By Joe Fields

GIAM (a light creation to the paintings of arithmetic) is a unfastened, open-source textbook -- the present model is 3.1. GIAM covers numerous subject matters within the foundations of arithmetic (logic, units, kinfolk, capabilities and cardinality) and introduces the reader to many thoughts of mathematical facts (direct, oblique, contradiction, contrapositive, mathematical induction, combinatorial proofs and magic).

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**Sample text**

How would you represent 8 and 9 as octal numbers? What octal number comes immediately after 7778 ? What (decimal) number is 7778 ? 4. One method of converting from decimal to some other base is called repeated division. One divides the number by the base and records the remainder – one then divides the quotient obtained by the base and records the remainder. Continue dividing the successive quotients by the base until the quotient is smaller than the base. Convert 3267 to base-7 using repeated division.

Absolutely not! ” We often need to first approach a topic by thinking visually or intuitively, but when it comes to proving our assertions, nothing beats the power of having the “right” definitions around. It may be surprising to learn that the “right” definition often evolves over the years. This happens for the simple reason that some definitions lend themselves more easily to proving assertions. In fact, it is often the case that definitions are inspired by attempts to prove something that fail.

The procedure we give is unusually inefficient – with very little thought one could devise an algorithm that would produce the desired answer using many fewer operations – however the main point here is purely to show that division can be accomplished by essentially mechanical means. The Euclidean algorithm is far more interesting both from a theoretical and a practical perspective. The Euclidean algorithm computes the greatest common divisor (gcd) of two integers. The gcd of of two numbers a and b is denoted gcd(a, b) and is the largest integer that divides both a and b evenly.