mathematics

Mathematics Made Difficult

Mathematics Made Difficult is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Mathematics Made Difficult rather than just read about it. In short: Mathematics Made Difficult is a book by Carl E. Linderholm that uses advanced mathematical methods to prove results normally shown using elementary proofs.

Key takeaways

Mathematics Made Difficult belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mathematics Made Difficult to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mathematics Made Difficult from memory before moving on to harder problems.

Reference excerpt

Mathematics Made Difficult is a book by Carl E. Linderholm that uses advanced mathematical methods to prove results normally shown using elementary proofs. Although the aim is largely satirical, it also shows the non-trivial mathematics behind operations normally considered obvious, such as numbering, counting, and factoring integers. Linderholm discusses these seemingly obvious ideas using concepts like categories and monoids. As an example, the proof that 2 is a prime number starts:

It is easily seen that the only numbers between 0 and 2, including 0 but excluding 2, are 0 and 1. Thus the remainder left by any number on division by 2 is either 0 or 1. Hence the quotient ring Z/2Z, where 2Z is the ideal in Z generated by 2, has only the elements [0] and [1], where these are the images of 0 and 1 under the canonical quotient map. Since [1] must be the unit of this ring, every element of this ring except [0] is a unit, and the ring is a field ...

References

Worked examples

Example 1 — a first encounter with Mathematics Made Difficult

Start with the simplest possible case. Write down what Mathematics Made Difficult claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mathematics Made Difficult before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mathematics Made Difficult ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mathematics Made Difficult

In research: Mathematics Made Difficult appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mathematics Made Difficult in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mathematics Made Difficult is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1972 non-fiction books, Mathematics books, Satirical books, so understanding it makes those chapters shorter.
In everyday life: Look for Mathematics Made Difficult outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Mathematics Made Difficult in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mathematics Made Difficult means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mathematics Made Difficult out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mathematics Made Difficult in simple terms?

Mathematics Made Difficult is a book by Carl E. Linderholm that uses advanced mathematical methods to prove results normally shown using elementary proofs.

Why does Mathematics Made Difficult matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mathematics Made Difficult?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mathematics Made Difficult.