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Introduction Roman Numerals Counting and Arithmetic Converting from Base 10

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Quit We use a base 10 system with 10 digits, they are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This is the decimal place – value system. 437 = 4 × 10 2 + 3 × 10 1 + 7 × 10 0 10’s place100’s place1’s place

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Quit Roman Numerals There is no place-value The letters have fixed values They are ordered from largest to smallest If a letter representing a smaller value comes before a larger one it is subtracted

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Quit The letters should be arranged from largest to smallest. 1510 is written MDX, largest to smallest Only powers of ten can be repeated. Don’t repeat a letter more than three times in a row. 100 is written LL, not XXXXXXXXXX Roman Numerals Rules

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Quit Numbers can be written using subtraction. A letter with a smaller value precedes one of the larger value. The smaller number is then subtracted from the larger number. Only powers for ten (I, X, C, M) can be subtracted. The smaller letter must be either the first letter or preceded by a letter at least ten times greater than it. CCXLIII = 100 + 100 + (50 – 10) + 1 + 1 + 1= 243 Roman Numerals Rules

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Quit Write in Roman Numerals 21 32 515 900 1005 1954 3592 XXI XXXII DXV CM MV MCMLIV MMMDXCII

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Quit A. 5353 B. 6363 C. 113113 Click on the number that matches the Roman Numeral LXIII

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Quit OOPS! Try again!

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Quit You are correct! LXIII = 50+10+3 = 63 L= 50; X=10; III=3 Remember:

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Quit A. 624624 B. 16241624 C. 55245524 DCXXIV Click on the number that matches the Roman Numeral

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Quit OOPS! Try again!

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Quit You are correct! DCXXIV = 500 + 100 + 20 + 4 = 624

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Quit A. 150150 B. 250250 C. 550550 CCL Click on the number that matches the Roman Numeral

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Quit OOPS! Try again!

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Quit You are correct! CCL = 100 + 100 + 50 = 250

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Quit Roman vs. Indo-Arabic Numerals Indo-Arabic Numerals are the numbers that we use today. 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 Roman Numerals are used today, but not in everyday writing. I, V, X, L, C, D, M Roman Numerals don’t have a symbol for zero.

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Quit Write down the two numbers you are adding right next to each other Rearrange the letters so they start with the largest and end with the smallest. Then start combining similar letters. Check your answer by adding the Indo-Arabic numbers. Adding Roman Numerals

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Quit 23 + 58 Adding Roman Numerals

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Quit Adding Roman Numerals Step 1. 23 + 58 Step 2. XXIII + LVIII Step 3. XXIIILVIII Step 4. LXXVIIIIII Step 4. IIIIII = VI Step 5. LXXVVI Step 6. VV = X Step 7. LXXXI = 81 Step 8. 23 + 58 = 81 Roman NumeralNumber I1 V5 X10 L50 C100 D500 M1000

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Quit 1. 10 + 15 2. 225 + 130 3. 5 + 4 4. 100 + 215 5. 30 + 50 6. 100 + 200 Adding Roman Numerals

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Quit Counting and Arithmetic Decimal or base 10 number system Origin: counting on the fingers “Digit” from the Latin word digitus meaning “finger” Base: the number of different digits including zero in the number system Example: Base 10 has 10 digits, 0 through 9 Binary or base 2: 2 digits, 0 and 1 Octal or base 8: 8 digits, 0 through 7 Hexadecimal or base 16: 16 digits, 0 – 9 and A – F

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Quit Decimal, Binary, Octal, Hexadecimal Binary (base 2) The number system is used directly by computers Hexadecimal (base 16) The number system that is used by computers to communicate with programmers eg colouring of webpages Octal (base 8) The number system that is used by either human or by computers to communicate with programmers Decimal (base 10) The number system that we are using

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Quit Decimal 14 15 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Hexadecimal E F D C B A 9 8 7 6 5 4 3 2 1 0 1110 1111 1101 1100 1011 1010 1001 1000 111 110 101 100 11 10 1 Binary 0 Octal 16 17 15 14 13 12 11 10 7 6 5 4 3 2 1 0

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Quit OnOff +– TrueFalse 6V0V YesNo 10 NorthSouth Why Binary? Early computer design used decimal John von Neumann proposed binary data processing (1945) Simplified computer design Used for both instructions and data

