WEBVTT
00:00:02.370 --> 00:00:10.270
In this video, weโll learn how to identify rational numbers and how to find the position of a rational number on a number line.
00:00:10.630 --> 00:00:15.050
But letโs begin by thinking about the definition of a rational number.
00:00:15.710 --> 00:00:26.540
A rational number is a number that can be expressed as a fraction ๐ over ๐ where ๐ and ๐ are integers and ๐ is not equal to zero.
00:00:26.950 --> 00:00:29.230
So letโs break this down a little bit.
00:00:29.490 --> 00:00:37.220
Weโre told that a rational number must be able to be written as a fraction ๐ over ๐ where ๐ and ๐ are integers.
00:00:37.460 --> 00:00:43.310
Remember that integers are the natural numbers, including zero, and the negatives of the natural numbers.
00:00:43.860 --> 00:00:54.060
So we could say, for example, that a fraction like two-thirds is a rational number because two and three are integers and three is definitely not equal to zero.
00:00:54.460 --> 00:00:59.830
We could also say that a number like negative three and a quarter is a rational number.
00:01:00.230 --> 00:01:08.210
This is because it can be written as a top-heavy fraction negative 13 over four, and negative 13 and four are both integers.
00:01:08.720 --> 00:01:11.890
But what about an actual integer value like 10?
00:01:12.070 --> 00:01:13.310
Would it be rational?
00:01:13.690 --> 00:01:25.540
Well, yes, because remember that we can write any integer as a value over one, in which case we can see very clearly that this is in the form ๐ over ๐ where ๐ and ๐ are integers.
00:01:26.320 --> 00:01:36.470
Although a value like 4.75, which is a decimal, doesnโt immediately appear to be rational, we can, in fact, remember that we could write this as a fractional form.
00:01:36.790 --> 00:01:44.430
4.75 can be written as four and seventy-five hundredths or, alternatively, four and three-quarters.
00:01:44.890 --> 00:01:51.770
19 over four would be this value as a top-heavy fraction, fitting the form of a rational number.
00:01:52.300 --> 00:02:04.690
In addition to terminating decimals like 4.75, we also have repeating or recurring decimals such as this one, 0.131313 and so on.
00:02:05.070 --> 00:02:09.020
Any recurring decimal can also be written as a fraction.
00:02:09.260 --> 00:02:13.330
For example, here, this is the fraction of 13 over 99.
00:02:13.760 --> 00:02:18.960
So terminating decimals and repeating decimals will be rational numbers.
00:02:19.670 --> 00:02:24.350
You might begin to wonder โwhat we would say if a number is not rational?โ
00:02:24.690 --> 00:02:28.750
Well, a number thatโs not rational is called an irrational number.
00:02:29.100 --> 00:02:36.650
Itโs one which canโt be written as a fraction ๐ over ๐ where ๐ and ๐ are integers and ๐ is not equal to zero.
00:02:37.150 --> 00:02:45.080
Although in this video weโre going to be focusing on rational numbers, it can be helpful to get an idea of some numbers which are irrational.
00:02:45.560 --> 00:02:54.290
Perhaps, the most famous irrational number is ๐, which is a value which cannot be expressed as a terminating or repeating decimal.
00:02:54.680 --> 00:03:00.100
It can also not be expressed as a fraction, so thatโs why ๐ is irrational.
00:03:00.550 --> 00:03:09.010
Decimals which donโt terminate or repeat and also values such as the square root of numbers, which are not perfect squares.
00:03:09.720 --> 00:03:12.230
But letโs have a look at some questions.
00:03:12.570 --> 00:03:17.000
As we go through each question, weโll recall this definition of a rational number.
00:03:17.230 --> 00:03:21.530
So hopefully, by the end of this video, youโll be able to recall it easily.
00:03:22.040 --> 00:03:26.670
In our first question, weโll identify if a number is rational or not.
00:03:29.110 --> 00:03:32.460
Is 12 and five-sixths a rational number?
