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Fundamental Principles 1@tyno4u — Transcript

by Matheminez · 2,335 words · 568 segments · language en · Watch on YouTube

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0:02Okay, good evening.
0:04And welcome to the Math Minds channel.
0:08It's been a while
0:11and now to move on
0:14I would like to
0:17go through once again the fundamental
0:19principles
0:21of mathematics.
0:23This is mainly for those
0:25who
0:27find maths
0:29those common concepts
0:31the basics
0:33who find the basics
0:36overwhelming or
0:39difficult.
0:42I would like to begin from there and
0:45then we move on to other different
0:47topics.
0:48These videos will not be taking long.
0:51So, we will try to make them as
0:56snappy as they could ever be.
0:59So, we'll move on now to
1:02the fundamental principles
1:04of mathematics.
1:07Or the fundamental principles in
1:09mathematics.
1:12The fundamental principles.
1:30These are the principles that will
1:33help one
1:35to
1:38navigate
1:39some problems
1:41carefully
1:43and fear and
1:45without fear.
1:50They are those principles
1:53which
1:55are applied in
1:59simple calculations
2:01simple arithmetics
2:03and the concepts actually
2:05um involved in those arithmetics.
2:09And the first I will talk about
2:13is I will quickly go through the
2:17um
2:18numbers
2:20numbers then the the number line.
2:22And then we move on to the
2:25common
2:26problems
2:28uh associated with the
2:30numbers on the number line.
2:33Okay.
2:34So,
2:35the first thing we do here is
2:39numbers.
2:45The mathematician is not the only person
2:47who makes use of numbers.
2:50In our everyday lives, we use numbers to
2:53count. And that is one thing as one
2:57im- important
3:00one importance we put on numbers. We use
3:03them to count.
3:05And we don't come here and
3:08get all the characters of numbers, no.
3:10We just focus on what we have come to
3:12do.
3:14Talking about numbers, there are
3:15different ways to look at numbers.
3:18Different sets of numbers. We call them
3:20sets of numbers.
3:22So, numbers then we call them sets of
3:25numbers.
3:26So,
3:27in the sets,
3:29we have
3:32the
3:36We have two major sets of numbers.
3:47Okay.
3:49We have two major sets of numbers.
3:53Actually,
3:55I would want to start with
3:59what we call complex numbers.
4:10Just to know
4:11that this is a set of numbers
4:15comprising of all any other number.
4:19Any other number, uh I won't put
4:20anything here.
4:22I will rather come here and
4:28write the real numbers.
4:33The real numbers are large enough
4:36and they're not
4:37as complete as the complex numbers. The
4:40complex numbers are complete
4:43as the case may be.
4:44Maybe in in the future, maybe tomorrow,
4:47a new set of numbers could be
4:49identified. Right, but now, this is the
4:51most complete of all the other sets of
4:53numbers.
4:54After the real numbers,
4:56we have
4:58what we call
5:04the rational and irrational numbers.
5:09And we have the irrational.
5:16And here,
5:17we have the rational.
5:19What are they?
5:24Rational, irrational.
5:26Rational
5:28numbers could be expressed as fractions,
5:30A over B.
5:32Where A and B are integers. We'll meet
5:35integers.
5:363 over 2,
5:39minus 1 over 5,
5:41etc.
5:43Even whole numbers are rational numbers
5:45because
5:47the denominator there is one.
5:50Why irrational numbers can never be
5:52expressed
5:53in the in the form A over B.
5:56They can't be expressed in that form.
5:58For example, root two.
6:01No two
6:03integers
6:04will form a fraction which will be equal
6:06to to exactly what root two.
6:09But,
6:10A
6:11and so and so forth.
6:13So,
6:14um these are
6:17coming from the real numbers.
6:19And the real numbers also, under the
6:21real numbers,
6:23we also have
6:25Okay, um let me put that under the
6:30rational numbers. Under the rational
6:31numbers,
6:33we also have
6:34what we call
6:36the integers. Let me call the integers.
6:43Integers. The integers
6:46are represented by
6:47Z.
6:55The integers are represented by Z.
6:59Rational numbers are represented by Q.
7:00Irrational numbers are represented by Q
7:02prime.
7:04Um right now, those terms are not big
7:06for you since you are just beginning.
7:09For the integers, Z,
7:12these are
7:13whole numbers which could be negative or
7:15positive.
7:17They are whole numbers which could be
7:18negative or positive. So, for example,
7:22minus two,
7:24minus one, 10,
7:2715,
7:28etc.
7:30These are integers.
7:35And then,
7:36after the integers, we have
7:39what we call
7:42the natural
7:44I think we will
7:46stop it at that, the natural numbers.
7:51The natural numbers are just whole
7:53numbers
7:55and which are always positive.
7:58Okay?
7:58They're also called called counting
8:00numbers, 1 2 3
8:034 and so on and so forth.
8:06One thing with these numbers
8:09is that they're all
8:11infinite.
8:13There's no end to them.
8:14That's one property of of numbers you
8:17will have to learn.
