# Higher K-Theories proceedings of the conference held at the by H. Bass By H. Bass

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72 7ß*l EXAMPLES 1. We have formally defined the meaning of the extended sum 2 + 3 + 4 by writing 2 + 3 + 4 = (2 + 3) + 4, where the use of the parentheses indicates that the 2 and the 3 are to be added first and the 4 is then to be added to the sum resulting from the first addition. Compare the value of (2 + 3) + 4 with that of 2 + (3 + 4). Explain why these two values are related as they are. (In your explanation be careful that you do not merely assert that something is true. ) 2. Compare the conceptual nature of the product 2 · 3 with that of the product 3-2.

Construct the essential multiplication table for the binary notation. 6. Construct the essential multiplication table for the octal notation. 7. Construct the essential multiplication table for the base-six notation. 8. Explain carefully what meaning (or meanings) might be associated with the expression 4 - 5 - 7 . 9. Evaluate the expressions: (a) 2 · (4 + 3); (b) (2 · 4) + (2 · 3). [Note: The use of the parentheses in (a) indicates that it is the result of adding 3 to 4 that is to be multiplied by 2.

All numbers in this problem are to be considered as given in octal notation, (a) What number must the symbol b represent if 2 · b = 112? (b) What number must the symbol c represent if 12 · c = 346? (c) Is there a natural number, say x, such that 3 · x = 42? If so, what is it? (d) Draw a representation of a set associated with the product 4-12. 4. All numbers in this problem are to be considered as given in binary 42 NUMBER SYSTEMS AND OPERATIONS OF ARITHMETIC notation, (a) What number must the symbol b represent if 11 · b = 1111?