0:00:01.360,0:00:07.732 Mathematics has its own[br]language, most of which which 0:00:07.732,0:00:14.104 were very familiar with. For[br]example, the digits 012. 0:00:14.120,0:00:21.590 345678 and nine are part[br]of our everyday lives, and 0:00:21.590,0:00:27.566 whether we refer to this[br]digit as zero. 0:00:28.100,0:00:31.040 Nothing. 0:00:31.570,0:00:38.045 Note[br]Oh oh, as in the 0:00:38.045,0:00:42.350 telephone code 0191 we[br]understand its meaning. 0:00:43.650,0:00:47.775 There are many symbols in[br]mathematics and most of them are 0:00:47.775,0:00:50.025 used as a precise form of 0:00:50.025,0:00:54.962 shorthand. We need to be[br]confident using the symbols and 0:00:54.962,0:00:59.142 to gain that confidence, we need[br]to understand their meaning. 0:01:00.460,0:01:06.576 To understand that meaning,[br]we've got two things to help us, 0:01:06.576,0:01:11.947 there's context. What else is[br]with the symbol? What's with it? 0:01:11.947,0:01:13.839 What's it all about? 0:01:14.690,0:01:19.184 And this convention and[br]convention is where 0:01:19.184,0:01:23.036 Mathematicians and[br]scientists have decided that 0:01:23.036,0:01:26.888 particular symbols will[br]represent particular things. 0:01:28.340,0:01:29.852 Let's have a look at some 0:01:29.852,0:01:36.672 symbols. That sign words[br]associated with it 0:01:36.672,0:01:39.250 a plus. That 0:01:40.260,0:01:43.580 Increase. 0:01:44.430,0:01:51.092 And positive. As[br]it stands, it has some form of 0:01:51.092,0:01:56.958 meaning, but we really need it[br]in a context. For example, 2 + 3 0:01:56.958,0:02:02.405 or two ad three. We know the[br]context, we add the two numbers 0:02:02.405,0:02:04.081 and we get 5. 0:02:05.300,0:02:10.173 Let's have a look at another[br]context. Now, if you've gone 0:02:10.173,0:02:15.489 abroad, say to Europe, and you'd[br]gone to a conference, you might 0:02:15.489,0:02:20.362 hand out some business cards,[br]and on your business card so 0:02:20.362,0:02:24.792 that people could contact you.[br]You might have your telephone 0:02:24.792,0:02:30.551 number, so chances are it will[br]be written as plus 44, and let's 0:02:30.551,0:02:32.766 have a phone number 1911234567. 0:02:33.930,0:02:40.890 And in this context, that plus[br]means that the person dials the 0:02:40.890,0:02:45.530 necessary code for an[br]international line from their 0:02:45.530,0:02:47.270 country, then 441911234567. 0:02:47.920,0:02:53.002 So it still means In addition[br]to, but not in the same way as 0:02:53.002,0:02:56.995 that plus sign were not actually[br]adding the number on. We're 0:02:56.995,0:02:58.447 doing it In addition. 0:03:00.000,0:03:03.872 Let's have a look 0:03:03.872,0:03:08.418 now. At this[br]sign. 0:03:09.630,0:03:14.990 Well, it's associated with this[br]one and minus. 0:03:14.990,0:03:18.955 Subtract. Take 0:03:18.955,0:03:20.430 away. 0:03:20.430,0:03:23.760 Negative. 0:03:23.760,0:03:30.792 And decrease.[br]And again, we need a context. 0:03:30.792,0:03:36.083 Let's six takeaway four or six.[br]Subtract 4 and we all know the 0:03:36.083,0:03:37.711 answer to be 2. 0:03:38.290,0:03:43.678 In a different context, if we[br]have minus 5 degrees C for 0:03:43.678,0:03:48.138 example. That means a[br]temperature of five 0:03:48.138,0:03:49.752 degrees below 0. 0:03:51.490,0:03:54.410 How about this one? 0:03:54.960,0:04:02.148 Words associated with[br]this symbol multiply. 0:04:02.150,0:04:09.340 Lots of. Times Now[br]this one is really just a 0:04:09.340,0:04:11.095 shorthand for adding. 0:04:12.120,0:04:17.064 Let's have a look at the[br]context. For example, six at 6, 0:04:17.064,0:04:19.536 at 6, at 6 at 6. 