How Did We Find Out About Numbers?

This book would be an excellent introduction to the world of numerals and indicating to students that there is a history to even numbers. This book connects numbers to various historical ages as well as people. This book would be an excellent book to analyze during studying numerals as well as a simply exploration further into the importance of the origin of things we study. I found that this book is easy to understand, but gives information as a textbook might. I would recommend this book to any middle grades classroom!!

Simplified discussion of ancient number systems. Numbers systems that insured that no number contained more than one instance of a numeral should have a special name. There is no discussion of binary numbers, though there is discussion the Babylonian base 60 and other bases.

This subject is not really appropriate to the series, I guess Asimov just couldn't resist a subject so quick, easy, and relatively simple.

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201808 Detailed Review

1. Numbers and Fingers
Numbers for speaking, not for writing. Using the fingers to represent numbers by a one-to-one equivalence. Discovering that ten names for the different possible numbers of fingers held up is a bit more succinct and handier than actually holding up the fingers. Every power show more of ten requires a new name, but that's fine, because talkers do not need numbers bigger than a thousand.

2. Numbers and Writing
When writing is invented it becomes necessary to invent symbols and a way of arranging them to represent numbers, i.e., to invent numerals. A number system with only "1" is inconvenient for large numbers. The Egyptians used a method of writing numbers that relied on summation and that required a distinct digit for ever power of ten. 999 might thus be HHHHHHHHHTTTTTTTTT111111111.

3. Numbers and the Romans
Some cultures liked 20 (a score), others 12 (a dozen). A gross is 12 * 12. Probably this is the reason why the times table required to be learned by children in my youth went up to 12. The Sumerians liked 60. The Romans liked 5, but liked to think in base 10. So they had a hybrid system which gave 5 and 10 special importance, so that 5 * X and 10 * X where X is some smaller symbol both received separate symbols, as V, X and L, C and D, M. They could have used a base five system if they had wanted to go all the way. It's reasonable to describe their system as base-10, with abbreviations. Wishing to save space, they also added rules about ordering, so that their system was not based purely on summation. The Roman number system did not have an infinite number of symbols, so it could not represent an arbitrary number. Assuming we pretend that it did, though, it would require an infinitely large grammar to represent it. Would it be possible to write a CFG to represent a smaller subset of this imaginary invented language and to calculate the value of any numeral? I think so, you could use sub-languages for pairs, like "X" and "L".

R1 => "I"
R1 =>"II"
R1 => "III"
R1 => "IV"
R1 => "V"
R1 => "VI"
R1 => "VII"
R1 => "VIII"
R1 => "IX"
R1s => R1 | ""

R10 => "X" R1s rule: 10 + v(R1s)
R10 => "XX" R1s rule: 20 + v(R1s)
R10 => "XXX" R1s
R10 => "XL" R1s
R10 => "L" R1s
R10 => "LX" R1s
R10 => "LXX" R1s
R10 => "LXXX" R1s
R10 => "XC" R1s

R => R10 R1

So, there is a recursive function that can generate a CFG for a Roman numeral system that can represent all numbers up to some bound. It must take as an argument the necessary number of distinct digits. For 1 - 9, that number is 2, for 1 - 99 it is 5 for 1 - 999, it is 7, and so forth.

4. Numbers and the Alphabet
Using letters for numbers and then do numerology. Boring! Asimov thinks so too.

5. Numbers and "NOTHING"
Zero, positional number system, all good. The grammar:

S => "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"
D => S | "0"
H 1 => D H1 | "" rule: v(D) * n(H1) + v(H1) n (D H1) = 1 + n(H1)
H => S H1 rule: v(S) * n(H1) + v(H1) n (S H1) = 1 + n(H1)

6. Numbers and the WORLD
These numbers finally catch on and now everybody uses them. show less

¿Cómo se las arreglaba el hombre para contar, antes de la aparición de los números? Piensa en lo que tardaría un pastor en contar las cabezas de su rebaño si tuviera que marcar un palote por cada una. Ahora usamos las centenas, las decenas y las unidades, sin esfuerzo. Isaac Asimov nos explica en este libro cómo descubrió el hombre el increíble mundo de los números.

¿Cómo se las arreglaba el hombre para contar, antes de la aparición de los números? Piensa en lo que tardaría un pastor en contar las cabezas de su rebaño si tuviera que marcar un palote por cada una. Ahora usamos las centenas, las decenas y las unidades, sin esfuerzo. Isaac Asimov nos explica en este libro cómo descubrió el hombre el increíble mundo de los números.

Interesante manera de descubrir los número a los largo de la historia...