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Maths IA · Modelling · topic 74 of 97

Is Moore's law over? Fitting exponential growth · IB Maths IA idea: guide, sample graph and step-by-step plan

straight line = constant doubling time

Every two years or so, the number of transistors on a chip is said to double. That is a claim you can test with a ruler and a log scale, and you can look for the point where it stops being true.

Difficulty●●○ Medium
KitA spreadsheet, a table of chip transistor counts with a source, your GDC
TimeAbout 12 to 15 hours
FitsAA SL2 of 3AA HL2 of 3AI SL2 of 3AI HL3 of 3
  • ✓ you like a model you can check with one straight line
  • ✓ you want to ask whether a famous 'law' still holds
  • ✓ you are happy to argue from messy real data

The big idea

Exponential growth looks like a curve on ordinary axes, but taking logarithms turns it into a straight line. The slope of that line gives the doubling time, and the scatter around it tells you how good the model is. Collect transistor counts for real chips, linearise, fit, and read off the doubling time with its uncertainty. The marks are in the questions that come after the fit: do the residuals show a pattern, did the doubling time change, and does a piecewise or logistic model describe the data better than one straight line? You also have to decide what counts as one data point, because a chip, a package and a graphics card are not the same thing.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Linearise and fit

Take logarithms of the counts, fit a straight line against the year, and convert the slope to a doubling time.

→ An exponential model becomes a line you can test. Explain the conversion from slope to doubling time in your own algebra.
Check the residuals

Plot the residuals of the log fit against the year and look for runs of the same sign.

→ A high R² can hide a curve. A pattern in the residuals is evidence that one exponential is not enough.
Split the data

Fit two lines, one before a year you choose and one after, and compare the doubling times.

→ A piecewise model. You must justify the break year with the data or with history, not choose it to get the answer you want.
Try a logistic model

Fit growth that levels off at a limit and compare it with the straight-line model.

→ A model with a ceiling. Say what the ceiling would be physically and how you judge which model is better.
Change the quantity

Repeat with another measure, such as transistors per square millimetre, clock speed or the price of a million operations.

→ The same method on a different variable. Different quantities may not follow the same law, which is part of the finding.
Compare with the original claim

Use Moore's own published rates as a prediction and score it against the data that came after.

→ A forecast tested against evidence, including how the claimed rate was itself revised.

Your Maths IA research question: run the check

which chips? which years?

Write your own question first. Then make sure it ticks every box:

  • Names the quantity (transistors per chip, or another measure) and the years it covers.
  • Says where the data comes from and how you decide what counts as one data point.
  • Asks for a number you can find and defend, such as a doubling time or a change in it, not just 'does it fit'.
  • Compares at least two models, or two time periods, so there is something to evaluate.
  • States how you will judge the fit: residuals, not only R².
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The model

Count against year on a log axis. If growth is exponential the points sit near a line, and the line's slope gives the doubling time. Draw yours with your own data, and mark any point that leaves the line.

T years×2year ttransistors N (log scale)log2 N = t/T + cstraight on a log axis= exponential growthslope = 1/T, so T = 1/slopeone point far from the line:what is it? why?

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Aim and context~1 page
What Moore's law claims, why you test it, and what you expect the doubling time to be before fitting.
The data~1.5 pages
Where each count comes from, what is included, what is left out and why. A table with sources.
From exponential to a line~2 pages
Every symbol defined. N = N₀·2^(t/T), logarithms, the slope and the doubling time derived.
Fitting and residuals~2 pages
The fit, its doubling time, and a residual plot that you read in words.
Is it slowing down?~3 pages
A second model (two periods, or logistic), compared with the first for fit and for sense.
Conclusion~1 page
Back to your aim: the doubling time, whether it changed, with numbers.
Evaluation~2 pages, the biggest part!
What counts as one chip, which data are missing, how much one point can move the result. Say which limitation matters most.
Bibliographynot counted
The source of every count and every claim about the law.

