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I notice… Exercise 2: Make a Complement The complement of a number “xxx” is and what makes “999?” The complement of 123 is 876. Note 1 + 8 =9 and 2 + 7 = 9 and 3 + 6 = 9. The complement of 46 is 53. Note 4 + 5 = 9 and 6 + 3 = 9. Work out the complements of these numbers 1) 234 (2) 222 (3) 252 4) 1234 (5) 5555 (6) 0000 7) 18 (8) 45 (9) 99 What do you notice about finding complements? I notice Work out the complements of these numbers 10) 222 (11) 44 (12) 8 13) 101 (14) 32123 (15) 0006 16) 45 (17) 9999999 (18) 121212121 Do you notice anything more? I notice The number 246 is made up of 200 + 40 + 6. This can be written in expanded form as 2 x 100 + 4 x 10 + 6 x 1, and is part of learning about place value. Writing numbers like this should be something you have already learned and practised. Write these numbers showing the place values in expanded form. 19) 34 (20) 789 (21) 5555 Write these numbers showing the place values in expanded form. The different characters are being used for different digits (so think of (( as a two digit number, were each digit is different) 22) (( (23) ((( (24) (((( What is the complement of each of these numbers? 25) 34 (26) (( (27) 789 28) ((( (29) 5555 (30) (((( Exercise 3: Complementary Subtraction This is a cool trick. You are going to learn how to subtract two numbers by adding. The problem 57 – 21 is easily seen to have the answer 36. There is no “carry” or “renaming” to be done so the tens digits are subtracted and the unit digits are subtracted. In the tens place 50 – 20 = 30 and in the units or ones place 7 – 1 = 6. The problem 51 – 27 requires a bit more thinking and understanding of how numbers work. Fifty-one (51) can be renamed forty-eleven (40 + 11). This is a good choice for this problem but is not the only choice available. The answer is now 40 – 20 = 20 in the tens place and 11 – 7 = 4 in the ones or units place. The answer is 20 + 7 = 27. BUT… like most ideas in mathematics there is another way. 1) In the first problem 57 – 21 we calculate the complement of 21. The complement of 21 is 78. Now add 57 and 78. We add 50 + 70 to get 120 and then 7 + 8 to get 15. The answer is 120 + 15 = 135 What do you notice about 135 and the answer to 57 – 21? I notice 2) In the second problem 51 – 27 we do the same procedure. It is just as easy. The complement of 27 is 72. We add 51 + 72 and get 120 + 3 = 123. What do you notice about 123 and the answer to 51 – 27? I notice You might have noticed that the answer is always 100 bigger and 1 smaller than the real answer. Now for a really good question! 3. Why is it that the complement answer is always 100 bigger and 1 smaller than the actual answer to a 2-digit subtraction problem? I think it is because… Exercise 4: More Complementary Subtraction Complementary subtraction works with many digit numbers. In this exercise we use base 10 numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In a later exercise we use base 2 numbers 0, 1 and see it is very simple to do. Computers use complementary base 2 addition to subtract. First let’s practice our complementary addition in base 10. Here is a 3 digit subtraction problem solved using complementary arithmetic. 521 – 257 = 521 + 742 -1000 + 1 = 12 x 100 + 6 x 10 + 3 x 1 - 10 x 100 + 1 x 1 = 264 Your turn. Complete these 3 digit subtractions using complementary arithmetic. 1) 631 – 243 2) 876 – 387 3) 315 – 076 Make a comment about complementary subtraction. Do you notice anything? Humm… Now try and solve these problems in the same way. Make sure you add 0 place holders when you need. That is why the 0 is in the hundreds place in (3) above. 