0:04:20.390,0:04:24.010 What we've actually got a 0:04:24.010,0:04:31.080 123456 is. And in short[br]hand we want to write 5 sixes. 0:04:31.800,0:04:36.597 On the symbol that's used is the[br]multiply sign, so we've got five 0:04:36.597,0:04:39.180 lots of 6 or 5 * 6. 0:04:40.720,0:04:45.032 Now this shorthand becomes even[br]shorter when we use some letters 0:04:45.032,0:04:46.208 instead of numbers. 0:04:46.930,0:04:53.902 So for example, instead of six,[br]let's say we were adding AA 0:04:53.902,0:04:56.807 again, will have 5 days. 0:04:57.990,0:05:02.676 But because we've used the[br]letter, if we use the multiply. 0:05:03.210,0:05:06.594 It could be confused[br]with the letter X. 0:05:08.120,0:05:12.813 So we don't write it in.[br]Basically we miss it out and we 0:05:12.813,0:05:16.423 just write 5, eh? So the[br]shorthand becomes shorter, so 0:05:16.423,0:05:20.394 when letters are involved, the[br]multiply sign is missed out is 0:05:20.394,0:05:22.560 the only symbol that we miss 0:05:22.560,0:05:28.358 out. Let's look[br]at division. 0:05:29.960,0:05:34.459 Now this symbol can actually be[br]written in three different ways. 0:05:34.610,0:05:37.406 We could have 10 / 5. 0:05:38.170,0:05:44.820 10 / 5 as it's written in a[br]fraction or 10 with a slash 0:05:44.820,0:05:49.570 and five now this ones come[br]about mainly because of 0:05:49.570,0:05:53.845 printing and typing and using[br]a computer. When these 0:05:53.845,0:05:57.645 symbols are not readily[br]available on the keyboard. 0:05:58.780,0:06:05.605 So how many times does 5 go[br]into 10 or 10 / 5? 0:06:06.920,0:06:11.727 Another very common symbol that[br]we use all the time without 0:06:11.727,0:06:13.912 thinking is the equals sign. 0:06:14.820,0:06:19.929 Again, it doesn't mean anything[br]on its own. We need it in some 0:06:19.929,0:06:25.431 form of context. So if we have[br]the sum 1 + 2 equals 3. 0:06:26.010,0:06:32.346 What we're saying is whatever we[br]have on this side is exactly 0:06:32.346,0:06:35.000 equal to. Whatever we have on 0:06:35.000,0:06:38.980 this side. Or it's the same as? 0:06:39.960,0:06:43.188 Variations on the 0:06:43.188,0:06:49.832 equals sign. Of the[br]not equal sign, a line through 0:06:49.832,0:06:52.766 the equals which is not equal 0:06:52.766,0:06:58.464 to. For example, X is not[br]equal to two. 0:06:59.260,0:07:02.476 So we know that X does not take[br]that value of two. 0:07:03.560,0:07:09.552 Two WAVY lines without[br]equals means approximately 0:07:09.552,0:07:11.264 equal to. 0:07:12.290,0:07:18.626 So it may not be[br]exact, but it's approximately 0:07:18.626,0:07:21.442 equal to that value. 0:07:23.230,0:07:28.018 Are greater than[br]or equal sign? 0:07:29.060,0:07:36.206 An example here[br]might be X 0:07:36.206,0:07:43.352 is greater than[br]or equal to 0:07:43.352,0:07:49.418 two. That means X could take the[br]value of two, but it could be 0:07:49.418,0:07:50.998 any number larger than two. 0:07:52.530,0:07:56.445 And are less 0:07:56.445,0:08:01.129 than. Or equal[br]2. 0:08:02.270,0:08:08.757 I'm here for example. Why would[br]be less than or equal to 7? 0:08:09.530,0:08:13.989 Why could equals 7? But it could[br]be any number less than 7. 0:08:14.920,0:08:18.868 And an easy way of remembering[br]which is the greater and the 0:08:18.868,0:08:22.816 less than part is the greater[br]is the wider part of the 0:08:22.816,0:08:25.777 symbol, whatever's on the[br]wider part of the symbol, 0:08:25.777,0:08:29.725 whatever's on that side is the[br]greater than the one where the 0:08:29.725,0:08:31.699 point is on the other side. 0:08:33.600,0:08:39.536 Other forms[br]of mathematical 0:08:39.536,0:08:42.504 symbols are 0:08:42.504,0:08:49.265 variables. I'm[br]basing used where things 0:08:49.265,0:08:51.944 take different values. 