What your Maths IA graph could look like

residuals tell the real story

Made-up counts to show the form of a strong graph, not the answer: transistors per chip on a log axis with a fitted line. Yours will look different. example only

19701980199020002010202024681012Yearlog₁₀ of transistors per chipfitted line: doubling every 2 yearsyour counts (made-up)slope = 0.15 per year:T = log₁₀2 / slope = 2 yearsILLUSTRATIVE · MADE-UP DATA
  • Plot the logarithm of the count (or use a log axis), so exponential growth shows as a line.
  • Both axes carry units: year, and the count on a labelled log scale.
  • Show the fitted line and the points together, and add a residual plot beside it.
  • Calculate the doubling time from the slope, then state its uncertainty.
  • Another graph to consider: the doubling time fitted in a moving window of ten years.

The maths in action

worked through, not sketched!

The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.

Left: a log-scale plot of 14 chip counts rising along a straight fitted line from 1971 to 2022, with a steeper dashed line for one doubling a year. Right: bars of the residuals against year, mostly small, with a large positive bar in 2022.
(a) Transistors per chip for 14 processors on a log scale: the fit doubles every 2.11 years (R² = 0.995), against Moore's first claim of one year. (b) Residuals, in doublings, are within 1.3 of the line; the largest is the 2022 chip.Source: Data: published transistor counts for 14 chips (Intel, Apple), values to be checked against manufacturers' figures; fit computed here.

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 9 done

  1. Read and predictFind out what Moore actually claimed and how the claim was later revised. Before any fitting, write down the doubling time you expect.
  2. Collect the dataGather counts for at least 15 real chips across several decades, with a source for each. Decide what counts as one data point.
  3. LineariseTake logarithms, set up N = N₀·2^(t/T) and show how the slope of log N against t gives the doubling time.
  4. Trial run (1 lesson)Fit a line to five points by hand, then check it against your GDC or spreadsheet. Fix any mismatch in the units of time or the base of the logarithm.
  5. Fit and read the residualsFit all the points, plot the residuals against the year and describe any pattern in words.
  6. Test a second modelFit two periods, or a logistic curve, and compare with the single line. Decide how you will judge which is better.
  7. Go further, with a reasonFind the uncertainty in the doubling time, or derive the least squares line yourself. HL: this is where the rigour for criterion E comes from.
  8. Reflect as you goAfter every result, write one line on what it says about the real industry and what it assumes. These lines become your evaluation.
  9. Cut, check, then sendCheck every graph has labelled axes and units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.

The maths you'll use

  • Exponential models – Write and relate the doubling time to the growth rate .
  • Logarithms and linearisation – Take so that the slope of a line fitted to the logs is .
  • Regression and residuals – Fit by least squares, compute , and plot the residuals to look for runs that a single line misses.
  • Piecewise and logistic models – Fit two periods separately, or , and compare the models fairly.
  • Uncertainty in the doubling time – Use the standard error of the slope to give a range for , and see how one extreme point shifts it.

Key equations

Syllabus links

  • 1.5Exponents and an introduction to logarithms
  • 1.7Rational exponents and laws of logarithms
  • 2.9Exponential and logarithmic functions
  • 4.4Correlation and the regression line of y on x

Read up first

Data and sources to start from

Transistor counts for named chips from manufacturer data sheets or a published compilation; record the chip, year and source for every row.

References to cite

  1. Moore, G. E. (1965). Cramming more components onto integrated circuits. Electronics, 38(8), 114-117.
  2. Moore, G. E. (1975). Progress in digital integrated electronics. In Technical Digest of the International Electron Devices Meeting (pp. 11-13). IEEE.
  3. Our World in Data. (n.d.). Moore's law: The number of transistors on microchips doubles every two years. https://ourworldindata.org/moores-law

Cite what you actually use, in your own format, and say where each number came from.

Traps to avoid in this Maths IA

R² is not proof
  • Using only chips from one company or one type, then calling it a law for the whole industry.
  • Quoting R² as proof. A line can have a high R² and still miss a bend.
  • Fitting the exponential on ordinary axes and then fitting a line on the logs, as if they were the same least squares problem.
  • Mixing up the base of the logarithm when converting the slope to a doubling time.
  • Choosing the break year after looking at which one gives the neatest answer.
  • Counting a package of several dies as one chip without saying so.
  • Extrapolating far beyond your data and stating the result as a prediction.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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