4) 83 – 57 5) 8765 – 1875 6) 227 – 78 7) 2709 – 883 8) 66666 – 777 Can you think of a reason why humans use base ten? Is it the only base we have ever used? Martians have one finger on each hand and use base 2. Venusians have 3 fingers on one hand and 4 on the other. What base would they use? Answer here Earthling Exercise 5: Martian Maths 12 + 12 = 102 A very very old mathematician’s joke reads “There are 3 types of mathematicians; those that can count and those that can’t.” After you have done this activity you will understand a similar joke that says “There are 10 types of mathematicians; those that can count and those that can’t.” Counting can be done in a very large number of ways. The simplest is based on the number two. Martians count this way. All of the Martians I have met had one finger on each hand and one toe on each foot. They are very good at pointing. A Martian only uses 0’s and 1’s when counting. The following table shows the start of counting using Martian numbering. See if you can finish the pattern. Look carefully. HumanMartianHumanMartianZero000001410000115200010163000111710001400100185001011960011020721801000229231024112512011001311011 Explain the Martian counting system in your own words. My words! 1) Invent a way to make sure that numbers written in Martian (like 101) do not get confused with numbers written in Earthling (like 101). Discuss your convention with your teacher. 2) Explain the joke about 10 types of mathematician 3) If each finger on your hand is a new column of a Martian number, how many can you count to, Earthling, using the Martian System? (a) on one hand? (b) on two hands? (c) what is the largest number you can count to using your body bits?Exercise 6: Complements in Base 2 This is the easiest of all exercises. You could be a Martian and get these right. The complement of 0 is 1 and the complement of 1 is 0 in base 2. All we do to find the complement of a number in base 2 is to flip all the digits between 0 and 1. The complement of 00101011 is 11010100. You are more likely to loose track of where you are up to than get the “flipping thing” wrong. Try these problems. 1) 101 (2) 111 (3) 1010 4) 0110 (5) 01010101 (6) 11111111 7) 01100110 (8) 11000011 (9) 00000001 This is so easy that a little electronic circuit was made up to do this job. The circuit is called an inverter. It looks like this. The little o means the output is opposite of the input. These circuits come in sixes on the 4049 chip. Logic ChartInOut0110In electronic circuits a “0” means “off” or no voltage and a ”1” means “on” or 9 Volts in many circuits. To “flip” or invert an 8 bit binary number like 01101100 we use 8 inverters all working in parallel or “processed at the same time”. It looks like this. Experience Task Make sure the power is OFF. Find an old computer and open it up. Look for the highways of 8 and 16 tracks running around the circuit board. These look like highways joining little townships of computer chips. These “highways” are the parallel tracks that convey all the information in and around the main processor. The highways you are looking at probably connect hard disks or CD drives to the main circuit board. Look for a hex-inverter chip in Dick Smith Electronics. Exercise 7: Computers, Adding Another little electronic device computers use is called an AND gate. The AND gate is represented by the symbol as shown and behaves with this logic. Logic ChartInput AInput BCarryOutput0000010110011110 This all looks quite complicated. A little explanation and all will be well. In Base 2 arithmetic the basic addition facts are 0 + 0 = 00 0 + 1 = 01 1 + 0 = 01 and 1 + 1 = 10. Look at the Logic Chart and note how the adder circuit mimics this arithmetic. Making 8 adder chips work in parallel (see Exercise 6) makes a circuit that can work with numbers up to 111111112 in size. If you did Exercise 5 you will be able to see that this is the number 255 in base 10 which is not very big at all. But it is a start. Modern computers use 64 or 128 bit adding. How big are these base two numbers? Try adding these 4-bit numbers in base 2 1) 1110 + 0001 (2) 1100 + 0010 (3) 1001 + 0110 Add the corresponding parts just as you do in the decimal system. When you have 1 + 1 you need to “carry” just as you do in the decimal system. Try adding these 4-bit numbers with one carry in each. 4) 1110 + 0010 (5) 1100 + 0100 (6) 1001 + 0001 Here are some mega problems with lots of carries. 