursion and with the recursion theorem, you have to prove these addition definitions are the same.

      Given the above machinery, which requires order before addition in the Peano Axiom development if you don’t do addition, you could then define addition by the standard place value algorithm.

      You would have to first define addition for 0 to 9, which can be just a table.

      Most Peano Axiom treatments do addition first and then order based on addition. So in those treatments, you would already have addition by recursion and have some version of the recursion theorem.

      The above is preliminary. Place value is typically not discussed in the Peano Axiom framework that I am aware of.

      When I made my comment I was thinking of doing place value after doing addition and multiplication and then using recursion on affine forms to get the place coefficient between 0 and 9 inclusive.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s