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Numbers: Their Tales, Types, and Treasures
Did you grow up thinking math is boring? It’s time to reconsider. This book will teach you everything you ever wondered about numbers—and more.
How and why did human beings first start using numbers at the dawn of history? Would numbers exist if we Homo sapiens weren’t around to discover them? What’s so special about weird numbers like pi and the Fibonacci sequence? What about rational, irrational, real, and imaginary numbers? Why do we need them?
Two veteran math educators explain it all in ways even the most math phobic will find appealing and understandable.
You’ll never look at those squiggles on your calculator the same again.
How and why did human beings first start using numbers at the dawn of history? Would numbers exist if we Homo sapiens weren’t around to discover them? What’s so special about weird numbers like pi and the Fibonacci sequence? What about rational, irrational, real, and imaginary numbers? Why do we need them?
Two veteran math educators explain it all in ways even the most math phobic will find appealing and understandable.
You’ll never look at those squiggles on your calculator the same again.
400 pages, Paperback
First published January 1, 2015
20 people are currently reading
92 people want to read
About the author
Alfred S. Posamentier
98 books31 followersAlfred S. Posamentier (born October 18, 1942) is among the most prominent American educators in the country and is a lead commentator on American math and science education, regularly contributing to The New York Times and other news publications. He has created original math and science curricula, emphasized the need for increased math and science funding, promulgated criteria by which to select math and science educators, advocated the importance of involving parents in K-12 math and science education, and provided myriad curricular solutions for teaching critical thinking in math.
Dr. Posamentier was a member of the New York State Education Commissioner’s Blue Ribbon Panel on the Math-A Regents Exams. He served on the Commissioner’s Mathematics Standards Committee, which redefined the Standards for New York State. And he currently serves on the New York City schools’ Chancellor’s Math Advisory Panel.
Posamentier earned a Ph.D. in mathematics education from Fordham University (1973), a Master’s degree in mathematics education from the City College of the City University of New York (1966) and an A.B. degree in mathematics from Hunter College of the City University of New York.
Posamentier was born in Manhattan in New York City, the son of Austrian immigrants. He has one daughter (Lisa, born in 1970), and one son (David, born in 1978). He resides in River Vale, New Jersey and is the current Dean of the School of Education and professor of mathematics education at Mercy College, New York. He was formerly professor of mathematics education and dean of the School of Education at The City College of the City University of New York, where he spent the previous 40 years.
Dr. Posamentier was a member of the New York State Education Commissioner’s Blue Ribbon Panel on the Math-A Regents Exams. He served on the Commissioner’s Mathematics Standards Committee, which redefined the Standards for New York State. And he currently serves on the New York City schools’ Chancellor’s Math Advisory Panel.
Posamentier earned a Ph.D. in mathematics education from Fordham University (1973), a Master’s degree in mathematics education from the City College of the City University of New York (1966) and an A.B. degree in mathematics from Hunter College of the City University of New York.
Posamentier was born in Manhattan in New York City, the son of Austrian immigrants. He has one daughter (Lisa, born in 1970), and one son (David, born in 1978). He resides in River Vale, New Jersey and is the current Dean of the School of Education and professor of mathematics education at Mercy College, New York. He was formerly professor of mathematics education and dean of the School of Education at The City College of the City University of New York, where he spent the previous 40 years.
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Displaying 1 - 9 of 9 reviews
January 1, 2022
The first half was fairly interesting -- don't be put off by the dry chapter and section titles a la "7.4 Construction of a Magic Square of Order 3" -- but around the midpoint the focus settles on numbers as numbers rather than numbers as language to communicate ideas.
Some people will love that! For me, prime numbers just aren't that interesting; ditto palindromic numbers; ditto Armstrong numbers:
"The Armstrong numbers have the property that each number is equal to the sum of its digits, when each digit is taken to the power equal to the number of digits in the original number."
I'm more of a using math than musing math type, which I guess makes me not a true mathematician, but we already knew that! :)
So I've dropped out at the halfway point, after some interesting history (global, not only Western!), some poetry talk, and some visual representations of interesting patterns for calculation.
Some people will love that! For me, prime numbers just aren't that interesting; ditto palindromic numbers; ditto Armstrong numbers:
"The Armstrong numbers have the property that each number is equal to the sum of its digits, when each digit is taken to the power equal to the number of digits in the original number."
I'm more of a using math than musing math type, which I guess makes me not a true mathematician, but we already knew that! :)
So I've dropped out at the halfway point, after some interesting history (global, not only Western!), some poetry talk, and some visual representations of interesting patterns for calculation.
February 23, 2018
Reading about numbers was fun. The only thing I wish is that understanding math came easier for me. But, it was interesting to see that people look for patterns in types of numbers. More math practice for me.
June 7, 2024
The first half was good. Second half was just repetitive and not anything really worth reading again.
June 18, 2016
Not for the number-phobic!
