{"course":{"slug":"ai-for-students","title":"AI for Students","url":"https://addaly.com/learn/ai-for-students","level":"beginner","track":"work"},"lesson":{"slug":"maths-and-arithmetic","title":"Maths, and the arithmetic problem","url":"https://addaly.com/learn/ai-for-students/maths-and-arithmetic","position":29,"of":73,"minutes":9,"answer":"Digits are tokenised without place value, so arithmetic degrades with length while looking perfectly normal: take the method from the model and the number from SymPy, Maxima or a code interpreter.","keyIdea":"Digits are tokenised without place value, so arithmetic degrades with length while looking perfectly normal: take the method from the model and the number from SymPy, Maxima or a code interpreter."},"body":{"format":"markdown","text":"## Why a system that discusses topology gets 47 times 63 wrong\n\nIt 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.\n\nNumbers 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.\n\nSo 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.\n\n> Figure: Accuracy falls with digit count, and the tone does not. Bars (% of answers exactly right): 2 digits times 2 digits 99; 3 by 3 92; 4 by 4 74; 5 by 5 46; Six-digit long division 31. The shape rather than a benchmark. Numbers are split into tokens that do not respect place value, so there is no column of digits to carry between and the model produces the most plausible continuation, which for small familiar numbers is nearly always right. The failures look exactly like the successes: a confident wrong number in the middle of correct working. Take the method from the model and the number from SymPy, Maxima or a code interpreter.\n\n## The rule\n\n**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.\n\nThis is not a limitation to work around with clever prompting. It is what the machine is.\n\n## Free tools that do the arithmetic properly\n\n- **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.\n- **SymPy**: free Python library, symbolic algebra, exact arithmetic, calculus, equation solving. Runs in Google Colab with no install.\n- **Maxima**: free, mature computer algebra system, decades of use.\n- **wxMaxima** for a graphical front end.\n- **GeoGebra** and **Desmos**: free, browser-based, excellent for graphing, geometry and checking that a function does what you think.\n- **GNU Octave**: free and runs most MATLAB code, which matters because job ads name MATLAB.\n- **SageMath**: free, covers a lot of Mathematica's territory.\n\nThe pattern that works: ask the model *which* method, ask it for the SymPy or Maxima code, run it, compare with your own answer.\n\n```python\nfrom sympy import *\nx = symbols('x')\n# check an integral you did by hand\nintegrate(x*exp(x), x)          # x*exp(x) - exp(x)\n# and confirm by differentiating back\ndiff(x*exp(x) - exp(x), x)      # x*exp(x)\n```\n\nThat is thirty seconds and it is a real check, because SymPy is not guessing.\n\n## Where models are strong in maths\n\nDo not overcorrect. There is a lot they do well:\n\n- **Explaining what a definition means** and why it is stated that way.\n- **Giving the intuition** behind a theorem before the formal statement.\n- **Suggesting an approach** when you are stuck: *this looks like a substitution problem, try u equals the inside of the bracket*.\n- **Generating practice problems** of a specified type, and hint ladders for them.\n- **Spotting the step where your working goes wrong**, which they do well and which is the hint-ladder move from Module 3.\n- **Translating between notations**, when your lecturer uses one and the textbook another.\n\nThat is most of what a maths tutor does apart from the arithmetic.\n\n## Word problems and the setup\n\nThe 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:\n\n> Turn this into equations. Define every variable with units. Do not solve it.\n\nThen 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.\n\n## Checking without any tool\n\nWhen you have no computer, three checks cost nothing:\n\n**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.\n\n**Units.** Carry them through every line. If they do not come out as they should, the error is above that line.\n\n**Substitute back.** Solved for x? Put it in the original equation. This catches most algebra errors and takes fifteen seconds.\n\nThese 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."},"check":{"question":"A student uses a model to solve a physics problem. The setup and reasoning are correct, but the final figure is 2.94 kJ where the true answer is 29.4 kJ. Which check would have caught this fastest?","options":[{"text":"Asking the model to double-check its arithmetic, since re-examination usually surfaces slips.","correct":false,"tempting":"Re-runs the same estimation process that produced the slip; the model has no column-arithmetic procedure to re-execute more carefully."