Number System grade 10

Summary

This educational video lesson focuses on teaching interval notation, set builder notation, and number lines, which are often challenging for students. The instructor uses a step-by-step approach to explain how to represent numbers greater than three and less than seven on a number line, employing open circles to denote exclusion of the endpoints. The lesson covers interval notation with parentheses to indicate non-inclusive boundaries and set builder notation using brackets to specify conditions for variables. The instructor also introduces the concept of real numbers and infinity, using creative analogies like a hungry crocodile to help students remember the direction of inequalities. The lesson aims to demystify these mathematical concepts through practice and repetition.

Takeaways

• 📚 The lesson introduces interval notation, set builder notation, and number lines, which are often challenging for students.
• 🔢 It explains how to represent numbers greater than three and less than seven, including examples like 3.1, 4.2, 5.6, and 6.54.
• 📏 The number line is used to visualize intervals, with open circles indicating numbers not included (e.g., 3 and 7).
• 📝 Interval notation is introduced as a concise way to express ranges, using round brackets to denote non-inclusive endpoints.
• 🦎 A mnemonic is used to remember the notation: a 'hungry crocodile' facing the larger number signifies 'greater than' or 'less than'.
• 📋 Set builder notation is explained, using specific brackets and the letter 'x' to define the set of numbers that meet certain conditions.
• 🌐 The concept of real numbers is briefly touched upon, indicating that 'x e r' signifies 'x is an element of the real numbers'.
• 🔄 The lesson reiterates the importance of practice to understand and apply these mathematical notations effectively.
• 🔄 The instructor provides another example with numbers greater than or equal to 7 and less than 12, demonstrating how to adjust the notation accordingly.
• ♾️ The lesson concludes with an example of numbers greater than five, introducing the concept of infinity in interval notation.

Q & A

What are the three different ways to represent numbers between 3 and 7 in mathematics?

The three different ways to represent numbers between 3 and 7 are using a number line with open circles at 3 and 7, interval notation (x e [3,7)), and set-builder notation (x ∈ ℝ | 3 < x < 7).

What does the open circle on a number line represent?

An open circle on a number line indicates that the number at that point is not included in the set.

How do you represent numbers that are not included in the set using interval notation?

In interval notation, numbers that are not included in the set are represented with round brackets, for example, (3,7).

What does 'x e' stand for in interval notation?

In interval notation, 'x e' is an abbreviation for 'x is an element of', indicating that x belongs to the set described within the brackets.

What is set-builder notation and how is it used to describe numbers between 3 and 7?

Set-builder notation is a way to describe a set of numbers using a variable, a condition, and a colon. For numbers between 3 and 7, it would be written as {x | 3 < x < 7}.

What does 'x ∈ ℝ' mean in set-builder notation?

'x ∈ ℝ' in set-builder notation means that x is an element of the set of real numbers, indicating that x can be any real number that satisfies the given condition.

How do you represent a number that is bigger than 5 but not including 5 on a number line?

On a number line, a number that is bigger than 5 but not including 5 would be represented with an open circle at 5 and a continuous arrow extending to the right.

What is the interval notation for numbers that are bigger than 5 but not including 5?

The interval notation for numbers that are bigger than 5 but not including 5 is (5, ∞).

How do you represent numbers that are bigger than 5 using set-builder notation?

In set-builder notation, numbers that are bigger than 5 would be represented as {x ∈ ℝ | x > 5}.

Why is it important to practice these different notations for representing numbers?

Practicing different notations helps to solidify understanding of how to represent sets of numbers mathematically and can be useful in various mathematical contexts and problem-solving.

What is the significance of the direction of the inequality symbols in set-builder notation?

The direction of the inequality symbols in set-builder notation indicates whether the boundary numbers are included or excluded. For example, '>' means the number is strictly greater, excluding the boundary, while '≥' would include the boundary number.

