ByKuldeep Singhon 12 December 2008

Review of Yearning For The Impossible by John Stillwell

This is a wonderful book to read around the subject of mathematics. I learned a lot from this book which can be used in general class discussion to motivate students in a particular field of mathematics. There are some real gems in the book such as:

1. The word surd comes from the Latin surdus meaning deaf.

2. The product of sum of two squares is itself the sum of two squares.

3. A simple example of an elliptic function is the height of a point on an ellipse given in terms of the arc length.

4. Relationship between Gaussian primes and ordinary primes.

5. Hamilton's quaternion which is another number system.

6. 3-D symmetry is rare because only 5 regular polyhedra are fully symmetrical.

7. If 1 was prime then we would not have unique prime factorisation.

8. 27 is the only cube which is 2 more than a square number.

There are many others like this.

However this book is not for the layman because as the author says in his preface `many of the ideas are hard and there is no way to soften them'.

The author has made extensive and fantastic use of illustrations to soften the blow and give an intuitive view of the mathematics. In particular the author has made excellent use of diagrams in the chapter on curved space.

I really do like the historical context the author has placed his mathematics. I did not know that the Chinese had approximated ð by the fraction 355/113 which is accurate to 6 decimal places. Nor did I know that the Leibniz series was known to Indian mathematician(s) well before it was discovered by Leibniz. However the author does not give the name(s) of the Indian mathematician(s) who had discovered this series.

The chapter on the fourth dimension has an excellent historical context where the author describes Hamilton's search for the arithmetic of triples, quadruples etc. The author starts by saying that the great Irish mathematician Hamilton (1805 to 1865) was the first to envisage complex numbers as pairs of real numbers. He goes on to describe the addition of pairs as component-wise addition, that is,

(a, b) + (c, d) = (a+c, b+d) and multiplication as (a,b).(c,d) = (ac-bd, ad+bc) . This system with addition and multiplication of ordered pairs defined in this manner satisfies all the rules of ordinary algebra (field) and the absolute value is multiplicative. The author claims that Hamilton tried for 13 years to extend this idea to triples (a, b, c) but failed. From the ashes of triples he revived a definition for quadruples (a, b, c, d) which ensured that the absolute value was multiplicative and all the rules of ordinary algebra was satisfied apart from multiplication being commutative (skew field). Hamilton called the system of quadruples with addition and multiplication the quaternion.

There is a wonderful chapter on the relationship between primes and Gaussian primes. The Gaussian prime is defined as a Gaussian integer which has an absolute value greater than 1 and cannot be factored into Gaussian integers of smaller absolute value. Numbers such as 1+i and 1-i are Gaussian primes but 2 is not because

2=(1+i)(1-i)

3 is Gaussian prime and an ordinary prime. This means that the ordinary primes are not a subset of the Gaussian primes. The author perhaps should have included that an ordinary prime of the form 1 mod 4 can be written as the sum of two squares but this can be factorised into Gaussian primes because a^2+b^2 = (a+bi)(a-bi). Therefore an ordinary prime of the form 1 mod 4 is not a Gaussian prime.

The last chapter is a nice description of the different kinds of infinity and countable sets. There is a really interesting quote in this chapter ` Many people believe the continuum hypothesis put Cantor in a mental hospital'.

In general there are some really good statements in the book which can be used to hook students to study mathematics such as -

`Calculus as we know it today, is perhaps, the most powerful mathematical tool ever developed'.

`Infinitesimal calculus brought almost the whole physical world within the scope of mathematics'.

However I do have a number of minor quibbles.

Sometimes a proposition or theorem is not clearly stated and it is only when you are half way through a proof you realise what is happening.

There are statements which do not relate to mathematical reality such as `square root of a half a turn' which means a rotation though a right angle. How are the two related?

Are they related by the polar or exponential form of a complex number?

