Практика Shadowing: WHAT ARE NATURAL NUMBERS? - Изучайте разговорный английский по видео

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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

101

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

103

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

105

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

106

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.

108

However, the i is there.

109

That i means imaginary.

110

Okay?

111

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.

113

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

114

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

115

You should know what an even number is.

116

You should know what an odd number is.

117

And you should know what a prime number is.

118

Okay?

119

You should know what a fraction is.

120

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

121

but ultimately you should know what a rational number is.

122

That would be my final point.

123

Rational numbers.

124

Rational numbers a number is rational

125

if it can be written as a

126

fraction rational numbers are all numbers

127

that can be written

128

as a fraction let me change this

129

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

130

that can be written as a fraction

131

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

133

so what's a rational number well all integers

134

or whole numbers all those things can be written as a

135

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

136

that are not rational the most popular of them will be

137

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

139

when you write it in decimal

140

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

141

so you have to write as a decimal and

142

when you write it as a decimal it does not repeat

143

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

144

so pi is irrational because

145

when you start writing the digits of pi it has no

146

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

149

that is not a perfect fourth okay so

150

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

155

and there's some other mathematical constants

156

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

158

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

160

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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Цели разговорной практики для этого клипа

Этот видеокlip помогает улучшить практику разговорного английского, а именно освоить темп речи и уверенность при объяснении абстрактных понятий (в данном случае — свойств чисел). Цель: научиться выражать мысли последовательно, используя простые примеры, как это делает говорящий в видео. Это ключевой этап для развития логicznego изложения в английском.

Банк фраз

"Do you know what... is?" — "Знаете ли вы, что такое...?" (удобно для постановки вопросов)
"If a number is..., it means..." — "Если число..., это означает..." (описание свойств)
"Let's say they are... plus..." — "Допустим, это... плюс..." (упрощение объяснения)
"Just know if we change... to..." — "Просто помните, если мы изменим... на..." (демонстрация преобразования)
"All the way to the end" — "Вплоть до бесконечности" (описание последовательности)

Как улучшить произношение и ритм речи

В этом видео особенно важно обратить внимание на натуральную паузу между идеями. Говорящий использует короткие паузы после вопросов ("Do you know what real numbers are?") и перед важными определениями, чтобы слушатель мог усвоить информацию. Это отличная возможность для shadow speech — повторяйте за говорящим, копируя его темп и интонацию. Обратите внимание на ударения в словах like "properties" (['prɒpətiz]) и "integers" (['ɪntɪdʒəz]) — неправильное ударение может испортить понимание. Практикуйте shadowspeaks daily: запишите себя, сравните с оригиналом и исправляйте ошибки.

Этот клип — отличный материал для учить английский с YouTube, так как сочетает простой язык с практической информацией. Используйте фразы из банка в своих разговорах о математике или других предметах, и вы увидите, как улучшится ваша разговорная речь!

Грамматика в этом видео

Конструкции, которые говорящий использует чаще всего, с точными словами из видео:

Конструкция	В видео
Пассивный залог be + причастие прошедшего времени — важно, что происходит, а не кто это делает	is called · can be written
Present Perfect have/has + причастие прошедшего времени — прошлое действие, важное сейчас	you've seen · you've known · we've talked

Что такое техника Shadowing?

Shadowing — это научно обоснованная техника изучения языка, изначально разработанная для подготовки профессиональных переводчиков и популяризированная полиглотом доктором Александром Аргуэльесом. Метод прост, но эффективен: вы слушаете аудио на английском от носителей языка и немедленно повторяете вслух — как тень, следующая за говорящим с задержкой в 1–2 секунды. В отличие от пассивного прослушивания или грамматических упражнений, Shadowing заставляет мозг и мышцы рта одновременно обрабатывать и воспроизводить реальные речевые паттерны. Исследования показывают, что это значительно улучшает точность произношения, интонацию, ритм, связную речь, понимание на слух и беглость речи — что делает его одним из самых эффективных методов для подготовки к IELTS Speaking и реального общения на английском.

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Цели разговорной практики для этого клипа

Банк фраз

Как улучшить произношение и ритм речи

Грамматика в этом видео

Что такое техника Shadowing?