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KS3: Bases Dr J Frost ([email protected]) Objectives: 1. To appreciate how we can have different number systems using different ‘bases’. 2. Count in different bases. 3. To convert numbers from decimal to another base. 4. To convert numbers from any base to decimal. Last modified: 22nd June 2015 Starter What 3 numbers comes next in each sequence? Base 10 Base 2 Base 3 Base 5 Base 2 Base 16 5 6 7 8 9 10 11 12 0 1 10 11 ? 100 2 10 11 12 344 400 401? 402 111 1000 1001? 1010 99 9A 9B ? 9C ? ? Follow up questions: • What do you think it means for our ‘normal’ number system to be base 10? Each digit has 10 ? possible values. • What is the rule for counting (if say in ‘base 6’) Once the digit goes past 5, we jump back ? to 0 (and the digit to the left goes up by 1) Base ! The base of a number system is the number of possible values for each digit. Values for each digit 0 to 9 0 to 1 0 to F (A=10, B=11, ... F=15) Base 10 2 16 ? ? ? Name of number system Decimal? Binary ? Hexadecimal ? Counting Game! Everyone stand up. Take it in turns to count in ternary (base 3), starting at 0. If you get it wrong, you sit down. 0 1 2 10 11 12 20 21 22 100 101 102 110 111 112 120 121 122 200 201 202 210 211 212 220 221 222 1000 1001 1002 1010 1011 1012 1020 1021 1022 1100 1101 1102 1110 1111 1112 1120 1121 1122 1200 1201 1202 1210 1211 1212 1220 1221 1222 2000 2001 2002 2010 2011 2012 2020 Exercise 1 1 Write out the first ten numbers in each of these bases, starting at 1. a Base 2: 1, 10, 11, 100, 101, 110, ? 111, 1000, 1001, 1010 b Base 5: 1, 2, 3, 4, 10, 11, 12, 13, ? 14, 20 c Base 4: 1, 2, 3, 10, 11, 12, 13, 20,?21, 22 2 a b What number comes after 555 in: Base 6? 1000 ? Base 7? 556 ? 3 How many times does the digit 0 occur if you write out the numbers 1 to 111111 in binary? (Hint: consider all two-digit numbers, then three, and so on) 2 digit numbers: 1 occurrence 3 digit numbers: 4 occurrences 4 digit numbers: 12 occurrences ? 5 digit numbers: 36 occurrences 6 digit number: 108 occurrences Total = 161 Any Base → Decimal If we were to write out the digits of the decimal number “2493”, what is the value of each digit? (Hint: Think primary school!) 1000 100? 10 multiply 1 2 4 9 310 ? 2000 +400 +90?+ 3 = 2493 This means number is in base 10. We don’t include it if the base is obvious from the context. Any Base → Decimal Now suppose we had a number in base 5 instead. How do we convert it to decimal? 125 25 ? 5 multiply 1 4 3 0 15 500 + 75 + 0 ?+ 1 = 576? Test Your Understanding Copy and complete in your book. 8 4 ?2 64 16 ? 4 1 1 0 1 12 8 + 0 +? 2 + 1 = 11 27 9 ?3 1 3 3 0 24 192 + 48 + ? 0 + 2 = 242 1 1 2 2 03 27 + 18 +? 6 + 0 = 51 The Maya numeral system is base 20 (“vigesimal”). Use the approach you used for converting other bases to decimal to vote for the correct number. Example 20 60 3 30 1 + 0 = 0 60 300 Q1 20 120 66 126 1 + 6 = 6 105 156 Q2 123 243 53 223 Q3 123 243 53 223 Q4 239 144 129 1 Q5 400 400 121 20 + 211 0 1 + 11 = 411 111 411 Q6 490 1180 1980 1380 Q7 8000 152000 157784 400 + 5600 582984 20 + 180 = 396884 1 +4 196884 Exercise 2 1 Convert the following numbers from the indicated base to decimal. 11012 1112 1100112 10223 7348 2335 5306 2 13 7 51 35 476 68 198 ? ? ? ? ? ? ? What is the following Mayan number in decimal? 4 ? 5 When the number “a036” in base 7 is converted to decimal, the value is 1742. Determine the value of the digit 𝑎. 𝟑𝟒𝟑𝒂 + 𝟎 + 𝟐𝟏 + 𝟔 = 𝟏𝟕𝟒𝟐 𝒂=𝟓 ? In general, what is the largest number in decimal that can be represented by 𝑛 binary digits? Give your answer in terms of 𝑛. 𝟐𝒏 − 𝟏 ? 