By Andy Liu

Arithmetical Wonderland is meant as an unorthodox arithmetic textbook for college students in basic schooling, in a contents path provided via a arithmetic division. The scope is intentionally constrained to hide merely mathematics, although geometric components are brought at any time when 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 a little bit daunting. it's the authors trust that a lot of that's brought on now not by means of the topic itself, yet by way of the language of arithmetic. during this ebook, a lot of the dialogue is in dialogues among Alice, of Wonderland status, and the twins Tweedledum and Tweedledee who hailed from in the course of the having a look Glass. the lads are studying excessive mathematics or trouble-free 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 e-book has huge intensity. It examines many arcane corners of the topic, and increases fairly unorthodox questions. for example, Alice tells the twins that six divided via 3 is 2 purely due to an implicit assumption that department is meant to be reasonable, while equity doesn't come into addition, subtraction or multiplication. a few issues frequently no longer lined are brought quite early, reminiscent of the strategies of divisibility and congruence.

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**Example text**

In the second case,” said Tweedledum, “we have a = b and b ≤ c. ” “The third case is exactly the same thing,” observed Tweedledee. “Here we have a ≤ b and b = c. ” “In the fourth and last case,” said Tweedledum, “we have a < b and b < c. This means that there exist positive integers k and such that a + k = b and b + = c. Now c = b + = (a + k) + . ” “Now I see why the Closure Property for addition of positive integers is important,” remarked Tweedledee. “Yes,” Tweedledum agreed. “As a sum of two positive integers, k + is also a positive integer.

An instance in which A is true but B is false is called a counterexample. We are not interested in instances in which the hypothesis A is false. They may be called non-examples. 3. 2. Solution If we take a = 16, b = 32, c = 4 and d = 8, then 4|16 and 8|32, so that both clauses in the hypothesis are satisfied. Moreover, 4 + 8 = 12 and 16 + 32 = 48, and 12|48. So we have an example of the conditional statement. If we take a = 16, b = 32, c = 2 and d = 8, then 2|16 and 8|32, so that both clauses in the hypothesis are satisfied.

Roared the Red Queen. What was the correct answer? 10. The White Queen was giving Alice an arithmetic examination. “Multiply these two sixdigit numbers,” she roared. Alice wrote down the two numbers one after another, began to day-dream, and just handed in what she wrote down as a twelve-digit answer. " roared the White Queen. What was the correct answer? 38 Divisibility 11. Think of a 2-digit number. Write it down three times in a row to form a 6-digit number. Divide this number by 13. If you are not unlucky, the division will be exact.

### Arithmetical Wonderland by Andy Liu

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