00:03:33.260 --> 00:03:44.770
Letโs begin by remembering the definition of a rational number, which is a number that can be expressed as ๐ over ๐ where ๐ and ๐ are integers and ๐ is not equal to zero.
00:03:45.210 --> 00:03:53.020
Although we have a mixed number here, 12 and five-sixths, could we write this as a complete fraction ๐ over ๐?
00:03:53.500 --> 00:03:58.090
In other words, could we write this as a top-heavy or improper fraction?
00:03:58.610 --> 00:04:05.250
If we consider our value of 12, then how many sixths would we have in 12 whole ones?
00:04:05.520 --> 00:04:15.580
Well, weโd have seventy-two sixths plus the fractional value of five-sixths, which would give us a fraction of 77 over six.
00:04:16.000 --> 00:04:23.820
Another way, of course, to do this is to take our denominator six and multiply it by 12 and then add on the value of five.
00:04:24.330 --> 00:04:30.390
Either way, we can see that 12 and five over six is equivalent to 77 over six.
00:04:30.690 --> 00:04:33.370
So, is this value a rational number?
00:04:33.820 --> 00:04:35.650
And the answer is yes.
00:04:37.750 --> 00:04:39.970
Letโs have a look at another question.
00:04:42.020 --> 00:04:44.930
Is every rational number an integer?
00:04:45.770 --> 00:04:51.710
In this question, we have two very mathematical terms, rational and integer.
00:04:52.130 --> 00:04:54.270
Letโs take the word integer first.
00:04:54.560 --> 00:04:58.400
This will be defined as a number that has no a fractional part.
00:04:58.620 --> 00:05:06.170
It includes the counting numbers, for example, one, two, three, four, zero, and the negatives of the counting numbers.
00:05:06.780 --> 00:05:15.750
A rational number is defined as one, which can be expressed as ๐ over ๐ where ๐ and ๐ are integers and ๐ is not equal to zero.
00:05:16.450 --> 00:05:19.180
So letโs take some numbers which are rational.
00:05:19.400 --> 00:05:25.480
For example, we could have this fraction six over one, which fits the form of a rational number.
00:05:25.680 --> 00:05:31.970
Six and one are integers and the one, this ๐-value on the denominator, is not equal to zero.
00:05:32.400 --> 00:05:36.540
This would be equivalent to the value six, which is an integer.
00:05:37.080 --> 00:05:39.440
Letโs take another rational number.
00:05:39.490 --> 00:05:42.910
Here, we have the example negative two-fifths.
00:05:43.440 --> 00:05:50.540
We should ask ourselves if negative two-fifths is an integer, a number that has no fractional part.
00:05:50.990 --> 00:05:52.500
And this would be no.
00:05:52.630 --> 00:05:57.380
We couldnโt write this in any way as a number that has no fractional part.
00:05:57.770 --> 00:06:03.050
Therefore, the answer to the question โis every rational number an integer?โ is no.
00:06:03.440 --> 00:06:13.050
If we consider the Venn diagram where we have the set of rational numbers, then contained within this set will be the set of integers.
00:06:13.440 --> 00:06:22.680
Using this diagram is helpful to illustrate that every integer is a rational number, but not every rational number is an integer.
00:06:24.540 --> 00:06:30.340
In the next question, weโll see more formal notation for the set of rational numbers.
00:06:33.080 --> 00:06:35.080
Which of the following is true?
00:06:35.120 --> 00:06:38.710
Option (A) one is an element of the rational numbers.
00:06:38.970 --> 00:06:42.930
Option (B) one is not an element of the rational numbers.
00:06:43.750 --> 00:06:48.650
The symbols in this question represents some formal mathematical notation.
00:06:48.890 --> 00:07:00.460
The โ symbol can be read as an element of or belongs to or is a member of, and this โ symbol represents the set of rational numbers.
00:07:01.050 --> 00:07:12.910
So in order to establish if one is a member of the set of rational numbers or one is not a member of the set of rational numbers, weโll need to recall what the rational numbers are.
00:07:13.490 --> 00:07:21.950
A rational number is a number which can be expressed as ๐ over ๐ where ๐ and ๐ are integers and ๐ is not equal to zero.
