In this video, we introduce the real line, discuss different kinds of numbers that fall on the real line, and how they can be represented using decimal expansions. We also briefly discuss approximations, significant figures and scientific notation. All of the numbers you use or meet in this course may be regarded as falling on the real number line indicated like this, extending infinitely to the left and to the right. Somewhere in the middle is the number zero and you can see the counting numbers one, two, three going to the right, and the negative counting numbers going to the left. These points that I've indicated are called integers. And in between, there's lots of other numbers, for example, halfway between zero and one is the number a half, which has a decimal expansion 0.5. Halfway between one and two is the number one and a half, or three on two, which has the decimal expansion 1.5. Now, if we go say a third of the way between zero and a one, somewhere like here, we have the number a third. And if you divide three into one repeatedly, you just get lots and lots of threes that go on forever. We say a third is nought point three repeated. If you look close to one, there's a number here, say six sevenths, and if you divide seven into six repeatedly, what you get is 0.857142, and then the digits repeat, 857142 forever. So, we say six sevenths is 0.857142 repeated. Now, let's have a closer look at a third. A third is 0.3 repeated. So, what should two thirds be? Stands to reason that that should be 0.6 repeated. So, what should three thirds be? It stands to reason that three thirds should be 0.9 repeated. But notice three thirds is actually one. So, what we have is that one has its decimal expansion 0.9999 going on for ever. So, if I take one and I look at this decimal expansion and say, truncate let's say to the third decimal place, I'll get a 0.999 which is little bit less than one. If I go further along and add more nines and stop, I still get a number which is a little bit less than one. The further I go along this decimal expansion, the closer I get to one, and if I imagine going on forever, in the limit we actually reach one. This idea of looking at an approximation and seeing what happens in the limit is the central core concept in calculus, and you'll see it many times in this course. Now, there are lots of other numbers, interesting numbers on this real number line. If I look just below one and a half, there's a number called the square root of two. The square root of two has a decimal expansion 1.41421 and so on. Now, this differs from the other examples. No matter where you look in the decimal expansion of the square root of two, you never get a repeated pattern of digits that goes on forever. The square root of two turns out not to be a fraction involving counting numbers. It's said to be irrational, and that's an idea that we explore in the next video on the Theorem of Pythagoras. What is the square root of two? Well, it's number whose square is two. Literally, so if I take a square whose side lengths are this number, the square root of two, when I multiply these two numbers together, I get an area which is exactly two square units. Now, there's another number, very important number on this line, a little bit more than three. This number is called pi, and it's decimal expansion is 3.14159 and so on. Like the square root of two, no matter where you look on the decimal expansion of pi, you will never get a sequence of digits that repeats forever. Pi turns out also not to be a fraction. The verification of that in fact is quite sophisticated, and to see that you would need all the tools that we develop in calculus in this course. Pi is defined in terms of circles. If we take a circle and I create a diameter, and I'll look at the perimeter. Call that P. Pi is in fact defined to be the ratio of the perimeter to the diameter. The fraction formed by dividing P by D. Now this is not a fraction of involving counting numbers or integers, it's a fraction involving real numbers. It's an amazing fact that no matter what circle I choose, no matter how small or how large, this ratio P divided by D is the same. It's not obvious, it's a fact that most people take for granted. The reason it's true in fact is because all circles can be approximated arbitrarily well, using similar triangles, and it's a fact from geometry that ratios of corresponding sides of similar triangles are equal. And that fact when you take the limit, using a limiting process, that produces this fact that pi is constant for all circles. Now, in real life, we can't use infinite decimal expansions of numbers. When we come to apply numbers, we have to use a finite number of decimal places. For example in your calculator, you'll read a display with maybe eight, or nine, or ten digits, depending on how powerful your calculator is. You certainly don't see an infinite decimal expansion of a real number. For example, a third as an abstract real number is 0.3 repeated. If I decide to truncate, if I just read the first three decimal places, I get 0.333. So, instead of equals, I've actually got approximately equal. A third is approximately equal to 0.333 to three decimal places. Now, two thirds which is 0.6 