mathematics

Googolplex

Googolplex 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 Googolplex rather than just read about it. In short: A googolplex is the large number 1010100, that is, 10 raised to the power of a googol. If written out in ordinary decimal notation, it would be 1 followed by a googol (10100) zeroes‍—‍‌a physically impossible number to write explicitly.

Key takeaways

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

Reference excerpt

A googolplex is the large number 1010100, that is, 10 raised to the power of a googol. If written out in ordinary decimal notation, it would be 1 followed by a googol (10100) zeroes‍—‍‌a physically impossible number to write explicitly.

History In 1920, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term googol, which is 10100, and then proposed the further term googolplex to be "one, followed by writing zeroes until you get tired". Kasner decided to adopt a more formal definition because "different people get tired at different times and it would never do to have Carnera [be] a better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer". It thus became standardized to 10(10100), which is usually written as 1010100 using the conventional interpretation for serial exponentiation.

Size A typical book can be printed with one million zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires 1094 such books to print all the zeros of a googolplex (that is, printing a googol zeros). If each book had a mass of 100 grams, all of them would have a total mass of 1093 kilograms. In comparison, Earth's mass is 5.97×1024 kilograms, the mass of the Milky Way galaxy is estimated at 1.8×1042 kilograms, and the total mass of all the stars in the observable universe is estimated at 2×1052 kg. Thus, the mass of all such books required to write out a googolplex would be vastly greater than the mass of the observable universe by a factor of roughly 5×1040.

In the physical universe In the PBS science program Cosmos: A Personal Voyage, Episode 9: "The Lives of the Stars", astronomer and television personality Carl Sagan estimated that writing a googolplex in full decimal form (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than is available in the known universe. Sagan gave an example that if the entire volume of the observable universe is filled with fine dust particles roughly 1.5 micrometers in size (0.0015 millimeters), then the number of different combinations in which the particles could be arranged and numbered would be about one googolplex. 1097 is a high estimate of the elementary particles existing in the visible universe (not including dark matter), mostly photons and other massless force carriers.

Mod n The residues (mod n) of a googolplex, starting with mod 1, are:

0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 1, 10, 9, 0, 4, 12, 13, 16, 0, 16, 10, 4, 24, 10, 5, 0, 1, 18, 25, 28, 10, 28, 16, 0, 1, 4, 24, 12, 10, 36, 9, 16, 4, 0, ... (sequence A067007 in the OEIS) This sequence is the same as the sequence of residues (mod n) of a googol up until the 17th position.

Worked examples

Example 1 — a first encounter with Googolplex

Start with the simplest possible case. Write down what Googolplex 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 Googolplex 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 Googolplex 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 Googolplex

In research: Googolplex 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 Googolplex 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: Googolplex is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1920s neologisms, Integers, Large integers, so understanding it makes those chapters shorter.
In everyday life: Look for Googolplex 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 Googolplex in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Googolplex 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 Googolplex out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Googolplex in simple terms?

Why does Googolplex 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 Googolplex?

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 Googolplex.