So it would be very easy to spoil the book for those of you out there who actually like numbers and mathematics by giving a quick table of contents and what’s covered in each chapter. But if you were looking for that kind of spoilage, you could just go to any online bookseller and check out the table of contents and page sample. Instead, I’ll just hit some highlights on both sides of the spectrum: things I liked and things I didn’t.
Numbers themselves have always been interesting to me. In a way, you can look at the universe and say it’s made of numbers. The authors don’t quite take that viewpoint, but they do look at counting and mathematics not just as common everyday things, but also as huge cultural achievements.
On the side of enjoyment, hints and suggestions at the origins of numbers and counting and how we could conceivably have wound up with a base-12 mathematical system. Or base-5. Or base-20. Just because we have ten fingers, doesn’t mean that’s the only way we could have counted things, and it wasn’t. Bits of several systems are still hanging around in the way we talk about numbers today.
There are also chapters and sections that discuss numbers in fields as wide as mathematics, history, literature—particularly poetry—and philosophy.
While looking at properties of numbers and the search for them, the authors also seem to recognize that some of these properties stem from the base of the numerical system. That doesn’t change some of the properties being pretty neat, though. Again, if you like numbers, but if you’re not, why are you reading this book?
And along with a discussion of square numbers, we find out about rectangular numbers and triangular numbers. A neat idea I’ve never run across before, but the visualizations makes sense. I have to wonder how many people who struggled with multiplication in their earlier educational days might have been assisted by being shown things in a more visual way. (Pentagonal and hexagonal numbers follow, to extend the visualizations.)
There’s also a chapter on recreational math, something that probably seems horrifying to some people, but Sudoku falls into this, as well as things like magic squares and palindromes. Mixed in here is something I’ve never run across before (under the name of Box Multiplication and as a shortcut rather than recreational): Napier’s Rods or Napier’s Bones.
On the duller side, for me, primes and Fibonacci numbers.
I never realized that there were so many different kinds of prime numbers, but this chapter is very formulaic: here’s a neat idea, here’s the explanation, and here’s a list of prime numbers that fit the idea. It also seems to go on for a really, really long time.
Fibonacci numbers fall into the same not-so-exciting category for me. The sequence is neat, I suppose, but not something I’ve ever been interested in spending much time on.
And we finish up with a discussion of the philosophy of numbers. Were they discovered or invented? There are several major schools of thought, and we visit with them all.
Overall rating: 3 stars. Taken as a whole, I enjoyed the book, but most of what I liked was the discussion and the history rather than the delineation of types of numbers and their properties. Neat, but only until you start laying things out in tables of numbers. Those chapters I probably could have skipped and been happier with the reading experience overall.
So it would be very easy to spoil the book for those of you out there who actually like numbers and mathematics by giving a quick table of contents and what’s covered in each chapter. But if you were looking for that kind of spoilage, you could just go to any online bookseller and check out the table of contents and page sample. Instead, I’ll just hit some highlights on both sides of the spectrum: things I liked and things I didn’t.
Numbers themselves have always been interesting to me. In a way, you can look at the universe and say it’s made of numbers. The authors don’t quite take that viewpoint, but they do look at counting and mathematics not just as common everyday things, but also as huge cultural achievements.
On the side of enjoyment, hints and suggestions at the origins of numbers and counting and how we could conceivably have wound up with a base-12 mathematical system. Or base-5. Or base-20. Just because we have ten fingers, doesn’t mean that’s the only way we could have counted things, and it wasn’t. Bits of several systems are still hanging around in the way we talk about numbers today.
There are also chapters and sections that discuss numbers in fields as wide as mathematics, history, literature—particularly poetry—and philosophy.
While looking at properties of numbers and the search for them, the authors also seem to recognize that some of these properties stem from the base of the numerical system. That doesn’t change some of the properties being pretty neat, though. Again, if you like numbers, but if you’re not, why are you reading this book?
And along with a discussion of square numbers, we find out about rectangular numbers and triangular numbers. A neat idea I’ve never run across before, but the visualizations makes sense. I have to wonder how many people who struggled with multiplication in their earlier educational days might have been assisted by being shown things in a more visual way. (Pentagonal and hexagonal numbers follow, to extend the visualizations.)
There’s also a chapter on recreational math, something that probably seems horrifying to some people, but Sudoku falls into this, as well as things like magic squares and palindromes. Mixed in here is something I’ve never run across before (under the name of Box Multiplication and as a shortcut rather than recreational): Napier’s Rods or Napier’s Bones.
On the duller side, for me, primes and Fibonacci numbers.
I never realized that there were so many different kinds of prime numbers, but this chapter is very formulaic: here’s a neat idea, here’s the explanation, and here’s a list of prime numbers that fit the idea. It also seems to go on for a really, really long time.
Fibonacci numbers fall into the same not-so-exciting category for me. The sequence is neat, I suppose, but not something I’ve ever been interested in spending much time on.
And we finish up with a discussion of the philosophy of numbers. Were they discovered or invented? There are several major schools of thought, and we visit with them all.