8:19Numbers are infinite.
8:20So, why am I
8:22talking about these sets of numbers?
8:25I talk about them so as to
8:29help um
8:31let you know
8:33that
8:35such things
8:37numbers are classified in those ways,
8:39one
8:41and two.
8:47We'll start using them.
8:50So, after talking about numbers,
8:53I also have to talk about number lines.
8:57All these numbers can be
9:00represented on a number line, but
9:03we would just look at the natural
9:05numbers and the integers represented on
9:07the number lines. What is a number line?
9:10A number line
9:18is a line
9:22where numbers are put
9:24or could be
9:26found or placed.
9:28So,
9:30on the number line I want to talk about
9:34have the natural numbers and the
9:36integers on them.
9:38So, on this number line, we have zero,
9:41one,
9:43two,
9:44three,
9:46four, five, and so on.
9:49And also, so this is the number line for
9:51integers.
9:57First of all, you have to
9:59agree with me that zero is at the center
10:01of every number set.
10:04Good. And at the
10:07at both sides, at either sides of zero,
10:09you now have the negative and the
10:11positives,
10:13which all, when they combine, they all
10:15go back to zero, where all numbers come
10:17from.
10:19That's me saying that now. It's not
10:22So,
10:24the number lines are used
10:27to
10:29place numbers. We find numbers on the
10:32number line.
10:33And it's
10:35has
10:40a way of showing us how um
10:43calculations,
10:45arithmetics, are being done.
10:49First of all, you notice that from zero
10:51to the right,
10:54from zero to the right,
10:58numbers are increasing.
11:01The positive numbers are increasing.
11:05So, if you go from zero to the right,
11:07you're moving
11:11towards an increase on the
11:13on the positive numbers.
11:16If you move from zero to the left,
11:20actually, this is decreasing.
11:26And this is increasing.
11:29But you see,
11:31where we have minus four here,
11:34one may
11:36mistakenly think this is bigger than
11:38this. No.
11:39So, as you move towards the left,
11:41numbers generally decrease. And as you
11:43move towards the right, numbers
11:45generally increase.
11:47So, that should also help us
11:50to be able to do certain calculations
11:53involving
11:55small numbers, negative or positive.
11:58So, first of all, I would want to go
12:01through immediately through the
12:04the nitty-gritties of addition and
12:07subtraction
12:08of numbers on the number line. Addition
12:10and subtraction of numbers on the number
12:12line.
12:15Addition
12:17and subtraction
12:22of numbers.
12:29Okay.
12:30There are rules we
12:32we talked about in the previous video
12:35concerning the fundamental principles.
12:37Those rules, I will reinvoke them here
12:41to help us
12:42do what we have to do.
12:45First of all,
12:46number one.
12:54There is a kind of addition which
12:57will ask us to combine two or more
13:01numbers having the same sign. Actually,
13:03there are two signs in mathematics,
13:05the positive sign
13:09and the negative sign.
13:11Positive sign, we will have have plus.
13:14Negative sign, we will have minus.
13:17So, there the first rule will ask us to
13:20put
13:22positive and positive together or
13:24negative and negative together.
13:26So,
13:27that's the first rule we talked about.
13:28We said
13:30when two numbers
13:33have the same sign
13:36when two numbers have the same sign as
13:37in positive and positive or negative and
13:39negative
13:40then we just have to add the two numbers
13:43and give them the answer that sign.
13:45So,
13:46when two numbers
13:57have
13:59the same
14:02sign
14:05we add
14:08we add
14:09the numbers. For example,
14:14we have
14:15-15
14:199.
14:22I know how we have
14:24of course 13
14:26+6.
14:28Before we do this,
14:30always note that the sign comes before
14:33the number, always.
14:35So, the sign here we have -15. So, the
14:38negative What shows that this 15 is
14:40negative is this minus here, this one
14:42here.
14:43And what shows that this 9 is negative
14:45is this minus here.
14:47What shows that this 13 is positive is
14:50an invisible positive sign because it is
14:52coming first. So, we can either write a
14:55positive sign or we leave it out.
14:58But at the end of the day we know that
14:59there is a plus here.
15:02And what shows that this 6 is positive
15:04is this plus sign here.
15:07So, what is this rule telling us?
15:09Because these two numbers
15:11have the same sign, they are both
15:13negative.
15:15We add them together.
15:17When we add them together, we have
15:18something like this.
15:2015
15:21+ 9
15:25The answer will be negative because both
15:26of them are negative. Your answer will
15:28now be
15:30negative 24.
15:32That's how we will do this.
15:34Of course, we already know
15:36this is positive and you're adding
15:39them together.
15:42What do you have here? I have positive
15:44what? 19.
15:47So,
15:48it comes
15:50in this way in these ways, actually.
15:54The second one we talk about is
15:58when
16:01we are limiting ourselves to two numbers
16:04now.
16:05When two numbers
16:09have different signs.
16:16Have different signs.
16:18For example,
16:21Okay, we do what? Sorry, we subtract.
16:27For example,
16:31negative 15
16:33and positive 9.
