跟读练习: WHAT ARE NATURAL NUMBERS? - 通过视频学习英语口语

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Hello, welcome back to another video.

I would like you to know today what the properties of numbers are.

Okay?

I know we all use numbers every day, but do you know what real numbers are?

Do you know what a natural number is?

If a number is not natural, what would you say it is?

Unnatural or supernatural?

Well, let's find out.

so properties of numbers i'm just going to

start with what i just asked natural numbers

if a number is natural it means you can find it in nature

and when humans try to count or try to talk about numbers They do this a lot.

One, two, three.

See the first number?

That's the first natural number.

One.

Not zero.

There's no zero among natural numbers.

So the set of natural numbers are the numbers you can count on your fingers.

One, two, three, four, five.

All numbers are positive.

So if you want to know natural numbers, you just need to look at your fingers.

Whatever you find is a natural number.

Okay, so set of natural numbers will be positive one, positive two, like that, three, four, all the way to the end.

Don't stop.

If it's a natural number, it keeps on going forever and ever.

Okay, let's go to number two.

So if a number is not a natural number, what is it?

It must be on the other side of the negative.

Okay, so, but before you cross to the other side of the negative, there's a number just before positive 1.

That number is called 0.

0 is not a natural number, but it belongs to the set of numbers called whole numbers.

So, our next definition is the set of whole numbers.

If a number is a whole number, well it is either a natural number or it's a zero.

Zero plus the set of natural numbers.

I love this!

numbers

so it's a whole numbers are just natural numbers including zero

that's it that's all we got to say let's move on to the next set let's talk about integers

why am I writing like this?

Come on.

Integers.

Okay, what are integers?

Well, integers are all these things you've seen.

Whole numbers plus whole negative.

I don't want to say negative whole numbers, but if that's how you're going to understand it,

Oh, let's say they are whole numbers plus all negative.

Oh, there's no negative natural number.

Just know if we change these signs to negative, just look at your number line.

Okay?

You walk into a classroom, you see the number line from negative infinity to zero to positive infinity.

That's what you call integers.

Put that in your head.

I'm just going to write the number line from left to right.

no fractions no decimals we don't call them whole numbers

but they are whole numbers

but still have negative portion of the whole that's confusing just

stick to it just look at the number line all the numbers you see are what you call integers keep

that in your head or in your notebook

and then later in your head let's move on the next thing let's go to real numbers whoa

so real numbers are real okay

if a number is a real number it means it is any of these things

that you've known or maybe fractions decimals rational or irrational numbers just know

that every number you know is a real number unless there's a part of it

that is not real

and the part of a number is not real you say it is an imaginary number

or a complex number i'm going to explain what an imaginary number is

or what a complex number is but just know

that every number you know is real all numbers fractions decimals

whole numbers negative positive zeros infinity All those numbers are real, okay?

Real numbers.

All numbers that are not imaginary.

Okay?

So, if a number is not imaginary, it's a real number.

This is the largest group of numbers you'll be dealing with, okay?

So if a number is not real, it's going to be imaginary.

Now what's an imaginary number?

That's the next thing I'm going to do.

Let's get rid of this.

I know this is in your head already.

Let's go.

Come on.

So, what's an imaginary number?

Imaginary number.

Let me give you a simple example.

The square root of 4 is positive 2.

Because positive 2 times positive 2 is going to give you 4.

So the root of 4 is going to be 2.

Now let's look at, what if I said that the square root of 4 is negative 2?

Will I be correct?

Yes, I will be because negative 2 times negative 2 is still going to be positive 4.

So, what if I say the square root of negative 4?

And that becomes impossible because it cannot be negative 2 and it cannot be positive 2

because both of these are answers to square root of 4.

So what's the square root of negative 4?

Yeah, it's a bit twisted.

What if we rewrite this as the square root of negative 1 multiplied by 4?

Negative 1 multiplied by 4 gives you the square root of negative 1 multiplied by the square root of 4.

Ah, now we can come back to getting our answer positive to or negative to whatever we choose,

100

will be fine but what again is the square root of negative one it is not one

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and it is not negative one

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because those two numbers multiply negative one times negative one gives you one not negative one

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so how do we resolve this well we just need to imagine

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that there is an answer is it real no that's why it's called an imaginary number

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so this is called the imaginary portion of the answer and it is written as i.

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The square root of 4 is 2.

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So our answer here is 2i or negative 2i, depending on which of these options you pick.

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However, the i is there.

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That i means imaginary.

110

Okay?

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So if a number doesn't have this kind of behavior, you know the number is real.

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If it has an I in it, it's imaginary.

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Now you know your sets of numbers, you know the properties of numbers.

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There's some other properties that you should know by now.