7) 1100110011 + 0011111111 (8) 00001111 + 00001111 (9) 1111 + 1111 Note: Many of these problems show what happens to a computer when you run out of room …”overload error”. Task: Make up some of your own base 2 addition problems and work out the answers. Exercise 8: Computers, Multiplying Look at the following base 10 multiplication problems. 4 x 6 = 6 + 6 + 6 + 6 3 x 20 = 20 + 20 + 20 Write a statement about what you see. If you can see that multiplying is just a better and more efficient way of adding then you are exactly correct. Multiplying can be thought of as “repeated addition”. This is exactly how computers multiply. 4 x 6 becomes 6 (once) 12 (twice) 18 (thrice) 24 (four-times)….stop, display the answer. Inside is a little counting circuit that counts the number of times and stops when it gets to 4. Here is an algorithm or programme that will multiply two numbers. Start Check TableABCCount661241263 10 Get first number (4) 20 Store in memory Count 30 Get second number (6) 40 Store in memory A 50 Store in memory B 60 Add memory A and memory B and store in memory C 70 Move C to A 80 Decrease Count by 1 90 Is count equal to 1 (NO…Goto 60, YES Goto 100) 100 Display C End TASK A The Check Table lets you think through the logic in steps just as a computer would do very quickly. Your task is to finish the chart. Start from the beginning. TASK B Make a new check table and solve these problems. 1) 3 x 7 (2) 8 x 9 (3) 1012 x 112 Exercise 9: Computers, Dividing Dividing two numbers is one of the slowest operations that a computers does. This is because it is a multi-step procedure and takes longer. How do you think a computer would divide 12 by 3. My answer Here is a programme or algorithm that a computer could use to do 12 divided by 3. Check Logic TableABCDCount123618096151NO66122NO3693NO04YESStart 10 Get first number Numerator 20 Store in A 30 Get second number Denominator 40 Store in B 50 Set Count to 0 60 Find complement of B 70 Store in C 80 Add A to C and store in D 90 Take off 10 and add 1 to D and store in A 100 Increase Count by 1 110 Test if A = 0 (NO… Goto 80, YES…Goto 120) 120 Display Count as the answer. End TASK A Work through the programme and see if you can repeat Check Logic Table correctly. TASK B Make a new check table and solve these problems. 1) 20 divided by 4 (2) 75 ÷ 15 (3) 10002 ÷ 102 Exercise 10: Computers, Powers and other Functions Computers can only do the operation of addition and they only do this using binary numbers. They can also find the complement of a number. How does a computer do these mathematical operations?AdditionUses addition ability of computerSubtractionUses complementary additionMultiplicationUses repeated additionDivisionUses repeated subtraction and countsPowersUses repeated addition repeatedly The computer does all of this in binary and at about half of the speed of the main processing chip oscillator. This is more than 200Mhz or 100 million calculations per second. Modern computers use several processors running together and are at least 10 times faster again. TASK Choose one of the following topics and investigate more about it on the internet and in your school library. Make up a colourful A3 poster recording what you find out. Include pictures, diagrams and questions. Your teacher will be able to use this for wall displays in the classroom and you may be asked to present you project to other students. LIST (select one) History of computers Binary arithmetic Cost of memory chips – ‘yesterday and today’ Types of computers Computers in Spacecraft The Shuttle Computers Systems Writing a programme to sort three numbers from smallest to largest Write a program to show how a computer would work out 23. Create a check logic table to show that it works properly Other Topic…check with teacher. Computers Can Only Add Answers Exercise 1 1) 7 (2) 6 (3) 2 4) 8 (5) 4 (6) 9 7) 3 (8) 5 (9) 1 10) 79 (11) 49 (12) 89 13) 87 (14) 15 (15) 32 16) 66 (17) 45 (18) 76 19) 777 (20) 555 (21) 111 22) 876 (23) 678 (24) 753 25) 545 (26) 984 (27) 992 28) 8765 (29) 3210 (30) 4444 31) 7531 (32) 6420 (33) 8282 34) 81 (35) 9979 (36) 6857 37) 87654 (38) 43210 (39) 88888 40) 97531 (41) 96969 (42) 