Add two two-digit numbers - with regrouping

Learning Objectives Represent any two-digit number as a first-degree polynomial in terms of its base. Model the process of 'regrouping' using floor functions and modular arithmetic. Deconstruct the addition of two-digit numbers into a formal algebraic algorithm. Define a recursive sequence where each term is generated by adding a constant two-digit number with regrouping. Analyze the function that maps two two-digit numbers to their sum, focusing on the non-linear 'carry' operation. Generalize the regrouping process to any base, connecting it to polynomial operations. Why does 99 + 1 create a new digit place, but 45 + 52 doesn't? Let's explore the hidden functions and sequences behind this simple operation. 🧐 In this lesson, we will revisit th...

TermDefinitionExample Polynomial Representation of IntegersA method of expressing a number as a polynomial evaluated at its base. A two-digit number 'ab' is represented as the polynomial P(x) = ax + b, where x = 10.The number 74 can be represented as the polynomial P(x) = 7x + 4. When evaluated at its base (x=10), P(10) = 7(10) + 4 = 74. Regrouping (Carry Operation)The algorithmic process in base-10 addition where a sum in a place value column that is 10 or greater is decomposed. The ones digit of the sum remains in the column, and the tens digit is 'carried' over to the next column of higher value.In 8 + 7 = 15, the '5' is the result in the ones column, and the '1' is the carry value transferred to the tens column. Floor FunctionThe function that t...

Algebraic Addition of Two-Digit Numbers (10a + b) + (10c + d) = 10(a+c) + (b+d) This formula represents the sum of two two-digit numbers, 'ab' and 'cd', by separating their place values. This is the initial step before accounting for the regrouping operation. Functional Definition of Regrouping Given a sum of digits in a column, S. The resulting digit is D = S \pmod{10} and the carry is C = \lfloor S/10 \rfloor. This pair of formulas precisely defines the regrouping process. The modulus operator gives the digit to write down, and the floor function gives the digit to carry to the next column. Complete Sum Algorithm Let S = (10a+b) + (10c+d). Let c_1 = \lfloor (b+d)/10 \rfloor and d_0 = (b+d) \pmod{10}. Let c_2 = \lfloor (a+c+c_1)/10 \rfloor and d_1...