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You should know what an even number is.

116

You should know what an odd number is.

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And you should know what a prime number is.

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Okay?

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You should know what a fraction is.

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and you should know what a decimal is, and then all kinds of decimals,

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but ultimately you should know what a rational number is.

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That would be my final point.

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Rational numbers.

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Rational numbers a number is rational

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if it can be written as a

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fraction rational numbers are all numbers

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that can be written

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as a fraction let me change this

129

so the grammar is correct it's any number any number any number

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that can be written as a fraction

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so every number we've talked about today is supposed to be

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a part well all rational numbers will be real numbers

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so what's a rational number well all integers

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or whole numbers all those things can be written as a

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fraction for example three is the same thing as three over one well there's some few numbers

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that are not rational the most popular of them will be

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so will be pi so just know

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that every number you come across is supposed to be rational unless it has a behavior like this

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when you write it in decimal

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or as a you can't write it as a fraction

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so you have to write as a decimal and

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when you write it as a decimal it does not repeat

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okay it does not it behaves like pi this is a good example of an irrational number okay

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so pi is irrational because

145

when you start writing the digits of pi it has no

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end that's number one another good example is the square root of any number

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that is not a perfect square or the cube root of any number

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that is not a perfect cube or the fourth root of any number

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that is not a perfect fourth okay so

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if you try to take the square root of say eight

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for example well this eight is not a perfect square you can't really tell

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so when you write the answer you're going to have decimals

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that keep on going you can check it in your calculator

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so the square root of a is also irrational okay

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and there's some other mathematical constants

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that you never find them um to end there's no you just approximate you just approximate you just approximate

157

or you assume a fraction which is not real

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so pi is irrational the square root of a number

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that is not a perfect square is also irrational okay i

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believe you've learned something you've gotten gotten something to keep okay

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if you ever forget any of these things come back to

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the video okay send me an email send me uh put

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a comment in the description um in the comment section

164

and please share like

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and subscribe to my channel it encourages me to put up more videos thank you

166

and don't stop learning god bless you bye Thank you.

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本片段的口语学习目标

本视频适合想要提升雅思口语练习中“解释概念”流畅度的学习者。通过模仿数字分类的讲解逻辑，能锻炼清晰表达抽象定义的能力，同时积累学术口语中常用的“下定义”“作区分”句型，为口语考试中的说理类话题打好基础。

实用短语库

“如果用这种方式理解的话”：用于简化复杂概念，适合口语中灵活转换表达
“只需想象一下数轴”：借助具象工具辅助说明，增强表达的画面感
“一直延伸到无穷”：描述数量范围时的常用表达，替换生硬的“很多”
“包括……以及所有的负数”：用于列举集合的构成，逻辑清晰

攻克口语弱点

视频中的讲解节奏自然，包含大量英语口语练习中关键的“停顿”和“连接词”，非常适合shadowspeak训练。例如，“好吧？”“你知道吗？”等口语化插入语能帮助学习者摆脱“背书式”表达，提升流畅度。此外，讲解数字概念时的升调（如“自然数是1、2、3……对吗？”）和降调（如“0不是自然数”）对比明显，可有效提高英语发音的语调准确性。建议反复跟读，重点模仿“逐步解释”时的语速变化，让口语更接近母语者的表达习惯。

通过看YouTube学英语，结合视频中的实际语境记忆这些短语，再用shadowspeak方法强化练习，能快速突破“概念表达不清晰”的瓶颈，为雅思口语考试中的学术话题讨论做好准备。

视频中的语法

说话人最常用的结构，并附上视频中的原话：

结构	视频中的用法
被动语态 be + 过去分词 — 强调发生了什么，而不是谁做的	is called · can be written
现在完成时 have/has + 过去分词 — 过去发生但与现在仍有关联的事	you've seen · you've known · we've talked

什么是跟读法？

跟读法 (Shadowing) 是一种有科学依据的语言学习技巧，最初开发用于专业口译员的培训，并由多语言者Alexander Arguelles博士普及。这个方法简单而强大：您在听英语母语原声的同时立即大声重复——就像是一个延迟1-2秒紧跟说话者的影子。与被动听力或语法练习不同，跟读法强迫您的大脑和口腔肌肉同时处理并模仿真实的讲话模式。研究表明它能显着提高发音准确性，语调，节奏，连读，听力理解和口语流利度——使其成为雅思口语备考和真实英语交流最有效的方法之一。

影子跟读法: 阅读完整分步指南 →

跟读练习: WHAT ARE NATURAL NUMBERS? - 通过视频学习英语口语

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本片段的口语学习目标

实用短语库

攻克口语弱点

视频中的语法

什么是跟读法？