Robert’s Remainder

Exercise

Robert’s Remainder records intermediate remainders from long division and turns them into another number. To make that operation precise, we will use the following rules.

Use base-ten notation, a nonnegative whole-number dividend, and a positive whole-number divisor. Write the dividend without leading zeros, except that zero itself is written as 0. Begin with the shortest prefix of the dividend that is at least as large as the divisor. Divide that prefix, record its remainder, then bring down the next digit and continue. Record the remainder at every step, including steps with a quotient digit of zero. Stop after processing the dividend’s last digit; do not continue into decimal places.

Write the recorded remainders next to one another, in order, without adding them. If a remainder has more than one digit, keep all its digits. Keep zeros within the resulting string, then read the string as a whole number, discarding any leading zeros. If the dividend is already smaller than the divisor, define Robert’s Remainder to be that dividend. Write this operation as RR(a, b), where a is the dividend and b is the divisor.

For example, if the recorded remainders were 4 and 10, we would form 410, not 14. Other ways of combining remainders could define other operations; concatenation is the convention used here.

What are the integer quotient, ordinary remainder, and Robert’s Remainder of 1234 divided by 5? Once we have calculated the example, what properties might this operation have?

 

 

Facts

1234 divided by 5 gives an integer quotient of 246 and an ordinary remainder of 4. To find Robert’s Remainder, begin with 12, because the initial digit 1 is smaller than 5. Dividing 12 by 5 leaves 2. Bringing down the 3 gives 23, which leaves 3. Bringing down the 4 gives 34, which leaves 4. Writing the remainders 2, 3, and 4 together gives RR(1234, 5) = 234.

The diagram also shows the ordinary division answer as 246 and 4/5. The number 234 is a different output: it answers our newly defined RR question about the same dividend and divisor.

Thoughts and Opinions

This operation is named for Robert, who has since graduated from MIT. This page is a reflection on his idea and an invitation to explore it. Sometimes you can be inspiring without even trying.

Long division gives us a quotient and an ordinary remainder, but its intermediate steps contain other information. Robert’s Remainder asks us to collect some of that information and see what happens. Giving the operation a name and clear rules lets someone else try it too.

For ordinary division, 1234 = 5 × 246 + 4. The remainder tells us that 1230 divides evenly into five whole-number groups, with four units left over. Those four units can also be divided into fractional shares, giving the familiar answer 246 and 4/5. Robert’s Remainder does not change that answer; it gives us another question to ask about the procedure.

When we learn a procedure, we inherit some of the work of the people who came before us. We can also notice something in its steps that we had not considered. Here, 234 is the answer to RR(1234, 5), and the name honors Robert’s contribution.

We can take the exploration further by applying RR repeatedly while keeping the divisor fixed. For divisor 5, the example becomes 1234 → 234 → 34 → 4. Applying RR to 4 gives 4 again, because 4 is smaller than 5. A value that remains unchanged by another application is called a fixed point.

When this repeated process reaches a fixed point, call that value Robert’s Reduced Remainder, written RRR(a, b). Thus RRR(1234, 5) = 4. This definition does not assume that every starting pair reaches a fixed point. If a sequence enters a cycle containing different values or never settles, it has no reduced remainder under this convention.

Here are some questions to explore:

1. How does the final recorded remainder appear at the end of Robert’s Remainder? What happens when that remainder is zero?
2. What changes if we define a different operation by adding the intermediate remainders instead of concatenating them? Try both conventions on the same examples.
3. Whenever Robert’s Reduced Remainder exists, is it the same as the ordinary remainder of the original division? If not, can you find an example?
4. Can repeated application of RR produce values that grow without bound? How would that differ from a sequence that continues forever while repeating values?
5. Apart from dividends already smaller than the divisor, are there inputs for which RR(a, b) = a? What would make such a fixed point possible?
6. Can the sequence return to its starting value after visiting a different value? What would a cycle of that kind look like?

Feel free to email me with your thoughts and opinions on this matter at [email protected].