0:08:52.710,0:08:56.100 For example, imagine your[br]journey to work in your car. 0:08:56.670,0:09:01.565 And imagine the speed that[br]you're traveling at as you go 0:09:01.565,0:09:05.570 along your journey, your[br]speed will change, so we 0:09:05.570,0:09:10.465 might refer to the speed with[br]a letter. Let's say the 0:09:10.465,0:09:14.025 letter V, because throughout[br]your journey the actual 0:09:14.025,0:09:15.805 numerical value is changing. 0:09:18.000,0:09:21.312 We usually use letters for[br]variables, letters of the 0:09:21.312,0:09:23.152 alphabet. You can call them 0:09:23.152,0:09:28.305 anything. But we try to use[br]letters of the alphabet and we 0:09:28.305,0:09:32.925 try to have those letters giving[br]us some indication of what our 0:09:32.925,0:09:34.465 variable is all about. 0:09:35.370,0:09:40.866 For example, we might use[br]deep for distance. 0:09:41.800,0:09:44.828 And T for time. 0:09:48.310,0:09:54.025 By convention, we use you to[br]be initial speed. 0:09:54.800,0:10:01.046 And V to[br]be our final 0:10:01.046,0:10:07.896 speed. But of course, we[br]might also refer to volume. 0:10:08.770,0:10:13.462 And This is why context is[br]important and we look to see 0:10:13.462,0:10:18.545 what a variable is with as to[br]what it might mean. So, for 0:10:18.545,0:10:22.846 example, if we were to CV is D[br]divided by T. 0:10:24.240,0:10:29.376 And we're told that Diaz[br]distance and T is time. Then we 0:10:29.376,0:10:31.944 know that that V is speed. 0:10:33.470,0:10:40.640 Whereas if we saw V[br]with Four Thirds Paillard cubed. 0:10:41.180,0:10:47.746 Where are is the radius of a[br]sphere. We know that that V is 0:10:47.746,0:10:50.560 also the volume of a sphere. 0:10:51.730,0:10:56.436 Now going back to our example[br]again of our journey to work, we 0:10:56.436,0:11:01.866 might want to record the time it[br]took us to go to work, and we 0:11:01.866,0:11:05.486 might want to do that over a[br]number of days. 0:11:06.900,0:11:11.580 The most sensible variable to[br]use would be a T for our time. 0:11:12.290,0:11:16.378 But of course, if we want to[br]record it for each day, how do 0:11:16.378,0:11:17.838 we distinguish between the TS? 0:11:18.530,0:11:23.978 Well, in this case we[br]use a subscript. 0:11:23.980,0:11:27.766 So for day 0:11:27.766,0:11:31.234 one. We might 0:11:31.234,0:11:38.282 use T1. The[br]next day to 0:11:38.282,0:11:45.271 223-2425. Or we might want[br]to be a little bit clearer and 0:11:45.271,0:11:50.683 actually put perhaps TM for the[br]time on Monday TT for Tuesday 0:11:50.683,0:11:56.546 TW. For Wednesday we would have[br]to do TH for Thursday so it 0:11:56.546,0:12:01.507 doesn't get confused with[br]Tuesday and TF for Friday, so we 0:12:01.507,0:12:06.468 can use these subscripts small[br]numbers or letters at the bottom 0:12:06.468,0:12:10.076 right of the variable to[br]distinguish between them. 0:12:11.150,0:12:16.372 Now, by convention,[br]mathematicians have decided that 0:12:16.372,0:12:23.832 we're going to use some[br]letters of the Greek alphabet 0:12:23.832,0:12:27.562 for some of our mathematical 0:12:27.562,0:12:33.210 symbols. For example,[br]we have pie. 0:12:33.980,0:12:40.840 Empires being chosen to equal[br]the number 3.14159 and it 0:12:40.840,0:12:46.328 goes on and on forever,[br]never repeats itself. 0:12:47.250,0:12:53.074 And so that we can precisely say[br]what we mean, rather than having 0:12:53.074,0:12:54.866 to round the value. 0:12:55.430,0:12:59.730 We use the letter Pi knowing[br]that it's that number. 