repeated, if I truncate at the third decimal place, instead of 0.666, I round up to 0.667. To three decimal places, this is closer to two thirds than 0.666. So, in that other example we had, six sevenths, remember that's 0.857142 repeated. If I truncate that to the third decimal place, I get 0.857. If I decided to truncate at the first decimal place, I'd actually round this up to 0.9. So, there are two different approximations to six-sevenths. One is more accurate than the other to three decimal places. But the most accurate approximation to one decimal place, to six-sevenths is 0.9. Now, the real number line is equipped with an arithmetic. You can do addition, subtraction, multiplication, and division. And there are various rules for handling approximations when you apply these arithmetic operations. The first rule, when adding or subtracting, quote your answer to the least number of decimal places that is used. The second rule, when multiplying or dividing, quote your answer to the least number of significant figures that are used. I'll explain what significant figures are in a moment. But let's just illustrate the first rule. So, suppose I have a length which I've measured correct to one decimal place, say 9.4 centimeters. Then, I want to add to it another length which I've measured carefully this time to two decimal places, which is 2.13 centimeters. When I combine the two lengths, I have to add these two numbers together. So, the combined length is 9.4 plus 2.13. And that's 11.53. Now, the three in the second decimal place can't be relied upon because the coarsest measurement is the 9.4. Because the coarsest measurement is to one decimal place, I'm not entitled to quote the answer to two decimal places. So, I approximate this to the nearest number to one decimal place, which is 11.5. Similarly with subtraction, suppose I take this length, 9.4 centimeters and I remove a segment which I've carefully measured to be 2.13 centimeters. So, what's left is the difference, 9.4 minus 2.13, which is 7.27. Once again, this seven in the second decimal place can't be relied upon because of the coarseness of the measurement of the 9.4, which is correct only to one decimal place. So again, you round your answer to the nearest number to one decimal place, which is 7.3. Let's do an example with multiplication. Suppose I take a rectangle, which is 9.4 centimeters wide and 2.13 centimeters high. So, the area is obtained by multiplying the numbers together, 9.4 times 2.13. If you type that into your calculator, it's a little bit hard to do in your head, you should get 20.022. But not all these digits are significant. You don't expect to be able to quote this answer to this degree of accuracy. The rule for multiplication is that you quote your answer to the least number of significant figures that appear in each of the numbers that are used in the arithmetic. The number of significant figures is the number of digits regardless of the decimal point. So, the 9.4 uses two significant figures, the 2.13 uses three. So, we quote our final answer to two significant figures, which gives you an approximation of 20 square centimeters. Now, if I just say 20 square centimeters without reference to the original problem, it's ambiguous. You don't know whether I'm saying this rectangle is 20 centimeters correct to the nearest unit, or possibly to the nearest ten units. Is it 20 to the nearest unit, or 20 to the nearest 10 square centimeters? Now, it may look a little bit pedantic. But to be more precise, you could write this as 2.0 times 10 square centimeters. And the fact that you've written 2.0 means that this zero in the expression is significant. So, this is correct to two significant figures. Now, if I wrote two times 10 square centimeters, because I've deliberately only written one digit, the two, you know that the answer is correct to the nearest 10 square centimeters. So, I'll give a more elaborate example. If you look say at the radius of the Earth. I believe that's 6,370,000 meters. Now, if I just quote that number to you, you don't know whether any of those zeros are significant in the accuracy of the radius of the Earth. And in fact, the Earth is not a perfect sphere. The radius changes as you move around the Earth. When I quote this radius, in fact only three digits are significant. It's not obvious in the way the numbers are represented. So to emphasize that only three digits are significant, we express this in what's called scientific notation. I only list three digits, and we put the decimal point between the first two digits. And to recover the original measurement, we multiply by an appropriate power of 10. So, the radius of the Earth in scientific notation is 6.37 times 10 to the six meters. And then the person receiving this information knows that you're only quoting the first three digits as accurate in this estimation of the radius of the earth. Now we've covered a lot of ground in just a few minutes, and there are many important ideas. Everything we've touched on is carefully explained and expanded in the notes accompanying this video. If any of these materials require clarification, please make a posting on the online discussion forum. We welcome your input and feedback.