Outlines

00:00

📚 Introduction to Interval Notation

The instructor begins by introducing the concepts of interval notation, set builder notation, and number lines, which are often challenging for students. The lesson aims to demystify these concepts through practice. A specific interval is discussed, which includes numbers greater than three and less than seven, exemplified by numbers like 3.1, 4.2, 5.6, and 6.54. The instructor emphasizes that these do not have to be whole numbers and that the lesson will cover when to use these notations. Three methods are introduced to represent this interval: using a number line with open circles to indicate non-inclusion of the endpoints, interval notation with parentheses to denote the same, and set builder notation with a mix of brackets and conditions to describe the set of numbers.

05:01

🔢 Exploring Number Lines and Notations

This section delves into the practical application of the three methods introduced earlier. The instructor uses a number line to visually represent numbers greater than or equal to 7 and less than 12, including examples of numbers within this range. The number line is marked with a solid dot at 7 (indicating inclusion) and an open dot at 12 (indicating exclusion). Interval notation is simplified with 'x e' to represent 'x is an element of,' using square brackets for inclusive and round brackets for exclusive intervals. Set builder notation is further explained with its specific bracket style and the inclusion of 'x e r' to specify that 'x' represents real numbers. The instructor uses a crocodile analogy to help remember the direction of the inequality symbols. The lesson concludes with an example of numbers greater than 5, using a continuous arrow on the number line to represent an infinite range, and infinity notation in interval and set builder notations.

Keywords

💡Interval Notation

Interval notation is a mathematical way of expressing a set of numbers within a specific range. In the video, it is used to describe numbers that are greater than a certain value and less than another. For example, the interval notation for numbers greater than 3 and less than 7 is expressed as (3, 7), where the parentheses indicate that 3 and 7 are not included in the set. This notation is crucial for understanding the concept of continuity and discreteness in mathematics.

💡Set Builder Notation

Set builder notation is a method used to describe a set of numbers by defining a property that all members of the set share. In the video, it is used to express the set of numbers greater than 3 and less than 7 as {x | x > 3, x < 7}. This notation is particularly useful for defining sets with complex conditions and is a fundamental concept in set theory.

💡Number Line

A number line is a visual representation of numbers in a straight line, with a defined origin (usually 0), direction, and unit length. In the video, the number line is used to graphically represent intervals, such as numbers greater than 3 and less than 7. The number line helps visualize the inclusion or exclusion of endpoints in intervals, with open circles indicating exclusion and filled circles indicating inclusion.

💡Open Circle

In the context of number lines, an open circle represents the exclusion of a particular number from the set. For instance, in the interval (3, 7), an open circle at 3 indicates that 3 is not included in the set. This visual cue is essential for understanding the boundaries of intervals and the nature of the sets being described.

💡Closed Circle

A closed circle on a number line signifies the inclusion of a number within the set. For example, in the interval [7, 12), a closed circle at 7 indicates that 7 is part of the set. This is the opposite of an open circle and is used to show that the endpoint is a member of the interval.

💡Infinity

Infinity is a concept that represents an unbounded quantity, larger than any number. In the video, it is used to describe intervals that extend indefinitely in one direction, such as numbers greater than 5, which can be expressed as (5, ∞). Infinity is a crucial concept in calculus and other areas of mathematics where limits and unbounded behavior are studied.

💡Real Numbers

Real numbers include all the points on the number line, encompassing integers, fractions, and irrational numbers. In the video, the term 'x e r' is used to denote that 'x' is an element of the set of real numbers. This notation is used to specify the domain of discourse when defining sets or intervals.

💡Element of (∈)

The symbol '∈' is used to denote membership in set theory. When 'x ∈ R' is used, it means that 'x' is a member of the set of real numbers. In the video, this notation is employed to clarify that the variable 'x' represents any real number within a defined interval or set.

💡Greater Than

The 'greater than' symbol (>) is used to indicate that one number is larger than another. In the video, it is used to define the upper bound of an interval, such as in the set of numbers greater than 3. This relational operator is fundamental in expressing inequalities and defining intervals.

