By Andy Liu

Arithmetical Wonderland is meant as an unorthodox arithmetic textbook for college kids in simple schooling, in a contents direction provided by way of a arithmetic division. The scope is intentionally limited to hide in basic terms mathematics, even if 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 just a little daunting. it's the authors trust that a lot of that's prompted no longer through the topic itself, yet via the language of arithmetic. during this e-book, 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 lads are studying excessive mathematics or easy quantity concept from Alice, and the reader is carried alongside during this educational exploration. therefore 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 particularly unorthodox questions. for example, Alice tells the twins that six divided by means of 3 is 2 merely due to 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 quite early, resembling the thoughts of divisibility and congruence.

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**Sample text**

Consider now the case where d|b and d|c. The first statement means that b = dq for some integer q, and the second statement means that c = dr for some integer r . Hence a = c − b = dr − dq = d(r − q). Now r − q is an integer since subtraction is closed in the set of integers. It follows that we have d|a. The third case where d|c and d|a is analogous. The first statement means that c = dr for some integer r , and the second statement means that a = dp for some integer p. Hence b = c − a = dr − dp = d(r − p).

Compute 7 3. Solution “Well, 7 3 = (7 + 3)(7 − 3) = 40,” said Alice. “I see that I have just been wasting my time. ” Ambition, distraction, uglification and derision are artificially defined operations and have no practical value. However, because we are not familiar with them, they can be used to test our understanding of the basic properties of operations. 5. (a) Determine whether ambition is commutative. (b) Determine whether uglification is commutative. We have a ⊕ b = ab + a + b = ba + b + a = b ⊕ a because ordinary addition and multiplication are commutative.

The number Alice wrote down was 100a + b = 3ab, according to the Queen of Hearts. By the Basic Divisibility Theorem, a divides b, so that b = na for some positive integer n. From 100a + na = 3na 2 , we have 100 + n = 3na. Hence 3 divides 100 + n and by the Basic Divisibility Theorem again, n divides 100. Since a and b are both two-digit numbers, n < 10. It follows that n = 2 or 5. If n = 5, then a = 7 is already not a two-digit number. Hence n = 2, a = 17, b = 34 and the correct answer is 1734 ÷ 3 = 578 = 17 × 34.