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ζ Bases Dr Frost Objectives: 1.To appreciate how we can have different number systems using different ‘bases’. 2.To convert numbers from decimal to another base. 3.To convert numbers from any base to decimal.

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1001100101 11010001 96E854 FFBA4D 000000 12958 23 8409284 These are all examples of numbers! What connects each group of numbers? 10 942

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Base The base of a number system is the number of possible values for each digit. Values for each digit BaseName of number system 0 to 910Decimal 0 to 12Binary 0 to F (A=10, B=11,... F=15) 16Hexadecimal ?? ?? ??

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Numbers in decimal If we were to write out 2493, what is the value of each digit? 2 4 9 3 1000 100 10 1 2000 +400 +90 + 3 = 2493 multiply ? ? ?

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Numbers in decimal Now suppose we had a number in base 5 instead. How do we convert it to decimal? 4 3 0 1 125 25 5 1 500 + 75 + 0 + 1 = 576 multiply ? ? ?

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Numbers in decimal Copy and complete in your book. 1 0 1 1 2 8 4 2 1 8 + 0 + 2 + 1 = 11 ? ? 3 3 0 2 4 64 16 4 1 192 + 48 + 0 + 2 = 242 ? ? The small number indicates the base. 1 2 2 0 3 27 9 3 1 27 + 18 + 6 + 0 = 51 ? ?

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(Switch to ‘I’m a Mayan’ slides)

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Exercises Original numberIn base 10 (decimal) 1101 2 13 111 2 7 110011 2 51 1022 3 35 734 8 476 234 5 69 530 6 198 Base 2 ? ? ? ? ? ? ?

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Converting FROM decimal to other bases Do the opposite! Convert 18 to binary. 1 0 0 1 0 2 16 8 4 2 1 ? 16 + 0 + 0 + 2 + 0 = 18 ? ? ? ? ?

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Converting FROM decimal to other bases Convert 272 to base 5. 2 0 4 2 5 125 25 5 1 ? 250+ 0 + 20+ 2= 272 ? ? ??

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Converting FROM decimal to other bases It can help to write out multiples of your various powers. Below is base 6. Multiples of 6Multiples of 6 2 Multiples of 6 3 6 12 18 24 30 36 72 108 144 180 216 432 648 864 1080 x 1 x 2 x 3 x 4 x 5 Therefore what is 800 is base 6? 3412 ?

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Exercises DecimalBinaryBase 6 3113 810012 10101014 771001101205 1021100110250 1051101001253 13651010101010110153 ? ? ? ? ? ? ? ? ? ? ? ? ? ?

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Decimal to Hexadecimal The most well-known usage of hexadecimal is to represent colours. Each colour can be composed of red, green and blue light, each of intensity varying between 0 and 255....which can be represented using just 6 digits in hexadecimal, 2 for each of the three colour components. A means 10, B means 11,... F means 15

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02550 000 REDGREENBLUE 255 0 75172198 2551280 ??? ??? ??? ??? HEXADECIMAL FF, FF, FF 00, 00, 00 00, FF, 00 FF, FF, 00 4B, AC, C6 FF, 80, 00 ??? ? ? ? ? ? 0: 0 1: 16 2: 32 3: 48 4: 64 5: 80 6: 96 7: 112 8: 128 9: 144 A: 160 B: 176 C: 192 D: 208 E: 224 F: 240 Multiples of 16:

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Exercises Provided worksheet.

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Adding in other bases 1 0 0 1 + 1 1 0 1

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Adding in other bases 1 0 1 0 + 1 1 0 1 1

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Multiplying in other bases 1 0 1 0 x 1 0 1

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QQQ Time The number of possible values each digit can have. 1a 1b Because each digit must be between 0 and one less than the base/the digits must be less than the base. 2a 2 2b 178 3 551 4 5 6 7 8 3900 11011 240 100100 a = 2, b = 4 ? ? ? ? ? ? ? ? ? ?

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Multiplication X 1 1 x 1 = 1 2 x 1 = 2 3 x 1 = 3 4 x 1 = 4 5 x 1 = 5 6 x 1 = 6 7 x 1 = 7 8 x 1 = 8 9 x 1 = 9 10 x 1 = 10 11 x 1 = 11 12 x 1 = 12 X 2 1.

Multiplication X 1 1 x 1 = 1 2 x 1 = 2 3 x 1 = 3 4 x 1 = 4 5 x 1 = 5 6 x 1 = 6 7 x 1 = 7 8 x 1 = 8 9 x 1 = 9 10 x 1 = 10 11 x 1 = 11 12 x 1 = 12 X 2 1.

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