00:07:22.550 --> 00:07:27.680
We might then look at this number one and think, โWell, itโs already an integer.
00:07:27.870 --> 00:07:30.650
How could I express this as a fraction?โ
00:07:31.170 --> 00:07:35.890
Well, any integer value can be written as that value over one.
00:07:36.260 --> 00:07:43.460
Looking at the definition, we can confirm that the one on the numerator is an integer, and so is the one on the denominator.
00:07:43.760 --> 00:07:48.180
And crucially, this number on the denominator is not equal to zero.
00:07:48.620 --> 00:07:53.630
This means that one is a member of the set of rational numbers.
00:07:54.040 --> 00:08:00.860
Therefore, our answer is that given in option (A); one is a member of the set of rational numbers.
00:08:02.820 --> 00:08:07.440
In the next question, weโll find the position of a rational number on a number line.
00:08:10.160 --> 00:08:14.900
Which of the numbers ๐, ๐, ๐, and ๐ is four-tenths?
00:08:16.130 --> 00:08:23.540
If we take a look at this number line, we can see that weโve got measurements of negative one, zero, and one.
00:08:23.970 --> 00:08:29.590
Weโve also got these four letters, which represent different values on this number line.
00:08:29.980 --> 00:08:33.660
We need to work out which of these represents four-tenths.
00:08:34.360 --> 00:08:39.640
The first thing we might think about is the fact that four-tenths is a positive value.
00:08:40.100 --> 00:08:44.880
We could, therefore, rewrite any options given that are below zero.
00:08:45.490 --> 00:08:50.740
Secondly, we might recognize that four-tenths must be between zero and one.
00:08:51.040 --> 00:08:56.830
If it was a fraction over 10 thatโs larger than one, then the numerator would be higher than 10.
00:08:57.500 --> 00:09:01.290
Letโs consider this section then between zero and one.
00:09:01.670 --> 00:09:07.540
One method of finding the position is to consider that this is divided into five sections.
00:09:07.900 --> 00:09:11.050
Ideally, however, we would like 10 sections.
00:09:11.530 --> 00:09:21.040
So if we split this stretch into 10 sections, then four-tenths along would be at the value which is represented by ๐.
00:09:21.540 --> 00:09:31.630
As an alternative method, we could have simplified the fraction of four-tenths by taking out the common factor of two in the numerator and denominator.
00:09:31.940 --> 00:09:42.420
Out of the five sections between zero and one, weโre looking for two-fifths, which would mean that it does confirm that the value ๐ represents four-tenths.
00:09:42.890 --> 00:09:44.790
So, ๐ is the answer.
00:09:45.040 --> 00:09:50.210
Notice that this value of ๐ would represent four-fifths on the number line.
00:09:52.360 --> 00:09:57.770
In this question, weโll need to find the position of a number without being given a number line.
00:09:59.060 --> 00:10:05.080
Find the rational number lying halfway between negative two-sevenths and four thirty-fifths.
00:10:05.990 --> 00:10:10.960
Here, we have two fractions, negative two-sevenths and four thirty-fifths.
00:10:11.280 --> 00:10:15.700
And weโre asked to find the rational number which lies halfway between.
00:10:16.190 --> 00:10:26.400
We can recall the rational number is a number which can be expressed as ๐ over ๐ where ๐ and ๐ are integers and ๐ is not equal to zero.
00:10:26.870 --> 00:10:32.730
The best way to start a question like this is to see if we can make the denominators the same value.
00:10:33.280 --> 00:10:38.340
We should be able to write negative two-sevenths as a fraction over 35.
00:10:38.640 --> 00:10:48.840
Observing that we multiply the denominator by five, then our numerator will also be multiplied by five, which gives a value of negative 10 over 35.
00:10:49.370 --> 00:10:56.300
It may help if we could visualize negative 10 over 35 and four over 35 on a number line.
00:10:56.760 --> 00:11:01.580
So at the lower end of this number line, we have got negative 10 over 35.