16:3713
16:39and negative 6.
16:42Okay.
16:44So, let's see what this is.
16:4815 is negative and 9 is positive. So,
16:51the
16:52the rule says we subtract. So, we have
16:54to subtract.
16:58When we subtract,
17:02sign
17:05will be for bigger number.
17:13Okay, I will talk more on that.
17:15When we sub- we subtract between 15 and
17:17When we do that, we will have
17:20we subtract.
17:24We have
17:266.
17:29When we get 6, we now look
17:32at the numbers and find out the bigger
17:34the bigger one. What is the sign of the
17:36bigger number?
17:38The bigger one is 15. We don't have the
17:41sign, then the sign is what? Negative.
17:43That is how this is -6.
17:46These two
17:4913 - 6, this is positive, this is
17:52negative. We subtract again.
17:57When we subtract, we have 7.
18:00We check the sign of the bigger number.
18:03It's positive, so
18:06whether you write it or not, you know
18:07it's positive.
18:11Another thing we will have to
18:14um take note of is
18:19when we see stuff like
18:2513 - 6,
18:29we
18:30immediately should know that it's the
18:31same as -6 + 13.
18:36So,
18:37it's done the same way.
18:39The one that comes first
18:43doesn't actually matter
18:45in this regard.
18:47But, they must have their various signs.
18:50So, we can't write this as -13 + 6 or it
18:55will be a different thing altogether.
18:57So, this and this are the same.
19:00That's what we're trying to say.
19:03And because they are the same, they are
19:04also treated the same way.
19:06So, what we we we are talking about
19:08there was
19:10because they have different signs, we
19:11have to subtract.
19:15When we subtract,
19:17we have seven.
19:19We now go back and check the bigger
19:21number. 13 is a bigger number. We pick
19:24the sign of that bigger number.
19:26That's +7.
19:28Let's take another example.
19:33We have a
19:42+6
19:44-11
19:49These two numbers
19:52have different signs.
19:55And to deal with them, we have to
19:58subtract because they have different
19:59signs.
20:00So, when we subtract,
20:04we have five.
20:06What sign will it will it have? It's
20:08going to have the sign of the bigger
20:09number.
20:10So, the 11 is the bigger number and the
20:12sign is negative.
20:13So, we have negative five. All right,
20:15negative five.
20:18The number three, the third one we talk
20:20about is
20:22several numbers
20:28Several numbers
20:30with different signs.
20:40When you see several numbers having
20:41different signs and you have to add them
20:43together,
20:44what we we do is this.
20:48Select them, group them according to
20:50their signs.
20:51That's the first one.
20:55Group
20:59according to
21:02signs.
21:04And apply
21:07apply
21:09the two rules.
21:14I'm home.
21:17So, for example,
21:19look at this, -2
21:23+5
21:25-7
21:27+6
21:29-9
21:32+10.
21:35Look at this.
21:36We have um
21:39numbers
21:40having different signs.
21:42And we want to add all of them together.
21:44The first thing we do here is find out
21:47the ones that are negative. There are
21:48three of them.
21:50So, you group. First of all, group.
21:53When you group, you have -2
21:56-7, -9 being together.
22:00And
22:01uh again, these other ones having
22:04positive sign,
22:05positive five,
22:10positive six,
22:12and positive 10 also being together.
22:15So, here, we apply rule
22:21rule one.
22:24Rule one, the first rule.
22:26Rule one.
22:28Both they have the same sign, you add
22:29all of them together. This is negative
22:332 + 7 + 9.
22:36The answer will be negative. And then
22:37this is positive
22:395 + 6 + 10.
22:42What is your answer here?
22:44You have something like
22:46this is -18, -18.
22:51Negative sign. This one is
22:53+21.
22:56So, it has come back to
22:58us using rule two.
23:01In that rule be two, you subtract.
23:06When you subtract, we have
23:08three.
23:10Because 21 is the bigger number and its
23:12sign is plus.
23:14We have the answer as positive.
23:18So,
23:20um this is not all we talked about in
23:22the first video.
23:24But,
23:26we will
23:28talk about other things, too,
23:30which will lead up to
23:32the first topic we will treat here.
23:35But, before we go,
23:37I would like you to
23:41take note of these problems.
23:43I mean, solve
23:45these problems and see if
23:48you have understood what we talked
23:51about.
23:54Minus 13
23:57plus 65
24:01two
24:04five
24:05minus nine
24:07minus three
24:10plus four
24:12minus one
24:14plus six
24:16minus 10
24:19plus
24:2112
24:24number three
24:3056
24:32minus
24:3419
24:36minus 45
24:40plus two
24:44number four
24:48negative 16 negative 31
24:52number five
24:55six plus three plus 19
24:59Number 6
25:05Number 3
25:07+ 2
25:09+ 4
25:11+ 5
25:13+ 6
25:15+ 7
25:20+
25:30Okay.
25:32So, um you can try this
25:35and
25:37send your feedback
25:39if you're having problems with them.
25:42See you in the next class.
25:46Thank you.

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