0:13:01.120,0:13:07.444 From the Greek alphabet we also[br]use some other letters such as 0:13:07.444,0:13:10.585 Alpha. 's 0:13:10.585,0:13:14.305 pizza. And 0:13:14.305,0:13:21.682 feature. Now, these[br]are often used as variables to 0:13:21.682,0:13:27.880 represent angles. So you might[br]see an angle marked in that way, 0:13:27.880,0:13:29.970 and the Greek letter Theta. 0:13:31.470,0:13:36.760 Another one that is commonly[br]used is a capital Sigma. 0:13:37.580,0:13:41.991 Now you should be familiar with[br]this one from any spreadsheet 0:13:41.991,0:13:46.402 program on a computer, because[br]you'll find it somewhere on the 0:13:46.402,0:13:48.808 toolbar because it means the sum 0:13:48.808,0:13:55.630 of. And it's a shorthand for[br]adding up a column or a row of 0:13:55.630,0:14:00.030 numbers on a spreadsheet. So it[br]means the sum of. 0:14:01.880,0:14:06.350 The positioning of where we put[br]letters or figures without 0:14:06.350,0:14:13.350 symbols. Has gives meaning to[br]it, just like our subscripts we 0:14:13.350,0:14:15.750 can have superscripts now 0:14:15.750,0:14:20.260 superscripts. Instead of going[br]at the bottom right, go at the 0:14:20.260,0:14:24.992 top right. So for example, for[br]the little two at the top right 0:14:24.992,0:14:26.812 hand side, that's a superscript. 0:14:27.600,0:14:33.918 And again, it's a shorthand as[br]most of our symbols are and that 0:14:33.918,0:14:35.376 means 4 squared. 0:14:35.910,0:14:40.050 4 *[br]4. 0:14:41.840,0:14:44.630 If we have 4 cubed. 0:14:45.150,0:14:52.570 We've got three of them,[br]4 * 4 * 4. 0:14:53.230,0:14:57.757 Alright, a number that means to[br]the power of. 0:15:00.190,0:15:07.260 So we've raised 4 to the power[br]of three 4 * 4 * 4. 0:15:08.930,0:15:13.130 32 0:15:14.270,0:15:19.166 With the 0 as a superscript now[br]this is actually got two 0:15:19.166,0:15:23.455 meanings. And we need the[br]context to be able to know which 0:15:23.455,0:15:29.690 meaning it is. It could[br]be 32 degrees. 0:15:30.540,0:15:33.588 And that would mean an angle. 0:15:34.140,0:15:35.889 Of 32 degrees. 0:15:37.460,0:15:40.790 But it could mean 32. 0:15:41.290,0:15:43.129 To the power. 0:15:44.130,0:15:47.670 Of 0. 0:15:48.840,0:15:50.100 Which is actually one. 0:15:50.850,0:15:56.323 Which are very different meaning[br]to an angle. So you need to know 0:15:56.323,0:16:00.954 the context to be able to decide[br]which one it means. 0:16:00.960,0:16:05.624 Staying without superscript,[br]we could have 32 degrees 0:16:05.624,0:16:12.620 again, but this time with a[br]capital C after it, and that 0:16:12.620,0:16:17.284 would mean a temperature of[br]32 degrees Celsius. 0:16:19.580,0:16:22.540 Let's have a look at a couple of 0:16:22.540,0:16:26.024 numbers now. It could be 6, 0:16:26.024,0:16:28.540 three. Or it could be 63. 0:16:29.300,0:16:30.776 Let's put a little bit more 0:16:30.776,0:16:33.115 context there. If I put a comma 0:16:33.115,0:16:38.875 between them. Then I know it's[br]not 63, but I'm still not sure 0:16:38.875,0:16:44.825 what the meaning is. It could be[br]a number of things, but if I 0:16:44.825,0:16:49.925 then put brackets around it, I[br]know straight away that it's a 0:16:49.925,0:16:51.200 pair of coordinates. 0:16:51.270,0:16:58.790 And what it represents is[br]the position 6, three on 0:16:58.790,0:17:01.046 my coordinate axis. 0:17:01.810,0:17:09.670 So I would go six[br]in the X Direction, 3 0:17:09.670,0:17:17.530 in the Y direction and[br]that is the position that 0:17:17.530,0:17:20.674 that coordinate is representing. 0:17:22.380,0:17:24.424 Now brackets may be used in a 0:17:24.424,0:17:31.728 different way. If for example,[br]I saw a pee and 0:17:31.728,0:17:34.692 brackets H equals 1/2. 