6 A three-digit number is 100 in decimal. What’s the smallest the base can be? In base 4 the biggest number in decimal is 𝟒𝟑 − 𝟏 = 𝟔𝟑, whereas in base 5 it’s 𝟓𝟑 − 𝟏 = 𝟏𝟐𝟒. So 5 is the smallest base. ? N 160?001 3 In computing, a byte consists of 8 bits, where each bit is a binary digit. What is the largest possible number in decimal that a byte can represent? 𝟐𝟓𝟓 The number with digits "𝑎1𝑏", where 𝑎 and 𝑏 are unknown digits, is 107 in decimal if the number was originally in base 5, and 205 in decimal if it was originally in base 7. Determine 𝑎 and 𝑏. 𝟐𝟓𝒂 + 𝟓 + 𝒃 = 𝟏𝟎𝟕 𝟒𝟗𝒂 + 𝟕 + 𝒃 = 𝟐𝟎𝟒 Solving, 𝒂 = 𝟒, 𝒃 = 𝟐. ? N 432 is in an unknown base, but when converted to decimal, gives 164. Determine the base. Let the base be 𝒃. Then 𝟒𝒃𝟐 + 𝟑𝒃 + 𝟐 = 𝟏𝟔𝟒. Solving this quadratic equation gives 𝒃 = 𝟔. ? Summary So Far We have learnt that the numbers we use in everyday life are in “base 10”. ? But numbers can be in any ‘base’ such as base 2 (binary). ? The base of a number system is the number of possible ? values for each digit. To convert a number to decimal, we just consider the value of each digit, just like in decimal each digit represents “units”, “tens”, “hundreds” and so on. 14035 = 𝟏𝟐𝟓 + 𝟏𝟎𝟎?+ 𝟑 = 𝟐𝟐𝟖 Decimal → Any Base Do the opposite! Convert 18 from decimal to binary. 16 8 ?4 2 1 1? 0? 0? 1 0 2 ? ? 16 + 0 + 0 + 2 + 0 = 18 Bro Tip: Start with the highest multiple possible of the highest power (in this case 16). Then see what’s left and continue to get the digits. Another Example Convert 272 for decimal to base 5. 125 25 ? 5 1 2? 0 4 2 ? ? ? 5 250+ 0 + 20+ 2= 272 Test Your Understanding Convert 100 from decimal to base 4. ? 64 16 4 1 1? 2? 1? 0? 4 64 +32 + 4 + 0 = 100 Decimal → Any Base It can help to write out multiples of your various powers. Below is base 6. Multiples of 6 x1 x2 x3 x4 x5 6 12 18 24 30 c. We can only have 1 lot of 6. Multiples of 62 Multiples of 63 36 72 108 144 180 216 432 648 864 1080 b. We can have 4 lots of 62. Therefore what is 800 is base 6? ? 3412 a. We can have 3 lots of 63. Exercise 3 1 Copy and complete the following table. Decimal Binary (Base 2) 3 ? 8 10 77 102 105? 1365 2 11 ? 1010 ? 1001101 ? 1100110 ? 1101001 ? 10101010101 ? 1000 3 ? 12 ? 14 ? 205? 250? N 10153 ? a horizontal line means 5, a dot 1 and a shell 0). 1000 is a four-digit decimal number whose first digit one. In what other bases can this can be converted to such that we still have a four-digit number which starts with 1? If the base is 𝒃, 𝒃𝟑 has to be between 500 and 1000 (if it were less, the first digit wouldn’t be 1). Only 8 and 9 satisfy this. ? 253 Convert 123 in decimal to Mayan numerals (recall that Mayan is base 20, and that ? ? Base 6 3 The decimal number “7a2” is 10322 in base 5. Determine the digit 𝑎. 𝒂=𝟏 N Prove that there is no base 𝑏 such that 123 in decimal can be converted to: i. 45 in that base. 𝟒𝒃 + 𝟓 = 𝟏𝟐𝟑 𝟓𝟗 𝒃= 𝟐 ii. 456 in that base. 𝟒𝒃𝟐 + 𝟓𝒃 + 𝟔 = 𝟏𝟐𝟑 Solving gives 𝒃 = 𝟒. 𝟖𝟐, −𝟔. 𝟎𝟕, neither or which are integers. ? ? Decimal → 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 Multiples of 16: 0: 1: 2: 3: 4: 5: 6: 7: 8: 9: A: B: C: D: E: F: 0 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 RED GREEN BLUE HEXADECIMAL 255 255 255 FF, FF, FF 0? 0? 0? ? 00 00, 00, 0? ? 255 0? ? 00 00, FF, ? 255 ? 255 0? ? 00 FF, FF, ? 75 ? 172 ? 198 ? C6 4B, AC, ? 255 ? 128 0? ? 00 FF, 80, Adding in decimal + 2 3 0 6 5 1 3 9 3 + 9 = 12 We’d use the 2 then carry the 1. Adding in other bases + 1 ? 1 1 0 ? 0 1 1 ? 0 0 1 ? 1 1 1 0 ? Another Example + 1 1 0 1 1 0 ? 0 0 1 1 1 0 0 1 1 Test Your Understanding 1 + 1 0 1 1 1? 0 1 0 1 1 0 2 + 3 3 2 0? 0 2 3 35 45 2 Exercise 4 Convert the following to hexadecimal. QQQ Time 1a The number of possible values each ?digit can have. 4 3900 ? 5 11011 ? 6 2400 ? 7 100100 ? 8 a = 2, b =?4 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 ? 551 ? 3