Overall rating: 3 stars. Taken as a whole, I enjoyed the book, but most of what I liked was the discussion and the history rather than the delineation of types of numbers and their properties. Neat, but only until you start laying things out in tables of numbers. Those chapters I probably could have skipped and been happier with the reading experience overall.
Want to read
July 20, 2020p21 Cardinal principle
3. Cardinal principle:
When we start counting with one, then the last number word reached after having counted all elements of the collection has a very special meaning. It not only is the counting tag of the last counted item but also describes a property of the collection as a whole. The last counting tag is
the result of counting. In everyday life we would call this the "number of objects in the collection" The property of a collection that is described by the last number word is sometimes called its numerosity, Mathematicians call it cardinality, hence the name of that principle. The cardinality
of the collection of disks in figure 1.4 is 19, or nineteen.
For young children, it is a difficult task, and a great achievement, to make the transition from the mechanical use of number words during the counting procedure (as expressed in the ordinal principle, where numbers just serve for tagging objects) to an understanding that a number can also denote a numerosity. They have to learn that the last number word is not just the name of the last thing but a property of the whole collection. It is the answer to teh question "How many?"
========================
p32
Any collection of arbitrary objects can be counted (abstraction principle).
Counting is a process of associating unique "counting tags with each of the objects. This one-to-one association between objects and counting tags is represented by the vertical double arrow (bijection principle).
The counting tags must have a fixed order.
The act of counting puts the objects to be counted in a certain order, but the ordering of the objects is irrelevant (invariance principle). Rearranging the shapes would not change the final outcome.
When we start counting at one, the cardinality (numerosity) of the set is described by the last of the number words (cardinal principle).
========================
*** P276 Divisibility of Numbers
3. Cardinal principle:
When we start counting with one, then the last number word reached after having counted all elements of the collection has a very special meaning. It not only is the counting tag of the last counted item but also describes a property of the collection as a whole. The last counting tag is
the result of counting. In everyday life we would call this the "number of objects in the collection" The property of a collection that is described by the last number word is sometimes called its numerosity, Mathematicians call it cardinality, hence the name of that principle. The cardinality
of the collection of disks in figure 1.4 is 19, or nineteen.
For young children, it is a difficult task, and a great achievement, to make the transition from the mechanical use of number words during the counting procedure (as expressed in the ordinal principle, where numbers just serve for tagging objects) to an understanding that a number can also denote a numerosity. They have to learn that the last number word is not just the name of the last thing but a property of the whole collection. It is the answer to teh question "How many?"
========================
p32
Any collection of arbitrary objects can be counted (abstraction principle).
Counting is a process of associating unique "counting tags with each of the objects. This one-to-one association between objects and counting tags is represented by the vertical double arrow (bijection principle).
The counting tags must have a fixed order.
The act of counting puts the objects to be counted in a certain order, but the ordering of the objects is irrelevant (invariance principle). Rearranging the shapes would not change the final outcome.
When we start counting at one, the cardinality (numerosity) of the set is described by the last of the number words (cardinal principle).
========================
*** P276 Divisibility of Numbers
February 20, 2017
This book was a gem. It tells the story of how numbers have evolved since their first documented use and provides a nice introduction to elementary number theory. In particular, I found the descriptions and introduction to the basic mechanics of various ancient number systems such as the sexagesimal (base-60) system of the Babylonians to be quite interesting. Several important number sequences are introduced such as the triangle number and fibonacci sequence along with a very nice discussion of some of the amazing patterns and series to be found in the so-called "Pascal's Triangle" (discovered and re-discovered centuries before by the Chinese, Indians and Persians). Important numeric constants like phi, e, and pi are covered in detail.
The actual math in this book is no more complicated than highschool algebra so it doesn't require much mathematical sophistication, instead this a book written with a sprit of curiosity and an appreciation of history.
The actual math in this book is no more complicated than highschool algebra so it doesn't require much mathematical sophistication, instead this a book written with a sprit of curiosity and an appreciation of history.
Read
July 26, 2016Boring
This entire review has been hidden because of spoilers.
October 26, 2015
This book reads like a text book and is good cure for insomnia.
April 23, 2017
This is about exploring the history of how numbers came to be and the terminology used in college mathematics and above. You talk about the most basic mathematics coming to be and discovering the formulas from various mathematicians have founded.
As to most math books, you don't have a particular structured plot, but the author summerizes all the necessary required knowledge to discuss certain arguments still going on today and other topics within math.
When the author referred, they referred more books that were more by them than other authors. Some other books were referred by other mathematicians, but it wasn't until the end. I was hoping to see a little more diversity.
As to most math books, you don't have a particular structured plot, but the author summerizes all the necessary required knowledge to discuss certain arguments still going on today and other topics within math.
When the author referred, they referred more books that were more by them than other authors. Some other books were referred by other mathematicians, but it wasn't until the end. I was hoping to see a little more diversity.
Displaying 1 - 9 of 9 reviews