Maths IA · Algebra and functions · topic 2 of 40

How does the base change a number's length, and which base is best for storing information? · IB Maths IA idea: guide, sample graph and step-by-step plan

gentle start, rich if you model carefully

Computers use base 2. We use base 10. Babylonians used base 60. Which one is best? You're going to find that the answer depends on what you count as cost.

Difficulty●○○ Gentle
KitCalculator or spreadsheet; software to graph
TimeAbout 12 to 15 hours
FitsAA and AI, SL and HL
  • ✓ you like logarithms and want to see a real use
  • ✓ you enjoy asking what 'best' means
  • ✓ you want maths you can graph and test

The big idea

← cost is the whole question

A number N written in base b needs about log N ÷ log b digits. A bigger base means fewer digits but more symbols per digit, so you need a model of cost. Choose one, such as the cost per digit being proportional to b, and find the base that minimises the total cost, using calculus. Then change the model and watch the answer move. A strong IA makes the modelling choice itself the point. At HL, rely on calculus with logarithms: differentiate, justify the minimum and test the model's sensitivity, rather than reading the minimum off a graph.

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 →

Choose your own cost model

Cost per digit proportional to b, to log b, or to b plus a fixed overhead. Decide which is realistic and defend it.

→ Your function changes and so does the best base. The evaluation compares what each assumption means in practice.
Prove the optimum

Use calculus on the continuous model, justify the minimum, then compare the best whole number.

→ The key step is differentiating with logarithms and checking the second derivative: the HL-level core.
Test a sensitivity

Vary one parameter in your cost model and see how far the best base moves.

→ You tell the reader how much to trust the answer, which is strong reflection.
Beyond whole bases

Allow negative or fractional bases, or balanced ternary, and decide whether they help.

→ A stretch, with new representations. You need to define clearly what a digit is.
Real storage

Compare the model with real systems: binary in a chip, decimal on a bank slip, hexadecimal in colours.

→ The model is tested against reality, with the limits of the model discussed.
Change the context

Phone numbers, barcodes, or the Babylonian system.

→ A personal starting point. The maths stays the same.

Your Maths IA research question: run the check

define 'best' first

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

  • Defines exactly what 'best' means: the quantity to be minimised.
  • States the cost model and the assumption behind it, and says what you will vary.
  • Includes a derivation with logarithms that you do yourself, not a result from a website.
  • Says what you will compare the answer with, such as other whole-number bases or a real system.
  • Is narrow enough for 12 to 20 pages: one cost idea developed, with a second for comparison.

Get your RQ checked · S$29

The picture to draw

The same number written in three bases. Draw your own, with your own N, and show how the length and the symbol count trade off.

base 21111101000base 101000base 163E810 digits, 2 states each4 digits, 10 states each3 digits, 16 states eachwhat counts as ‘cost’? that is YOUR modelling decisionN = 1000

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.

Introduction and aim~1.5 pages
A real question about storing numbers, what you mean by best, and what you predict.
Digits and logarithms~2.5 pages
How the number of digits depends on the base, with your own derivation using logarithms and the change-of-base rule.
A cost model~2 pages
Your model of cost, the assumptions behind it and why you chose it. A table of whole-number bases for one N.
Optimising the base~3 pages
The calculus: derivative, critical point, second-derivative test, with the algebra shown and the result checked on a graph.
Whole numbers and sensitivity~2.5 pages
The best whole-number base, how flat or sharp the minimum is, and what happens when you change the model.
A real system~1.5 pages
Compare the model with binary, decimal or hexadecimal in use, and say what the model leaves out.
Reflection and evaluation~1.5 pages
Which assumption would change the answer most, and what you would test next.
Conclusion~1 page
Answer your aim, with the model's limits stated.
Referencesnot counted
Sources for the cost idea and any real-system facts.

What your Maths IA graph could look like

state the assumption on the graph

Made-up model, to show you the form of a strong graph, not the answer. Yours will use your own cost model and N. example only

234567891011128090100110120130140Base bTotal cost C(b) (cost units), for N = 10⁶assumed model: cost per digit b + 5whole-number basesthe curve is flat near its lowest point:does the exact base matter much?ILLUSTRATIVE · MADE-UP DATA
  • Axis labels name the quantity and say what N is, and that the cost units are arbitrary.
  • The curve is the continuous model, the points are the bases that exist. Say why they differ.
  • The model assumption is written in the legend, so the reader sees what was assumed.
  • The y scale starts at 80 to show the shape of the minimum. Say so in the text.
  • Another graph to consider: plot the best base against the overhead you assumed, to show how sensitive the answer is.

Your game plan

try a second cost model!

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

0 of 10 done

  1. Explore and predictWrite the same number in several bases. Predict which base gives the lowest cost and what 'cost' should mean.
  2. Trial run (1 lesson)Convert a few numbers by hand and count digits. Check the digit-count formula, including the rounding up.
  3. Derive the digit formulaShow why a number needs about log N / log b digits, using the change-of-base rule. Be exact about the 'about'.
  4. Choose a cost modelWrite down what it assumes and why it is realistic. Keep a second model in reserve for the comparison.
  5. Tabulate and graphCompute the cost for whole-number bases. Graph the continuous version to see the shape.
  6. OptimiseDifferentiate and solve for the minimum with the working shown. Check that it is a minimum. Compare with the table.
  7. Test the assumptionsChange the cost model and see how the best base moves. Which change matters most?
  8. Compare with realityPick a real system and ask how well the model describes it, and why it might not.
  9. ReflectWhat did you assume that you could not test? What would a different model say?
  10. Write, cut, checkWrite in the layout order and keep one worked derivation. Run the checklist, then get your question checked before your teacher's draft.

Maths you'll need

Traps to avoid in this Maths IA

  • Using a cost model without saying why. The model is the heart of the project, so justify it.
  • Finding the best base by graph or table only, with no calculus, at HL.
  • Forgetting that the number of digits must be a whole number, so the formula needs rounding up.
  • Treating base e as 'the answer' without saying that e cannot be a real base here.
  • Not checking that the stationary point is a minimum.
  • Software output for the logarithm steps with no derivation.
  • A long history of number systems with no maths in it.

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.

Get your RQ checked · S$29