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Quit Numbers: Physical Representation Different numerals, same number of oranges Cave dweller: IIIII Roman: V Arabic: 5 Different bases, same number of oranges 5 10 101 2 11 4

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Quit Number System Roman: position independent Modern: based on positional notation (place value) Decimal system: system of positional notation based on powers of 10. Binary system: system of positional notation based on powers of 2 Octal system: system of positional notation based on powers of 8 Hexadecimal system: system of positional notation based on powers of 16

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Quit Positional Notation: Base 10 Place10 1 10 0 Value101 Evaluate4 × 103 × 1 Sum403 1’s place10’s place 43 = 4 × 10 1 + 3 × 10 0 43

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Quit Positional Notation: Base 10 Place10 2 10 1 10 0 Value100101 Evaluate5 × 1002 × 107 × 1 Sum500207 1’s place10’s place 527 = 5 × 10 2 + 2 × 10 1 + 7 × 10 0 100’s place 527

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Quit Positional Notation:Octal 624 8 Place8282 8181 8080 Value6481 Evaluate6 × 642 × 84 × 1 Sum 384164 64’s place8’s place1’s place = 404 10 404

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Quit Positional Notation: Hexadecimal 6,704 16 Place16 3 16 2 16 1 16 0 Value4,096256161 Evaluate6 × 4,0967 × 2560 × 164 × 1 Sum24,5761,79204 4,096’s place256’s place 1’s place 16’s place = 26,372 10 26372

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Quit Positional Notation: Hexadecimal 2B5 16 Place16 3 16 2 16 1 16 0 Value256161 Evaluate2 × 25611 × 165 × 1 Sum5121765 4,096’s place256’s place 1’s place 16’s place = 693 10 693

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Quit Positional Notation: Binary Place2727 2626 2525 2424 23232 2121 2020 Value1286432168421 Evaluate1 × 1281 × 640 × 321 × 160 × 81 × 41 × 20 × 1 Sum 128640160420 11010110 2 = 214 10

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Quit Binary Number Equivalent Decimal Number 8’s (2 3 )4’s (2 2 )2’s (2 1 )1’s (2 0 ) 00 × 2 0 0 11 × 2 0 1 101 × 2 1 0 × 2 0 2 111 × 2 1 1 × 2 0 3 1001 × 2 2 4 1011 × 2 2 1 × 2 0 5 1101 × 2 2 1 × 2 1 6 1111 × 2 2 1 × 2 1 1 × 2 0 7 10001 × 2 3 8 10011 × 2 3 1 × 2 0 9 10101 × 2 3 1 × 2 1 10

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Quit Converting from Base 10 Base 876543210 2 2561286432168421 8 32,7684,096 5126481 16 65,5364,096 256161 Power

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Quit 0 1 0 64 6 Integer Remainder 1101 Binary 24816 32 2 1 2 345 Base 22 10 22 Base 10 to Base 2 Power 22/16 6 6/8 6 6/4 2 0 2/2 0/1 0 = 10110 2

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Quit 22 Base 10 to Base 2 0 ( 112 ) 10110Base 2 ( 022 ) ( 152 ) ( 1 112 ) ( 0222 ) Remainder Quotient 22Base 10

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Quit 0612345 0 164 42/32 Integer Remainder 10101 Binary 24816 32 2 Base 42 10 42 Base 10 to Base 2 10 Power 10/16 10 10/8 2 2/4 2 0 2/2 0/1 0 = 101010 2

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Quit 42 Base 10 to Base 2 1 ( 022 ) 101010Base 2 ( 152 ) ( 0102 ) ( 1 212 ) ( 0422 ) Remainder Quotient 42Base 10

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Quit 1 Addition in Binary 1010 1010 + ––––––––– 010 Carry 10

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Quit 1 ––––– Multiplication in Binary 11 11 × 01 + 11 110 01 ––––––

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Quit 126 Base 10 to Base 8 176Base 8 0 ( 118 ) ( 7 158 ) ( 61268 ) Remainder Quotient 126Base 10

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Quit 126 Base 10 to Base 16 7EBase 16 0 ( 7 78 ) (12616 ) Remainder Quotient 126Base 10 14E

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