💡Less Than

The 'less than' symbol (<) is used to denote that one number is smaller than another. In the context of the video, it is used to define the lower bound of an interval, such as in the set of numbers less than 7. This symbol, like 'greater than,' is essential for expressing inequalities and is a basic concept in mathematics.

Highlights

Introduction to interval notation, set builder notation, and number lines.

Students often struggle with these concepts initially due to their complexity.

Explanation of representing numbers bigger than three and smaller than seven.

Demonstration of using a number line to represent intervals.

Use of open circles on a number line to indicate non-inclusive endpoints.

Introduction to interval notation with the format (a, b) to represent ranges.

Explanation of using 'xe' in interval notation to denote the variable x.

Clarification on the use of round brackets to indicate non-inclusive endpoints in interval notation.

Introduction to set builder notation and its format.

Explanation of using set builder notation to represent numbers within a range.

Discussion on the use of 'x e r' in set builder notation to denote real numbers.

Practical example of representing numbers bigger than 3 using set builder notation.

Creative analogy using a hungry crocodile to remember the direction of inequalities.

Representation of numbers bigger than and equal to 7 on a number line.

Use of a solid dot on a number line to indicate inclusive endpoints.

Interval notation for numbers bigger than and equal to 7 and smaller than 12.

Set builder notation for the same range, including the use of 'x e r'.

Explanation of representing numbers bigger than 5 using a continuous arrow on a number line.

Interval notation for numbers bigger than 5 using infinity as an endpoint.

Set builder notation for numbers bigger than 5, including the use of infinity.

Encouragement for students to practice these concepts to overcome initial confusion.

