Maths, and the arithmetic problem

Why a system that discusses topology gets 47 times 63 wrong

It is strange the first time you see it. A model that can explain the intuition behind a manifold will misplace a digit in a three-digit multiplication. The reason goes back to tokens.

Numbers get split into tokens in ways that do not respect place value. Depending on the tokenizer, 4763 might be 47 and 63, or 4, 76, 3. The model never sees a column of digits to carry between. It has learned enormous numbers of arithmetic examples and produces the most plausible continuation, which for small familiar numbers is almost always right and for larger or unusual ones is a good guess.

So performance degrades with digit count in a way no calculator’s ever would. Two-digit multiplication is usually fine. Four-by-four is unreliable. Long division, arithmetic with many decimal places, and anything requiring exact rational arithmetic are all risky, and the failures look completely normal: a confident wrong number in the middle of correct working.

The rule

Never accept a number a model computed if the number matters. The concept, the method, the setup, the interpretation: all fine. The arithmetic: check it.

This is not a limitation to work around with clever prompting. It is what the machine is.

Free tools that do the arithmetic properly

• A code interpreter, if your assistant has one. This is the real fix: the model writes Python and the Python runs. The answer then comes from an actual computation. Check that the code is right, because the code can be wrong; but if the code is right, the number is right.
• SymPy: free Python library, symbolic algebra, exact arithmetic, calculus, equation solving. Runs in Google Colab with no install.
• Maxima: free, mature computer algebra system, decades of use.
• wxMaxima for a graphical front end.
• GeoGebra and Desmos: free, browser-based, excellent for graphing, geometry and checking that a function does what you think.
• GNU Octave: free and runs most MATLAB code, which matters because job ads name MATLAB.
• SageMath: free, covers a lot of Mathematica’s territory.

The pattern that works: ask the model which method, ask it for the SymPy or Maxima code, run it, compare with your own answer.

from sympy import *
x = symbols('x')
# check an integral you did by hand
integrate(x*exp(x), x)          # x*exp(x) - exp(x)
# and confirm by differentiating back
diff(x*exp(x) - exp(x), x)      # x*exp(x)

That is thirty seconds and it is a real check, because SymPy is not guessing.

Where models are strong in maths

Do not overcorrect. There is a lot they do well:

• Explaining what a definition means and why it is stated that way.
• Giving the intuition behind a theorem before the formal statement.
• Suggesting an approach when you are stuck: this looks like a substitution problem, try u equals the inside of the bracket.
• Generating practice problems of a specified type, and hint ladders for them.
• Spotting the step where your working goes wrong, which they do well and which is the hint-ladder move from Module 3.
• Translating between notations, when your lecturer uses one and the textbook another.

That is most of what a maths tutor does apart from the arithmetic.

Word problems and the setup

The hardest part of an applied maths question is usually turning the words into equations, and this is a good use. Ask for the setup only:

Turn this into equations. Define every variable with units. Do not solve it.

Then solve it yourself. You have removed the part you find hard for the wrong reasons (parsing dense prose) and kept the part the exam is testing.

Checking without any tool

When you have no computer, three checks cost nothing:

Estimate first. Round everything hard and get an order of magnitude before you look at the answer. 47 times 63 is about 50 times 60, so about 3000. An answer of 296 or 29,610 is caught instantly.

Units. Carry them through every line. If they do not come out as they should, the error is above that line.

Substitute back. Solved for x? Put it in the original equation. This catches most algebra errors and takes fifteen seconds.

These are the same checks a good teacher has always taught, and they matter more now, not less, because the source of the wrong number has changed from tiredness to something that never gets tired and is wrong anyway.