0:17:35.450,0:17:39.861 I know that's actually nothing[br]to do with coordinates at all, 0:17:39.861,0:17:42.668 and that is most likely to be 0:17:42.668,0:17:49.288 probability. And that is[br]saying the probability of 0:17:49.288,0:17:51.880 event H happening. 0:17:52.640,0:17:54.080 Is equal to 1/2. 0:17:54.860,0:17:56.904 I don't know what that event is, 0:17:56.904,0:18:00.295 I could. Surmise that it[br]could be the probability, 0:18:00.295,0:18:04.195 perhaps of scoring ahead when[br]I toss a coin is equal to 0:18:04.195,0:18:07.445 half, but I don't know I'd[br]need more information, but 0:18:07.445,0:18:10.045 certainly that it's something[br]to do with probability. 0:18:11.630,0:18:14.998 Another symbol we use. 0:18:15.000,0:18:17.739 The percentage symbol. 0:18:18.320,0:18:25.184 This is familiar to us because[br]here we've got our divide sign. 0:18:25.760,0:18:28.984 And this means out 0:18:28.984,0:18:36.048 of 100. So if we[br]have for example 90%, it means 0:18:36.048,0:18:38.216 90 out of 100. 0:18:39.060,0:18:45.438 Let's look at[br]a few more 0:18:45.438,0:18:48.627 symbols, for example. 0:18:49.140,0:18:51.960 R square root sign. 0:18:53.720,0:18:59.770 An example here might be the[br]square root of 16. 0:19:00.590,0:19:05.842 And it's the number that when we[br]multiply it by itself gives us 0:19:05.842,0:19:08.670 16. So the answer could be full. 0:19:09.170,0:19:14.065 Or it could be minus four,[br]because minus four times minus 0:19:14.065,0:19:15.845 four gives us 16. 0:19:16.610,0:19:20.647 Now, often in printed material,[br]you might just see the square 0:19:20.647,0:19:24.684 root is just the tick without[br]the line across the top. 0:19:25.550,0:19:27.270 If you're writing it. 0:19:27.810,0:19:32.126 It's clearer if you put the line[br]across the top to include, so 0:19:32.126,0:19:36.110 it's clear what you're including[br]in that square root sign. But be 0:19:36.110,0:19:38.102 aware that you might just see 0:19:38.102,0:19:44.320 the tick sign. Another symbol[br]that's used as an X with a bar 0:19:44.320,0:19:47.365 across the top and it said X 0:19:47.365,0:19:51.262 Bar. And it means 0:19:51.262,0:19:56.560 the mean. So it's the mean[br]of a set of numbers. Instead 0:19:56.560,0:19:59.683 of writing the word mean, we[br]write X Bar. 0:20:00.970,0:20:07.495 Another symbol, let's say we[br]have one point 3. 0:20:08.170,0:20:10.098 And we see a dot over the three. 0:20:10.730,0:20:13.015 And this is our recurring 0:20:13.015,0:20:18.831 decimal sign. And what that[br]means is we've got 1.3 and 0:20:18.831,0:20:24.210 three 333 and it goes on[br]forever and ever reccuring, so 0:20:24.210,0:20:29.589 the dot over the decimal place[br]means it goes on forever. 0:20:31.000,0:20:36.160 We might have 1.317[br]for example. 0:20:37.370,0:20:39.386 And dots over the three on the 0:20:39.386,0:20:42.530 7. Similarly, it means it goes 0:20:42.530,0:20:48.432 on forever. But what it means[br]this time slightly differently? 0:20:48.432,0:20:53.572 Is this the 317 that's repeated[br]and goes on forever? 0:20:53.580,0:20:59.916 In summary, mathematical[br]symbols are precise 0:20:59.916,0:21:03.084 form of shorthand. 0:21:03.620,0:21:08.684 They have to have meaning for[br]you. You need to understand them 0:21:08.684,0:21:12.482 and to help with that[br]understanding. You have context. 0:21:12.482,0:21:17.546 What else is with them and you[br]have convention which is what 0:21:17.546,0:21:20.500 mathematicians and scientists[br]have decided. Certain symbols 0:21:20.500,0:21:23.268 will represent. And that's[br]all you need to know.