Place value notation and related algorithms are a complicated set of machinery based on the associative, and commutative laws of addition and multiplication.
21 = 2 * 10 + 1
The manipulation of such expressions is by the laws of arithmetic including the distributive law as appropriate to a calculation.
Represent and solve problems involving addition and subtraction.
- 2.OA.1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1
For grade 2, they have
Understand place value.
2.NBT.1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
Place value can only be invented after understanding the arithmetic laws of associativity, etc. These laws are really algebraic laws as well. If the elements of a set satisfy such laws, the set is called by some algebraic name like group, ring, or field. Such names indicate the set follows those laws. This is considered algebra.
So any set of objects satisfying the laws of associativity, etc. are an algebraic object. Thus the natural numbers are a set that satisfies such laws and so is an algebraic object. Thus natural numbers are an algebraic set of objects. So arithmetic is part of algebra. Moreover, this is true from the beginning. Arithmetic is never not algebra, because arithmetic is always dealing with a set of objects that satisfy algebraic laws of the type of associativity, etc. which make the natural numbers are an algebraic set.
Place value is a definition together with the standard algorithms is only possible within an algebraic set that satisfies sufficient of the laws of associativity, etc.
The purpose of this note is not to determine the minimum set of algebraic laws for place value notation to be minimally defined. Although this is an underemphasized topic.
The book Pre-Algebra New Math Done Right Peano Axioms goes a bit more into place value and using letters to represent the digits and carries for addition of one, two and three digit numbers.