By Engel A. (ed.)

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**Additional info for Mathematische Olympiadeaufgaben aus der UdSSR, mit Loesungen**

**Sample text**

Let’s examine some facts about zero. The concept of zero, which indicates the absence of a quantity, has most likely been understood since prehistoric times. Each time a hunter came home without any game, the number zero was experienced. Though the early Egyptian, Greek, and Roman civilizations understood the concept of zero, they had no symbol for zero. Their systems of numeration did not need a symbol for zero. Zero was, however, represented on ancient Babylonian tablets and Chinese counting boards as a blank space used for a missing place value in a numeral.

If you are not concerned about the decimal answer to a problem and simply work in binary, addition and multiplication can be done quite easily. EXAMPLE 3 Find the sum of the binary numbers, 101 111 101 and 10 011 010. 4 • The Numbers of Technology 37 and the normal “carrying” procedure as noted. Proceeding from the l’s place and working to the left give the following result. 1 111 1 1 101 111 101 ϩ 10 011 010 ᎏᎏᎏᎏᎏᎏ 1 000 010 111 ϭ 1 000 010 1112 Note: Writing the numbers in decimal form, we have 381 ϩ 154 ϭ 535.

In Problems 28–31, write the equivalent Hindu-Arabic fraction for each Roman fraction. 28. . 29. ϭ ϭ 30. S . 31. S ϭ Ϫ 22 Chapter 1 • NUMBERS—Old and New In Problems 32–35, write the equivalent Hindu-Arabic fraction for each Greek fraction. 32. kdЈ 33. jgЈ 34. ilbЈlbЈ 35. mufaЈfaЈ q Explore The sum of a whole number and a fraction is called a mixed number. In the Hindu-Arabic system, 5 ϩ 2ȋ3 is written as 5ᎏ23ᎏ. In Problems 36–39, find the mixed numbers represented by each ancient numeral. 36.