Who Can Name the Bigger Number



Week 10 English for Maths II

Task 1: Supposing you take part in the following contest what number would you write? Using standard math notation, English words, or both, name a single whole number–not an infinity–on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature

Task 2: Listen and take notes. Then answer the questions:

• How did the author introduce the topic of Big Numbers?

• Why does the author claim that a biggest number contest is pointless when contestants take turns?

• Why did the author declare the girl’s victory without bothering to count the 9’s on the cards?

Task 3: What is the viewpoint of the writer, who is the audience and how do you know?

Who Can Name the Bigger Number? Scott Aaronson, 1999

In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces “Eighty-three!” The second, mightily impressed, replies “You win.”

A biggest number contest is clearly pointless when the contestants take turns. But what if the contestants write down their numbers simultaneously, neither aware of the other’s? To introduce a talk on “Big Numbers,” I invite two audience volunteers to try exactly this. I tell them the rules:

You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number–not an infinity–on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.

So contestants can’t say “the number of sand grains in the Sahara,” because sand drifts in and out of the Sahara regularly. Nor can they say ”my opponent’s number plus one,” or ”the biggest number anyone’s ever thought of plus one”–again, these are ill-defined, given what our reasonable mathematician has available. Within the rules, the contestant who names the bigger number wins.

Are you ready? Get set. Go.

The contest’s results are never quite what I’d hope. Once, a seventh-grade boy filled his card with a string of successive 9’s. Like many other big-number tyros, he sought to maximize his number by stuffing a 9 into every place value. Had he chosen easy-to-write 1’s rather than curvaceous 9’s, his number could have been millions of times bigger. He still would been decimated, though, by the girl he was up against, who wrote a string of 9’s followed by the superscript 999. Aha! An exponential: a number multiplied by itself 999 times. Noticing this innovation, I declared the girl’s victory without bothering to count the 9’s on the cards.

And yet the girl’s number could have been much bigger still, had she stacked the mighty exponential more than once. Take 999, for example. This behemoth, equal to 9387,420,489, has 369,693,100 digits. By comparison, the number of elementary particles in the observable universe has a meagre 85 digits, give or take. Three 9’s, when stacked exponentially, already lift us incomprehensibly beyond all the matter we can observe–by a factor of about 10369,693,015. And we’ve said nothing of 9999 or 99999.

Place value, exponentials, stacked exponentials: each can express boundlessly big numbers, and in this sense they’re all equivalent. But the notational systems differ dramatically in the numbers they can express concisely. That’s what the fifteen-second time limit illustrates. It takes the same amount of time to write 9999, 9999, and 9999 –yet the first number is quotidian, the second astronomical, and the third hyper-mega astronomical. The key to the biggest number contest is not swift penmanship, but rather a potent paradigm for concisely capturing the gargantuan.

Such paradigms are historical rarities. We find a flurry in antiquity, another flurry in the twentieth century, and nothing much in between. But when a new way to express big numbers concisely does emerge, it’s often a by-product of a major scientific revolution: systematized mathematics, formal logic, computer science. Revolutions this momentous, as any Kuhnian could tell you, only happen under the right social conditions. Thus is the story of big numbers a story of human progress.

Task 4: Answer the following comprehension questions:

• Why do you think infinities were not allowed in the contest?

• Explain the following claim. “The key to the biggest number contest is not swift penmanship, but rather a potent paradigm for concisely capturing the gargantuan.”

• Define the bold words. Are they formal or informal? Are they technical, academic or colloquial?

Task 5: A paragraph deals with one topic and has a topic sentence which introduces the main idea of the whole paragraph. Are there topic sentences in every paragraph? Are paragraphs in this text well-structured?

Coherent order: Paragraphs may be organised according to a sequence of time (chronological order), space (describing something from top to bottom or foreground to background) or arranging information in order of importance. Alternatively, you might present the most general information first and then move on to focus on the more detailed, specific information.

Task 6: A paragraph has good coherence when ideas are arranged in a logical order. Is this the case for this text?

Cohesion: There is cohesion when sentences within a paragraph are linked together. There are various ways of linking one sentence to another:

• repetition of important words

• substitution of pronouns e.g. ‘this’, ‘it’, ‘these’

• substitution by synonyms (words with nearly the same meaning)

• using linking words or phrases which show the relationship between ideas, e.g. ‘however’ indicates a contrast, ‘in addition’

Task 7: Find examples in the text that indicate paragraph cohesion.

More on Large Numbers

• Is 109 a milliard or a billion?

• How do you say 1015?

• How many naming systems are there? Which one are you using here?

Task 8: Read the following text from Wikipaedia and complete the missing words. throughout, due to, particularly, including, whereas,

The long and short scales are two of several different large-number naming systems used ……………….the world for integer powers of ten. Many countries, ………………..most in continental Europe, use the long scale ……………… most English-speaking countries and Arabic-speaking countries use the short scale. In all such countries, the number names are translated into the local language, but retain a name similarity ………….. shared etymology. Some languages, ……………. in East Asia and South Asia, have large number naming systems that are different from the long and short scales.

Task 9: Complete the missing names from the table:

|Name |Short scale |Long scale |

| |(U.S. and |(continental Europe, |

| |modern British) |older British) |

| | | |

|Million |106 |106 |

|Milliard |  |109 |

| |109 |1012 |

| |1012 |1018 |

| |1015 |1024 |

| |1018 |1030 |

| |1021 |1036 |

| |1024 |1042 |

| |1027 |1048 |

| |1030 |1054 |

| |1033 |1060 |

| |1036 |1066 |

| |1039 |1072 |

| |1042 |1078 |

| |1045 |1084 |

| |1048 |1090 |

| |1051 |1096 |

| |1054 |10102 |

| |1057 |10108 |

| |1060 |10114 |

| |1063 |10120 |

| |10303 |10600 |

• How do you spell Google?

• How many zeroes are in a googol?

• How many zeroes are in a googolplex?

Task 10: Read the text and complete the gaps using the appropriate form of the words in brackets.

The googol family.

The names googol and googolplex ………………….(invent) by Edward Kasner's nephew, Milton Sirotta. Kasner and Newman's 1940 book ……………. (introduce) in Mathematics and the Imagination, in the following passage:

The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who …………(ask) to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he ……………..(suggest) "googol" he ………………(give) a name for a still larger number: "Googolplex". A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It ……………..(suggest) that a googolplex should be 1, ………….(follow) by writing zeros until you got tired. The googolplex is a specific finite number, equal to 1 with a googol zeros after it.

|Value |Name |Authority |

|10100 |Googol |Kasner and Newman, dictionaries (see above) |

|10googol = [pic] |Googolplex |Kasner and Newman, dictionaries (see above) |

Conway and Guy have suggested that N-plex …………..(use) as a name for 10N. This gives rise to the name googolplexplex for 10googolplex. This number (ten to the power of a googolplex) is also known as a googolduplex. Conway and Guy have proposed that N-minex be used as a name for 10−N, giving rise to the name googolminex for the reciprocal of a googolplex. None of these names are in wide use, nor ……….any currently …………….(find) in dictionaries.

Task 11: Suggested homework/Presentation topic: Search the Web and write a short essay/ make a short presentation of other large numbers used in mathematics and physics such as:

• Avogadro's number

• Graham's number

• Skewes' number

• Steinhaus–Moser notation

NOTICE:

Presentations will be taking place in week 13. You will be presenting for five minutes on a Math-related topic of your choice.

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