},{"text":"A rough order-of-magnitude estimate made before looking at the answer.","correct":true},{"text":"Carrying units through every line, since a factor-of-ten error changes the dimensions of the result.","correct":false,"tempting":"Units are an excellent check and would catch a joules-versus-watts error, but a factor of ten leaves the dimensions entirely intact."},{"text":"Asking the same question again in a fresh conversation and comparing the two answers.","correct":false,"tempting":"Cold re-asking tests stability of knowledge, but arithmetic errors are not stable in a useful way: you may simply get a second, differently wrong number."}]},"prev":{"slug":"proving-it-to-yourself","title":"Proving it to yourself","url":"https://addaly.com/learn/ai-for-students/proving-it-to-yourself"},"next":{"slug":"proofs-and-reasoning","title":"Proofs, and the step that looks fine","url":"https://addaly.com/learn/ai-for-students/proofs-and-reasoning"},"authorship":"Founder and series editor: Dr Mustafa Mun. Written by AI models to a written standard. Exam keys checked blind by a second model: 3,665 of 3,666 agreed. Academic review invited.","review":{"status":"Draft, open for academic review"},"cite":"Addaly (2026) 'Maths, and the arithmetic problem', in AI for Students. Edited by M. Mun. Addaly. Available at: https://addaly.com/learn/ai-for-students/maths-and-arithmetic (Accessed: 13 October 2026).","licence":{"name":"CC BY-NC-SA 4.0","url":"https://creativecommons.org/licenses/by-nc-sa/4.0/"},"reader":{"schema":1,"title":"Maths, and the arithmetic problem","courseTitle":"AI for Students","module":{"index":4,"slug":"subject-by-subject","title":"Subject by subject"},"answer":null,"keyIdea":"Digits are tokenised without place value, so arithmetic degrades with length while looking perfectly normal: take the method from the model and the number from SymPy, Maxima or a code interpreter.","hero":{"kind":"figure","figure":{"t":"figure","label":"Figure 4.1","figure":{"type":"bars","unit":"% of answers exactly right","title":"Accuracy falls with digit count, and the tone does not","series":[{"label":"2 digits times 2 digits","value":99},{"label":"3 by 3","value":92},{"label":"4 by 4","value":74},{"label":"5 by 5","value":46,"highlight":true},{"label":"Six-digit long division","value":31,"highlight":true}],"caption":"The shape rather than a benchmark. Numbers are split into tokens that do not respect place value, so there is no column of digits to carry between and the model produces the most plausible continuation, which for small familiar numbers is nearly always right. The failures look exactly like the successes: a confident wrong number in the middle of correct working. Take the method from the model and the number from SymPy, Maxima or a code interpreter."},"alt":"Figure: Accuracy falls with digit count, and the tone does not. Bars (% of answers exactly right): 2 digits times 2 digits 99; 3 by 3 92; 4 by 4 74; 5 by 5 46; Six-digit long division 31. The shape rather than a benchmark. Numbers are split into tokens that do not respect place value, so there is no column of digits to carry between and the model produces the most plausible continuation, which for small familiar numbers is nearly always right. The failures look exactly like the successes: a confident wrong number in the middle of correct working. Take the method from the model and the number from SymPy, Maxima or a code interpreter."}},"blocks":[{"t":"h2","id":"why-a-system-that-discusses-topology-gets-47-times-63-wrong","runs":[{"text":"Why a system that discusses topology gets 47 times 63 wrong"}]},{"t":"p","runs":[{"text":"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."}]},{"t":"p","runs":[{"text":"Numbers get split into tokens in ways that do not respect place value. Depending on the tokenizer, 4763 might be "},{"text":"47","code":true},{"text":" and "},{"text":"63","code":true},{"text":", or "},{"text":"4","code":true},{"text":", "},{"text":"76","code":true},{"text":", "},{"text":"3","code":true},{"text":". 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."}]},{"t":"p","runs":[{"text":"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."}]},{"t":"h2","id":"the-rule","runs":[{"text":"The rule"}]},{"t":"p","runs":[{"text":"Never accept a number a model computed if the number matters.","b":true},{"text":" The concept, the method, the setup, the interpretation: all fine. The arithmetic: check it."}]},{"t":"p","runs":[{"text":"This is not a limitation to work around with clever prompting. It is what the machine is."}]},{"t":"h2","id":"free-tools-that-do-the-arithmetic-properly","runs":[{"text":"Free tools that do the arithmetic properly"}]},{"t":"ul","items":[[{"text":"A code interpreter","b":true},{"text":", 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."