By Andy Liu

Arithmetical Wonderland is meant as an unorthodox arithmetic textbook for college students in simple schooling, in a contents direction provided via a arithmetic division. The scope is intentionally limited to hide simply mathematics, although geometric parts are brought every time warranted. for instance, what the Euclidean set of rules for locating the best universal divisors of 2 numbers has to do with Euclid is showcased. Many scholars locate arithmetic just a little daunting. it's the authors trust that a lot of that's triggered no longer through the topic itself, yet via the language of arithmetic. during this publication, a lot of the dialogue is in dialogues among Alice, of Wonderland reputation, and the twins Tweedledum and Tweedledee who hailed from during the having a look Glass. the men are studying excessive mathematics or undemanding quantity idea from Alice, and the reader is carried alongside during this educational exploration. hence many formal proofs are switched over to soothing daily language. however, the booklet has enormous intensity. It examines many arcane corners of the topic, and increases relatively unorthodox questions. for example, Alice tells the twins that six divided via 3 is 2 simply as a result of an implicit assumption that department is meant to be reasonable, while equity doesn't come into addition, subtraction or multiplication. a few subject matters frequently now not lined are brought fairly early, corresponding to the strategies of divisibility and congruence.

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**Example text**

If a is negative, I think −a should be positive. In other words, the inverse of an inverse is the original number. ” “Absolutely,” beamed Alice. 1. Prove that −(−a) = a for any integer a. Solution “Let me see,” said Tweedledum. “The definition of −(−a) is that it is the inverse of −a. Now −a is the inverse of a, so that a + (−a) = 0. Since addition obeys the Commutative Law, (−a) + a = 0 so that a is also the inverse of −a. We have already seen that a number can only have one inverse. ” “Wonderful,” said Alice.

I notice that all four corner pieces are shaded,” said the Lion. ” “Indeed, it is,” said the White King. ” asked the Unicorn. “This means that we have 4 more shaded squares than unshaded ones. Now wherever we cut a 1 × 4 piece from, it is bound to consist of 2 shaded squares and 2 unshaded ones. ” “Now I see why the cake cannot be cut completely into 1 × 4 pieces,” said Alice. ” “I want more plum-cake pieces,” roared the Lion. ” “I am kind of full already,” said the White King to Alice. “I will go with you to the pantry.

Each integer belongs to exactly one of these three subclasses. So now a number has two parts. The numerical part is called its absolute value, to which we append a positive or a negative sign. ” asked Tweedledum. “Yes,” replied Alice. ” “The negative integers,” said Tweedledum without thinking. “You have to be more cautious,” said Tweedledee. “It seems to depend on what a is. If it is a positive integer, then you are correct. If a is 0, then −a is also 0. If a is negative, I think −a should be positive.