00:11:01.800 --> 00:11:04.960
And at the top end, we have four thirty-fifths.
00:11:05.370 --> 00:11:10.670
This value of zero thirty-fifths would also just be equivalent to zero.
00:11:11.120 --> 00:11:22.580
If we were to think in terms of the distance from zero, on the left-hand side, we have a distance of ten thirty-fifths and on the right-hand side, a distance of four thirty-fifths.
00:11:22.890 --> 00:11:26.400
Thatโs equivalent to fourteen thirty-fifths in total.
00:11:26.690 --> 00:11:30.610
Half of that would give us seven thirty-fifths.
00:11:31.090 --> 00:11:42.460
Therefore, if we start from negative 10 and count one, two, three, four, five, six, seven, we would get a value of negative three thirty-fifths.
00:11:42.840 --> 00:11:50.350
As a check, counting down seven thirty-fifths from four thirty-fifths would also give us negative three thirty-fifths.
00:11:50.730 --> 00:11:54.850
Therefore, we could give the answer as negative three thirty-fifths.
00:11:55.170 --> 00:12:01.020
There is, of course, a different method if we donโt want to or couldnโt draw it on a number line.
00:12:01.430 --> 00:12:09.970
Letโs go back to our two original values: negative two-sevenths, which we could write as negative ten thirty-fifths, and four thirty-fifths.
00:12:10.320 --> 00:12:15.410
The halfway point between these two is equivalent to finding the median.
00:12:15.690 --> 00:12:19.170
We would begin by adding these two fractions.
00:12:19.460 --> 00:12:24.550
To add fractions, we must have the same denominator and we add the values on the numerator.
00:12:24.700 --> 00:12:28.700
In this case, negative 10 plus four would give us negative six.
00:12:29.210 --> 00:12:31.140
So, remember, weโre finding the median.
00:12:31.140 --> 00:12:33.240
So, weโve added our values.
00:12:33.450 --> 00:12:37.310
And then, as thereโs two values, we need to divide by two.
00:12:37.650 --> 00:12:46.700
In order to divide this fraction by two, we can consider it as the fraction two over one, and then we multiply by the reciprocal.
00:12:46.950 --> 00:12:51.720
So, we need to work out negative six thirty-fifths multiplied by one-half.
00:12:52.080 --> 00:12:57.840
Before we multiply, we can simplify this by taking out the common factor of two.
00:12:58.100 --> 00:13:02.950
So, weโll have negative three thirty-fifths multiplied by one over one.
00:13:03.240 --> 00:13:14.990
Multiplying the numerators and then multiplying the denominators separately, we get the value of negative three thirty-fifths, which confirms our earlier answer given by the first method.
00:13:16.590 --> 00:13:19.340
Letโs have a look at one final question.
00:13:21.340 --> 00:13:27.800
Which of the following expressions is rational given ๐ equals one and ๐ equals 34?
00:13:28.470 --> 00:13:42.870
Option (A) negative 39 over ๐ minus one, option (B) 39๐ over ๐ minus 34, option (C) 39๐ over ๐ minus one, or option (D) ๐ over ๐.
00:13:43.840 --> 00:13:49.910
In order to answer this question, letโs start by remembering the definition of a rational number.
00:13:50.420 --> 00:13:59.390
A rational number is a number that can be expressed as ๐ over ๐ where ๐ and ๐ are integers and ๐ is not equal to zero.
00:14:00.120 --> 00:14:09.280
In the four options here, we can see that weโve got four fractions that contain numerical values and also the algebraic terms ๐ and ๐.
00:14:09.760 --> 00:14:13.530
However, weโre given numerical values for ๐ and ๐.
00:14:13.860 --> 00:14:18.150
So, weโll take each expression in turn and plug in these values.
00:14:18.610 --> 00:14:21.620
Letโs start with the first expression in option (A).
00:14:21.830 --> 00:14:24.980
Weโll plug in the value that ๐ is equal to one.
00:14:25.410 --> 00:14:31.830
Weโll still have 39 on the numerator, and weโll have one subtract one on the denominator.