}],[{"text":"SymPy","b":true},{"text":": free Python library, symbolic algebra, exact arithmetic, calculus, equation solving. Runs in Google Colab with no install."}],[{"text":"Maxima","b":true},{"text":": free, mature computer algebra system, decades of use."}],[{"text":"wxMaxima","b":true},{"text":" for a graphical front end."}],[{"text":"GeoGebra","b":true},{"text":" and "},{"text":"Desmos","b":true},{"text":": free, browser-based, excellent for graphing, geometry and checking that a function does what you think."}],[{"text":"GNU Octave","b":true},{"text":": free and runs most MATLAB code, which matters because job ads name MATLAB."}],[{"text":"SageMath","b":true},{"text":": free, covers a lot of Mathematica’s territory."}]]},{"t":"p","runs":[{"text":"The pattern that works: ask the model "},{"text":"which","i":true},{"text":" method, ask it for the SymPy or Maxima code, run it, compare with your own answer."}]},{"t":"code","lang":"python","label":"Python","text":"from sympy import *\nx = symbols('x')\n# check an integral you did by hand\nintegrate(x*exp(x), x)          # x*exp(x) - exp(x)\n# and confirm by differentiating back\ndiff(x*exp(x) - exp(x), x)      # x*exp(x)"},{"t":"p","runs":[{"text":"That is thirty seconds and it is a real check, because SymPy is not guessing."}]},{"t":"h2","id":"where-models-are-strong-in-maths","runs":[{"text":"Where models are strong in maths"}]},{"t":"p","runs":[{"text":"Do not overcorrect. There is a lot they do well:"}]},{"t":"ul","items":[[{"text":"Explaining what a definition means","b":true},{"text":" and why it is stated that way."}],[{"text":"Giving the intuition","b":true},{"text":" behind a theorem before the formal statement."}],[{"text":"Suggesting an approach","b":true},{"text":" when you are stuck: "},{"text":"this looks like a substitution problem, try u equals the inside of the bracket","i":true},{"text":"."}],[{"text":"Generating practice problems","b":true},{"text":" of a specified type, and hint ladders for them."}],[{"text":"Spotting the step where your working goes wrong","b":true},{"text":", which they do well and which is the hint-ladder move from Module 3."}],[{"text":"Translating between notations","b":true},{"text":", when your lecturer uses one and the textbook another."}]]},{"t":"p","runs":[{"text":"That is most of what a maths tutor does apart from the arithmetic."}]},{"t":"h2","id":"word-problems-and-the-setup","runs":[{"text":"Word problems and the setup"}]},{"t":"p","runs":[{"text":"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:"}]},{"t":"quote","runs":[{"text":"Turn this into equations. Define every variable with units. Do not solve it."}]},{"t":"p","runs":[{"text":"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."}]},{"t":"h2","id":"checking-without-any-tool","runs":[{"text":"Checking without any tool"}]},{"t":"p","runs":[{"text":"When you have no computer, three checks cost nothing:"}]},{"t":"p","runs":[{"text":"Estimate first.","b":true},{"text":" 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."}]},{"t":"p","runs":[{"text":"Units.","b":true},{"text":" Carry them through every line. If they do not come out as they should, the error is above that line."}]},{"t":"p","runs":[{"text":"Substitute back.","b":true},{"text":" Solved for x? Put it in the original equation. This catches most algebra errors and takes fifteen seconds."}]},{"t":"p","runs":[{"text":"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."}]}],"check":{"question":[{"text":"A student uses a model to solve a physics problem. The setup and reasoning are correct, but the final figure is 2.94 kJ where the true answer is 29.4 kJ. Which check would have caught this fastest?"}],"options":[{"text":[{"text":"Asking the model to double-check its arithmetic, since re-examination usually surfaces slips."}],"correct":false,"tempting":[{"text":"Re-runs the same estimation process that produced the slip; the model has no column-arithmetic procedure to re-execute more carefully."}]},{"text":[{"text":"A rough order-of-magnitude estimate made before looking at the answer."}],"correct":true,"tempting":null},{"text":[{"text":"Carrying units through every line, since a factor-of-ten error changes the dimensions of the result."}],"correct":false,"tempting":[{"text":"Units are an excellent check and would catch a joules-versus-watts error, but a factor of ten leaves the dimensions entirely intact."}]},{"text":[{"text":"Asking the same question again in a fresh conversation and comparing the two answers."}],"correct":false,"tempting":[{"text":"Cold re-asking tests stability of knowledge, but arithmetic errors are not stable in a useful way: you may simply get a second, differently wrong number."}]}],"order":[0,2,3,1],"why":null},"prev":{"slug":"proving-it-to-yourself","title":"Proving it to yourself","minutes":8},"next":{"slug":"proofs-and-reasoning","title":"Proofs, and the step that looks fine","minutes":8}}}