Unit 1. Math 116. Number Systems

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1 Unit Math Number Systems

2 Unit One Number Systems Sections. Introduction to Number Systems Through out history civilizations have keep records using their own number systems. This unit will introduce some basic number systems that were used by past civilizations. Examples of Ancient Number Systems ) Egyptian ) Attic ) Roman ) Mayan ) Traditional Chinese ) Babylonian I) Egyptian Number System The Egyptian use symbols to represent the values that are multiples of ten. The symbols are written in figure - Figure. = staff = heel bone = coil of rope = lotus flower, = pointing finger, = tadpole,, = astonished man (This chart is provided by the Department of Mathematics and Statistics at Wichita State University)

3 These symbols were provided by =Millions =Hundred Thousands =Ten Thousands =Thousands =Hundreds =Tens =Ones Examples Write the following numbers as an Egyptian number. ) (Symbols courtesy ) (Symbols courtesy

4 ) (Symbols courtesy Example Write the following Egyptian Numbers as a decimal number. a) Answer: b) Answer: c) Answer:

5 II) Roman Numerals Roman Numerals are very similar to the Egyptian system, but are based on instead of. The Roman Numerals Symbol Number Value I V X L C D M V X L C D M Example Convert the following decimal number to a Roman numeral. XXV ) ) = CC = XL = VI So, the final answer would be CCXLVI

6 ) 989 = M 9 = CM 8 = LXXX 9 = IX Final Answer: MCMLXXXIX ) X, MMM, XX Final Answer is XMMMXX ) 8 M = C = XL = VIII = 8 Final answer: MCXLVIII

7 Section. Place-Valued Systems The Mayan System The Mayan system came into existence about BC. This system is based on 8 and. The Mayans were the first to use the concept of zero. The number zero was denoted by the symbol Mayan symbols Symbol Number Value 8 9 Number Systems

8 Section. Binary numbers Babylonians system was based on The Mayan system was based on Our number system is based on Computer use a number system based on (Binary) System Base Digits Place Values Binary,,,,8,, Quintary,,,,,,,, Octal 8,,,,,,,,8,,,9 Converting a based number other than base to base Write each of the following on a decimal numeral Example Convert each number to a base number ) 9 98 )

9 ) 8 ) ]

10 Binary Numbers (Base Two) Example Convert to a base number (Binary Number) ) : (,, 8, ) 8 : ( 8, 8 8 is number binary The or is remainder the until process the repeating Keep Note So is which divides that two of power greatest the Find Then Thus Note difference a as Write or use so big too is which divide that power of all Check First

11 Example Convert to a binary number 8, 8 8 rewrite Then divide that power of all Check First

12 Example Convert to a binary number, 8 8 rewrite Then divide that power of all Check First Example Convert to a base ten number Example Convert to a base number 8 \

13 Example Convert into a base number to converts This of powers List Example 8 Convert 9 into a base number to converts This of powers List

14 \Section. Factors and Prime Numbers A prime number is a number that is only divisible by the number itself and one. The Prime factorization of a number If a number is written as the product of it prime factors, this is called the prime factorization. Divisibility Rules Number Divisibility Rule The number must be even. The sum of the digits of the number must divide by. The last two digits divides by. The last digit is or. The number is divisible by and. 8 The last three digits are divisible by 8. 9 The sum of the digits divides by 9 The number last digit is zero. Examples of Prime Factorizations of Numbers Example Write the prime factorization of the following number in canonical form Canonical Form

15 Example Write the prime factorization of the following number in canonical form Canonical Form Example Write the prime factorization of the following number in canonical form. 9 Example Determine if the following numbers are prime. ) has factors other than the number itself and one, so 9 is not prime. ) is prime. It s only factors are and. ) Use the sieve method.

16 Now, try all numbers less than.,,,9,8,,,,,,, all fail to divide, so prime.

Math in Our World. Section 4.1. Early and Modern Numeration Systems

Math in Our World. Section 4.1. Early and Modern Numeration Systems Math in Our World Section 4.1 Early and Modern Numeration Systems Learning Objectives Define a numeration system. Work with numbers in the Egyptian system. Work with numbers in the Chinese system. Identify