There are a number of places where the book jumps from a simple introduction to the more advanced level within a few pages. An example of this is, we are introduced to the idea of a complex number and within 10 pages we are discussing Bezout's Theorem.

In places I am not to sure what the author means. For example on page 45 the author says `complex analysis is known for its regularity and order.' Clearly complex numbers don't have order. For example if i>0 then i^2>0 which is false and if i<0 then i^2>0 again this is false.

In addition there are a couple of typos such as on page 137 should read c1 not c2 in the formula of line 2. On page 169 it says substitute (7.6) in (7.1) but there is no (7.6).

This is a wonderful book to read around the subject of mathematics. I learned a lot from this book which can be used in general class discussion to motivate students in a particular field of mathematics. There are some real gems in the book such as:

1. The word surd comes from the Latin surdus meaning deaf.

2. The product of sum of two squares is itself the sum of two squares.

3. A simple example of an elliptic function is the height of a point on an ellipse given in terms of the arc length.

4. Relationship between Gaussian primes and ordinary primes.

5. Hamilton's quaternion which is another number system.

6. 3-D symmetry is rare because only 5 regular polyhedra are fully symmetrical.

7. If 1 was prime then we would not have unique prime factorisation.

8. 27 is the only cube which is 2 more than a square number.

There are many others like this.

However this book is not for the layman because as the author says in his preface `many of the ideas are hard and there is no way to soften them'.

The author has made extensive and fantastic use of illustrations to soften the blow and give an intuitive view of the mathematics. In particular the author has made excellent use of diagrams in the chapter on curved space.

I really do like the historical context the author has placed his mathematics. I did not know that the Chinese had approximated ð by the fraction 355/113 which is accurate to 6 decimal places. Nor did I know that the Leibniz series was known to Indian mathematician(s) well before it was discovered by Leibniz. However the author does not give the name(s) of the Indian mathematician(s) who had discovered this series.

The chapter on the fourth dimension has an excellent historical context where the author describes Hamilton's search for the arithmetic of triples, quadruples etc. The author starts by saying that the great Irish mathematician Hamilton (1805 to 1865) was the first to envisage complex numbers as pairs of real numbers. He goes on to describe the addition of pairs as component-wise addition, that is,

(a, b) + (c, d) = (a+c, b+d) and multiplication as (a,b).(c,d) = (ac-bd, ad+bc) . This system with addition and multiplication of ordered pairs defined in this manner satisfies all the rules of ordinary algebra (field) and the absolute value is multiplicative. The author claims that Hamilton tried for 13 years to extend this idea to triples (a, b, c) but failed. From the ashes of triples he revived a definition for quadruples (a, b, c, d) which ensured that the absolute value was multiplicative and all the rules of ordinary algebra was satisfied apart from multiplication being commutative (skew field). Hamilton called the system of quadruples with addition and multiplication the quaternion.

There is a wonderful chapter on the relationship between primes and Gaussian primes. The Gaussian prime is defined as a Gaussian integer which has an absolute value greater than 1 and cannot be factored into Gaussian integers of smaller absolute value. Numbers such as 1+i and 1-i are Gaussian primes but 2 is not because

2=(1+i)(1-i)

3 is Gaussian prime and an ordinary prime. This means that the ordinary primes are not a subset of the Gaussian primes. The author perhaps should have included that an ordinary prime of the form 1 mod 4 can be written as the sum of two squares but this can be factorised into Gaussian primes because a^2+b^2 = (a+bi)(a-bi). Therefore an ordinary prime of the form 1 mod 4 is not a Gaussian prime.

The last chapter is a nice description of the different kinds of infinity and countable sets. There is a really interesting quote in this chapter ` Many people believe the continuum hypothesis put Cantor in a mental hospital'.

In general there are some really good statements in the book which can be used to hook students to study mathematics such as -

`Calculus as we know it today, is perhaps, the most powerful mathematical tool ever developed'.

`Infinitesimal calculus brought almost the whole physical world within the scope of mathematics'.