00:14:32.200 --> 00:14:36.460
This, of course, simplifies to negative 39 over zero.
00:14:36.790 --> 00:14:39.440
You might think that this looks pretty good as a fraction.
00:14:39.570 --> 00:14:43.270
Weโve got a number on the numerator and a number on the denominator.
00:14:43.570 --> 00:14:50.520
But in fact, if youโve ever tried to divide a number by zero on your calculator, youโll get an undefined answer.
00:14:50.890 --> 00:15:00.210
And importantly, if we look at our definition of a rational number, the ๐, the value on the denominator, cannot be equal to zero.
00:15:00.630 --> 00:15:05.400
So, this value of negative 39 over zero is not a rational number.
00:15:05.550 --> 00:15:08.230
And so, we can exclude option (A).
00:15:08.760 --> 00:15:16.460
In the expression in option (B), weโll need to substitute in the value ๐ equals 34 twice as ๐ occurs twice.
00:15:16.890 --> 00:15:23.330
Weโll, therefore, have the calculation 39 times 34 over 34 minus 34.
00:15:23.690 --> 00:15:31.470
Before we rush to calculate 39 multiplied by 34, you might already notice whatโs going to happen on this denominator.
00:15:31.780 --> 00:15:36.100
Once again, weโre going to have a denominator that has a value of zero.
00:15:36.290 --> 00:15:41.560
So, we know that this expression when ๐ is equal to 34 would not be rational.
00:15:42.040 --> 00:15:47.600
We can use the same method of plugging in the ๐- and ๐-values into option (C).
00:15:48.010 --> 00:15:52.480
So, it have 39 times 34 over one minus one.
00:15:52.890 --> 00:15:57.980
You may have already noticed that this denominator will also give a value of zero.
00:15:58.310 --> 00:16:03.930
So, option (C) is not a rational number when ๐ is one and ๐ is 34.
00:16:04.390 --> 00:16:10.720
The expression in option (D) is ๐ over ๐ which will be is 34 over one.
00:16:11.190 --> 00:16:14.720
Letโs check if this fits the definition of a rational number.
00:16:14.890 --> 00:16:19.410
We have it as a fraction ๐ over ๐ where ๐ and ๐ are integers.
00:16:19.690 --> 00:16:22.220
And 34 and one are integers.
00:16:22.540 --> 00:16:26.460
And of course, the denominator one is not equal to zero.
00:16:26.850 --> 00:16:29.840
Therefore, 34 over one is rational.
00:16:30.030 --> 00:16:32.170
And so, our answer is (D).
00:16:32.450 --> 00:16:37.090
๐ over ๐ is rational when ๐ equals one and ๐ equals 34.
00:16:37.420 --> 00:16:43.250
Before we finish with this question, letโs just take a quick look at this expression in option (A).
00:16:43.570 --> 00:16:51.210
We saw that this expression negative 39 over ๐ minus one is not rational, but thatโs not always the case.
00:16:51.490 --> 00:17:05.530
If we had any other value other than ๐ equals one, for example, ๐ equals two, then weโll work out negative 39 over two minus one, which would give us a rational value of negative 39 over one.
00:17:05.890 --> 00:17:16.350
The expressions (A), (B), and (C) are only irrational because they had a denominator of zero, as they did in fact have integer values on the numerator and denominator.
00:17:17.680 --> 00:17:20.840
Letโs now summarize what weโve learned in this video.
00:17:21.400 --> 00:17:33.040
Firstly, we began with the definition of a rational number, which is a number that can be expressed as a fraction ๐ over ๐ where ๐ and ๐ are integers and ๐ is not equal to zero.
00:17:33.570 --> 00:17:41.480
Rational numbers include integers, fractions and mixed-number fractions, terminating decimals, and repeating decimals.
00:17:42.120 --> 00:17:46.190
Numbers that are not rational are called irrational numbers.
00:17:46.630 --> 00:17:56.000
And finally, we saw this more formal notation that this symbol, which looks like a โ with an extra line, represents the set of rational numbers.