02368 43) 99799 (44) 49 (45) 648 I notice… There are patterns in the digits of some numbers - like 18 become 81. Overall, each digit can be considered on its own without looking at the other digits. Exercise 2 1) 765 (2) 777 (3) 747 4) 8765 (5) 4444 (6) 9999 7) 81 (8) 54 (9) 00 10) 777 (11) 55 (12) 1 13) 898 (14) 67876 (15) 9993 16) 54 (17) 0000000 (18) 878787878 I notice… There is not much extra to notice than was the case for exercise 1 19) 3 ( 10 + 4 ( 1 (20) 7 ( 100 + 8 ( 10 +9 ( 1 21) 5 ( 1000 + 5 ( 100 + 5 ( 10 + 5 ( 1 22) ( ( 10 + ( ( 1 (23) ( ( 100 + ( ( 10 + (( 1 24) ( ( 1000 + ( ( 100 + ( ( 10 + ( ( 1 25) 65 (26) (9 - () ( 10 + (9 - () ( 1 (27) 210 For these next questions, square brackets have been invented to show what is happening in each column of the number, and to keep each column apart. 28) [9 - (][9 - (][9 - (] (29) 4444 30) [9 - (][9 - (][9 - (][9 - (] Exercise 3 1) You should notice the number is 99 more than the answer (100 bigger and 1 smaller). 2) You should notice the number is 99 more than the answer (100 bigger and 1 smaller). 3) The reason is not obvious. In the 1st problem 57 -21 we create the following chain. 57 - 21 becomes 57 + 99 – 21 = 57 + 100 – 21 - 1 = 100 + 57 - 21 - 1 = 100 bigger and 1 smaller than the original problem as suggested in the exercise. Exercise 4 1) 6 ( 100 + 3 ( 10 + 1 ( 1 + 7 ( 100 + 5 ( 10 + 6 ( 1-10 ( 100 + 1 = 388 2) 8 ( 100 + 7 ( 10 + 6 ( 1 + 6 ( 100 + 1 ( 10 + 2 ( 1 - 10 ( 100 + 1 = 489 3) 3 ( 100 + 1 ( 10 + 5 ( 1 + 9 ( 100 + 2 ( 10 + 3 ( 1 – 10 ( 100 + 1 = 239 I notice… Comment will vary, though and important one is that the system only works when subtracting numbers that have the same columns, so for 347 – 56, the complementary addition is 943, not just 43. Author notes the method is quite easy to do but a bit tedious. 4) 8 ( 10 + 3 ( 1 + 4 ( 10 + 2 ( 1 - 10 ( 10 + 1 = 26 5) 8 ( 10000 + 7 ( 1000 + 6 ( 10 + 5 ( 1 + 8 ( 10000 + 1 ( 1000 + 2 ( 10 + 4 ( 1 - 10 ( 10000 + 1 = 6889 + 1 = 6890 6) 2 ( 100 + 2 ( 10 + 7 ( 1 + 9 ( 100 + 2 ( 10 + 1 ( 1 - 10 ( 100 + 1 = 149 7) 2 ( 1000 + 7 ( 100+0 ( 10 + 9 ( 1 + 9 ( 1000 + 1 ( 100 + 1 ( 10 + 6 ( 1 - 10 ( 1000 + 1 = 1819 + 6 + 1 = 1826 8) 6 ( 10000 + 6 ( 1000 + 6 ( 100 + 6 ( 10 + 6 ( 1 + 9 ( 10000 + 9 ( 1000 + 2 ( 100 + 2 ( 10 + 2 ( 1 - 10 ( 10000 + 1 = 65889 Probably base 3.5 at a rough guess. Exercise 5 The pattern is obvious to Martians, but Earthlings may not have much of a clue. You could try a web search on the binary system, or discuss the place value system for Martian mathematics with your teacher 1) The convention used by mathematicians is to put a subscript after the number, so 1012 shows the number is in base 2, while 1017 means the answer is in base 7. We don’t put a number for base ten, as this is the ‘normal’ base we usually use, so we only have to show when we are NOT using base ten. 2) 2 in Martian is written as 10 3) (a) One hand 31 (one less than 25) = 111112 (b) Two hands 210 - 1 = 1023 = 11111111112 (c) Two hands and two feet and your head is 221 – 1 = 2097154 = 1111111111111111111112 Exercise 6 1) 010 (2) 000 (3) 0101 (4) 1001 (5) 10101010 6) 00000000 (7) 10011001 (8) 00111100 (9) 11111110 Exercise 7 64 bit gives 264 – 1 as the biggest number. Try doubling to work out this number – or have a look at it on your calculator, or on a computer. 128 bit gives 1128 - 1 1) 1111 (2) 1110 (3) 1111 4) 10000 (5) 10000 (6) 1010 7) 10000110010 (8) 00011110 (9) 11110 Exercise 8 Multiplications can be written as repeated subtractions Task A Check TableABCCount6612412618318624224241 Task B 1) (2) (3) Check TableCheck TableCheck TableABCCountABCCountABCCount3367881691011011010116396168248101010111111093125248327111101123154328406153183408485183212488564211568643648722721 Exercise 9 The computer would repeatedly subtract, so for 12 ÷ 3 it would repeatedly subtract three and see how many times this happened – if the computer could subtract! Instead it will have to repeatedly add the complement of three until it gets to the number 12 and then say how many times it had to do this. 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