However I do have a number of minor quibbles.

Sometimes a proposition or theorem is not clearly stated and it is only when you are half way through a proof you realise what is happening.

There are statements which do not relate to mathematical reality such as `square root of a half a turn' which means a rotation though a right angle. How are the two related?

Are they related by the polar or exponential form of a complex number?

There are a number of places where the book jumps from a simple introduction to the more advanced level within a few pages. An example of this is, we are introduced to the idea of a complex number and within 10 pages we are discussing Bezout's Theorem.

In places I am not to sure what the author means. For example on page 45 the author says `complex analysis is known for its regularity and order.' Clearly complex numbers don't have order. For example if i>0 then i^2>0 which is false and if i<0 then i^2>0 again this is false.

In addition there are a couple of typos such as on page 137 should read c1 not c2 in the formula of line 2. On page 169 it says substitute (7.6) in (7.1) but there is no (7.6).

ByCharles Ashbacheron 7 September 2015

This is one of those books where I dislike the title, yet love the content. Mathematicians generally go where the necessity and reasoning takes them, so it is a misnomer to claim that they have a yearning for the impossible. Also, to argue that mathematical results are impossible at some point then are considered routine later is incorrect.

As mathematical knowledge has expanded, new discoveries sometimes contradict and in all cases extend previous knowledge. What was thought to be impossible earlier is demonstrated to be possible, so the earlier “fact” was simply a “significant misunderstanding.”

Stillwell discusses several of these “significant misunderstandings” in this book. They are:

*) The discovery of irrational numbers.

*) The discovery of imaginary (complex) numbers.

*) The discovery of perspective and points at infinity.

*) The development of calculus and the use of the infinitesimal.

*) The discovery of non-Euclidean geometry and curved space.

*) The discovery of the quaternions and the fourth dimension.

*) The discovery of the algebraic ideal and the loss of unique factorization.

When each of these discoveries was made, that discovery changed some aspects of mathematics forever. The previous ideas were not somehow just rendered “inoperable” but were shown to be restricted cases of a more general result. Irrational numbers were found to be the limits of sets of rational numbers, real numbers were found to be a subset of complex numbers and Euclidean geometry was found to be just one possibility among multiple options. Stillwell does an excellent job laying the historical foundations for these discoveries; he is to be commended for his historical accuracy.

While the word impossible certainly is a powerful word to catch your eye, it is inappropriately placed in the title of this book. However, that is really the only complaint that I will lodge about it. The rest is first-rate.

Published in Journal of Recreational Mathematics, reprinted with permission

As mathematical knowledge has expanded, new discoveries sometimes contradict and in all cases extend previous knowledge. What was thought to be impossible earlier is demonstrated to be possible, so the earlier “fact” was simply a “significant misunderstanding.”

Stillwell discusses several of these “significant misunderstandings” in this book. They are:

*) The discovery of irrational numbers.

*) The discovery of imaginary (complex) numbers.

*) The discovery of perspective and points at infinity.

*) The development of calculus and the use of the infinitesimal.

*) The discovery of non-Euclidean geometry and curved space.

*) The discovery of the quaternions and the fourth dimension.

*) The discovery of the algebraic ideal and the loss of unique factorization.

When each of these discoveries was made, that discovery changed some aspects of mathematics forever. The previous ideas were not somehow just rendered “inoperable” but were shown to be restricted cases of a more general result. Irrational numbers were found to be the limits of sets of rational numbers, real numbers were found to be a subset of complex numbers and Euclidean geometry was found to be just one possibility among multiple options. Stillwell does an excellent job laying the historical foundations for these discoveries; he is to be commended for his historical accuracy.

While the word impossible certainly is a powerful word to catch your eye, it is inappropriately placed in the title of this book. However, that is really the only complaint that I will lodge about it. The rest is first-rate.

Published in Journal of Recreational Mathematics, reprinted with permission

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