Transcripts

hello everyone welcome to this lesson so

in this lesson we're going to start

talking about

interval notation set builder notation

and number lines now students usually

really struggle with this in the

beginning because it is weird it's super

confusing

but let's practice it and you'll see

it's actually not that bad so let's say

we have the numbers

bigger than three and smaller than

seven so those would be numbers such as

three point one four point

two five six

and six point five four for example

it's any number bigger than three

and smaller than seven it doesn't only

have to be

four five and six we don't always have

to work with

these natural numbers sometimes we will

sometimes we won't but i'll show you

when to do what so for now we're just

going to

take any number between three and seven

okay so kevin this is not an english

class class so we don't say numbers

bigger than three and smaller than seven

we need to write this in a fancy

mathematical way yes so there are three

different ways that you are going to be

doing this

the first one let's use a number line

okay so there we have our number line

what we would then do is just put the

two main numbers so you'd go

3 and seven if you want you can put some

numbers in between

that's fine and now we want to go

between three and seven so we want to

have numbers between three and seven

but they must be bigger than three so

we'll put a little open circle like this

and smaller than 7 so we'll put an open

circle like that

and then we'll just connect these two

together i'm going to do that in a

different color rather

and so the numbers that fit between 3

and 7

will be any of these on this pink line

over here

at the end i have used an open circle

what that means is that the number 3 is

not

included and the number 7 is not include

next we can use something called

interval notation

and so this one's quite easy you just

say x

e i'll explain now and then we're gonna

use round brackets

and we're gonna go from three to seven

the xe is just a way of saying that

we're using the numbers

x okay um yeah i'm actually not going to

explain too much about that you don't

really need to know you just need to

know

to you just need to remember to say xe

okay it'll become

more familiar to you as we go on

i've used a round bracket over

here and i've used a round bracket over

here what that means is that the number

seven and the number three are not

included if those numbers were included

you would use a

square bracket like that the third one

is called set

boulder notation and so the way this one

works is it typically uses these

funny brackets first step is to say what

letter we're working with you're working

with

x you then put a line like this you then

say that x is all the numbers

bigger than three and smaller than

seven if you're still confused as to the

direction that these

things should face that's okay i've got

grade twelves who still don't know

that direction some people just really

struggle with that but i will do my best

to

practice that as much as we can and then

your last step is to put a semicolon

here

and to say that we are working with all

the real numbers

so we'll say x e r so for now if this

seems like

what seems very strange it is strange i

mean why do we have to do all this weird

stuff

why can't we just say that x must be

bigger than three and smaller than seven

but trust me as we do this a few times

you'll start to get the hang of it

okay the xcr i didn't explain that part

that's just

saying that we are busy with real

numbers real numbers

are all numbers that you can see such as

three comma one one

one one three comma one four five six

eight two 0.148 it's

all the numbers that you can imagine if

we only wanted to use

numbers like 1 2

3 then we're gonna have to change that r

to something else but i'm going to

practice that as we go on

i just want to quickly practice this

part that we did here

if i want to say that a number is bigger

than 3 then i say

x which is the number is bigger than

3. so this means that the x is bigger

than the three so there's many ways that

students use to try remember this

so one way i explain to people is

imagine that this mouth here is a

crocodile and that crocodile is hungry

now if you are a

hungry crocodile are you gonna eat

something that's really big or something

that's really small

well if you're very hungry you're gonna

eat something that's big

and so the crocodile is clearly trying

to eat the x which means that the

x is bigger than the three so x is

bigger than three

this one says that it's smaller than

seven

so x must be smaller than seven that

means the 7 is bigger

and that's why the crocodile's facing

the 7. but now when you've got something

like this

and something like this you can just put

them together like this

okay so let's do another one so here

we're going to use numbers that are

bigger than

and equal to 7

and then it's also smaller than 12. so

give me some examples that would be

things like

numbers that are bigger than and equal

to 7 so that means you can use 7 as well

because it also says equal to seven

we can use numbers like seven point one

one we can use

eight we can use nine nine point one

four six eight three two one

ten eleven but we can't use 12 because

it says smaller than 12 it doesn't say

smaller than

and equal to 12. but we can get very

close to 12 we can go 11.99999

if we wanted to so what we do for the

number line is we use a

seven so we'll have a seven and a twelve

and if you want you can put a few

numbers in between you don't have to

so now for the seven it's equal to seven

so we're gonna use a

colored in dot it's not going to be an

open dot for the 12 that's going to be

an

open dot because we can't actually reach

the 12 we only get

close to it we can then connect the line

and so x which is the numbers can be any

number in between here which is these

ones that we said and we can do a whole

lot more we can do

every single number that we can see in

between those two

numbers interval notation very easy you

just say

x e just remember the x e we use a

square bracket now because we including

the 7

and a round bracket for the 12 because

we're not including that

set border notation has the funny

brackets we say we're working with

x you'd put a line we then say x can be

anything bigger than

7 but it can also include 7. so we show

it with that line like that

we then say smaller than 12 and then we

say

x e r at the end so there's three

different things with this

there's this part there's

this part and then lastly

there's that part the last one for this

lesson will be the numbers bigger than

five so that would mean 5.001

5.26 a thousand

ten thousand um

one million because it's any number

bigger than five

and you can go very close to the five as

well but you just can't touch the five

so what we'll do is we'll show one

number on our number line

and because now we don't have two

numbers we're not going to include the

five

now how do we show that it can be any

number bigger than five or you just show

a continuous arrow

like that and what that means is that

you can go on and on and on and on and

on

in interval notation you'll say xe

always remember the xe

round bracket because we're not

including the five

now what number do we call this one up

here

well a number that is so big that it

just keeps going on and on and on we're

going to use

infinity and infinity is always a round

bracket

because you can't reach infinity for the

interval notation

which always has three components you

start off by saying that we're working

with

x you can now say that x must be

bigger than 5 you could also say the

other way around

you could also do it like that it

doesn't matter x is bigger than 5 you

don't have to say smaller than infinity

you can just say bigger than 5

and then you must say x is an element of

r

so notice once again we have three

different components we've got this part

we've got this part and that part over

there

for this part over here some students

they do like to do the following

they like to say that x must be bigger

than 5

but it must be smaller than infinity you

can add that

over there if you wanted to so guys

that's all for this lesson in the next

lesson i will practice this a few more

times because i know it

it is quite weird at first