Roman Numerals and Additive v. Multiplicative Number Systems
Roman Numerals
Roman Numerals was the standard numbering system and method of Arithmetic in Ancient Rome and Europe until about 900 AD,when the Arabic Numbering System, which was originated by the Hindu's,came into use. The concept of "zero"did not exist in Europe until after 1000AD;thus,there was no Roman numeral symbol for "zero".
Counting:
I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XXI, XXII, XXIII, XIV, XV, XVI, XVII, XVIII, XXIX, XXX…..
• How many digits are there in Roman Numerals?
I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000 V = 5,000
X = 10,000 L = 50,000 C = 100,000 D = 500,000 M = 1,000,000
• What meaning does place value have?
• How are the numbers constructed?
Operations in Roman Numerals
• Conversions from base 10
Ø 7310 = ____
Ø 13910 = ____
• Addition
Ø 23 + 58 = XXIII + LVIII = LXXVIIIIII = LXXVVI = LXXXI = 81
• Subtraction???
Yikes! Use the abacus!!
Additive Base v. Multiplicative Base
• 22123 = 2(27) + 2(9) + 1(3) + 2(1) = 7710
Place Value: 33 32 31 30
To get the value of the next place to the left, we multiply by the base.
Can we use this multiplicative relationship with Roman numerals?? LXXIII = 50 + 2(10) + 3 = 73
What is enabled by a multiplicative base?
The Invention of Zero
The introduction of zero into the decimal system in 13th century was the most significant achievement in the development of a number system, in which calculation with large numbers became feasible. The notion of zero was introduced to Europe in the Middle Ages by Leonardo Fibonacci who translated from Arabic the work of the Persian (from Usbekestan province) scholar Abu Ja'far Muhammad ibn (al)-Khwarizmi. The word "algorithm," Medieval Latin 'algorismus', is a contamination of his name and the Greek word arithmos, meaning "number, has come to represent any iterative, step-by-step procedure. Khwarizmi in turn documented (in Arabic, in the 7th century) the original work of the Hindu mathematician Ma-hávíral as a superior mathematical construction compared with the then prevalent Roman numerals which do not contain the concept of zero. When these scholarly treatises were being translated by European accountants, they translated 1, 2, 3,.. upon reaching zero, they pronounced, "empty", Nothing! The scribe asked what to write and was instructed to draw an empty hole, thus introducing the present notation for zero.
Properties of Zero
a ∙ 0 = 0
a + 0 = a
0/a = 0
a/0 = undefined
Formal definition of division
a/b = c must meet two conditions: c ∙ b = a; c is unique
2/0 does not meet the condition that c ∙ b = a
0/0 does not meet the condition that c is unique