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M110SECTION 4.2PLACE VALUE SYSTEMSWhat do we mean by a place value system?Consider the number 47. What does the 4 represent?__________________________What does the 7 represent?__________________________Now consider the number 74. What does the 7 represent?__________________________What does the 4 represent?__________________________Thus we can see that where the symbol or numeral is placed or positioned in the numberchanges the value of that symbol. It is now important that a symbol for ______________ bepart of any place value numeration system in order to hold a particular place when none ofthat place is required.THE BABYLONIAN NUMERATION SYSTEMThe Babylonian number system began with tally marks just as most of the ancient mathsystems did. The Babylonians developed a form of writing based on cuneiform. Cuneiformmeans "wedge shape" in Latin. They wrote these symbols on wet clay tablets which werebaked in the hot sun. Many thousands of these tablets are still around today. TheBabylonians used a stylist to imprint the symbols on the clay since curved lines could notbe drawn. The Babylonians had a very advanced number system even for today's standards. It was abase 60 system (sexigesimal) rather than a base ten system (decimal).The Babyloniansdivided the day into twenty-four hours, each hour into sixty minutes, and each minute tosixty seconds. This form of counting has survived for four thousand years. There were two symbols: _______ which represented one and_______ which represented ten. To write small numbers (less than 60), this system worked much like the Egyptian system.For example:25 = 42 = = =To represent larger numbers, the Babylonians used groups of these symbols, separated byspaces, and multiplied the value of these groups by increasing power of 60.Example There are two groups. The right most groups represents the units or1s 600 and the next group to the left represents the 60s 601 . Tocalculate the value of this number:Because the early Babylonians did not have a symbol for zero, it could be hard to tellexactly how many spaces were between groups of symbols. Examples: Convert to a Hindu-Arabic number.The Babylonians also used the symbol _______ to indicate subtraction.Thus to find the value of: Write each of the following as a Babylonian numeral.216 188412,221THE MAYAN NUMERATION SYSTEMThe Mayan Indians lived on the Yucatan Peninsula in central America from about 200 B.C.to 1540 A.D. The Mayans used a vigesimal system, which had a base 20. This system isbelieved to have been used because the Mayans lived in such a warm climate and therewas rarely a need to wear shoes, thus 20 was the total number of visible fingers and toesmaking the system workable. Therefore two important markers in this system are 20,which relates to the fingers and toes, and five, which relates to the number of digits on onehand or foot. The Mayan system used a combination of three symbols.There were two symbols: _______ which represented one _______ which represented five and_______ which represented zero. Unlike the Babylonians, the Mayans did have a symbol for zero.The Mayans wrote their numbers vertically as opposed to horizontally with the lowestdenomination on the bottom. After 19, the Mayan used positioning of symbols torepresent larger numbers. The lowest position represented the units or 1's, the next higherrepresenting the 20's 1×20 , the next position representing the 360's 1×20×18 , the nextrepresenting the 7200's 1×20×18×20 , etc.Examples: Convert to Hindu-ArabicExamples: Convert to Mayan152 1751246 13,339THE HINDU-ARABIC PLACE VALUE SYSTEMThe Hindu-Arabic system is a place value system based on 10 (unlike the Babylonian systemwhich is based on 60 and the Mayan system which is based on 20). This means that the columns(like the areas in Babylonian or the levels in Mayan) are powers of 10.106105104103102101100millionshundredthousandstenthousandsthousands hundreds tensones orunitsWHAT IS THE BASE? __________WHAT NUMERALS ARE USED? _______________________________________________________THE MEANING OF A DIGITWhat does the digit 7 mean in the number 247,189? _____________________________________What does the digit 2 mean in the number 247,189? _____________________________________What does the digit 9 mean in the number 247,189? _____________________________________WRITING A NUMBER IN EXPANDED FORM3, 592 = _______________________________________________________________________________5,208,610 = ____________________________________________________________________________(7 x 103) + (6 x 102) + (0 x 101) + (9 x 100) = ___________________(3 x 107) + (6 x 103) + (6 x 102) + (3 x 100) = ___________________MULTIPLYING USING THE GALLEY METHODAn important belief of Hindu-Arabic is that we can do basic numerical calculation verysimply using pencil and paper. One method of multiplication that preceded our currentmethod was "The Galley Method".Given: 45 × 83 YOU TRY:926 × 75 807 ×


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IUS M 110 - PLACE VALUE SYSTEMS

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