In this lesson, we're going to learn some really important things

在本課中，我們將學習一些非常重要的內容

about a whole branch of math called Algebra.

關於一門名為 "代數 "的數學分支。

The first thing you need to know is that Algebra is a lot like arithmetic.

首先您需要知道的是，代數與算術很相似。

It follows all the rules of arithmetic

它遵循所有算術規則

and it uses the same four main operations that arithmetic is built on:

它使用與算術相同的四種主要運算：

addition, subtraction, multiplication and division.

加法、減法、乘法和除法。

But Algebra introduces a new element...

但代數引入了一個新元素

the element of the unknown.

未知因素。

When you were learning arithmetic,

在你學算術的時候

the only thing that was ever unknown was the answer to the problem.

唯一未知的就是問題的答案。

For example, you might have the problem 1 + 2 = what?

例如，你可能會遇到 1 + 2 = 什麼的問題？

The answer isn't known until you go ahead and do the arithmetic.

答案只有等你去算一算才知道。

Now the important thing about Algebra is that when we don't know what a number is yet,

現在，代數的重要之處在於，當我們還不知道一個數字是什麼的時候、

we use a symbol in its place.

我們用一個符號來代替。

that symbol is usually just any letter of the alphabet.

該符號通常只是字母表中的任何一個字母。

A really popular letter to choose is the letter 'x'.

一個非常受歡迎的字母是'x'。

So in arithmetic, we would just leave the problem like this: 1 + 2 = "blank"

是以，在算術中，我們只需這樣處理問題：1 + 2 = "空白&quot；

and we'd write in the answer when we did the addition.

在做加法運算時，我們會寫入答案。

But in Algebra, we'd write it like this: 1 + 2 = x

但在代數學中，我們會這樣寫： 1 + 2 = x

The 'x' is a place holder that stands for the number that we don't know yet.

'x'是一個佔位符，代表我們還不知道的數字。

What we have here is a very basic algebraic equation.

我們現在看到的是一個非常基本的代數方程。

An equation is just a mathematical statement that two things are equal.

等式只是兩個事物相等的數學表述。

An equation says, "the things on this side of the equal sign

等式說，等號這一邊的事物

have the same value as the things on the other side of the equal sign."

與等號另一側的事物具有相同的價值；

In this case, our equation is telling us that the known values on this side (1+2)

在這種情況下，等式告訴我們這邊的已知值 (1+2)

are equal to what's on the other side,

與另一邊的東西相等、

which happens to be the unknown value that we are calling 'x'.

這恰好是我們稱之為 'x' 的未知值。

One of the main goals in Algebra

代數的主要目標之一

is to figure out what the unknown values in equations are.

就是找出方程中的未知值。

And when you do that, it's called "solving the equations".

當你這樣做的時候，就叫做 "解方程"。

In this equation, it's pretty easy to see that the unknown value is just 3.

在這個等式中，很容易看出未知值只是 3。

All you have to do is actually add the 1 and 2 together on this side of the equation

你要做的就是把等式這邊的 1 和 2 相加

and it turns into 3 = x, which is the same as x = 3.

就變成了 3 = x，與 x = 3 相同。

So now we know what 'x' is! It's just 3.

現在我們知道什麼是'x'了！ 就是3。

That almost seems too easy, doesn’t it?

這似乎太容易了，不是嗎？

And that's why in Algebra, you are usually given an equation in a more complicated form

這就是為什麼在代數中，通常會給你一個更復雜形式的等式

like this: x - 2 = 1.

就像這樣：x - 2 = 1。

This is exactly the same equation as 1 + 2 = x,

這與 1 + 2 = x 的等式完全相同、

but it has been rearranged so that it's not quite as easy to tell what 'x' is.

但經過重新編排後，&#39；x&#39；是什麼就不那麼容易分辨了。

So in Algebra, solving equations is a lot like a game

是以，在代數學中，解方程很像一場遊戲

where you are given mixed-up, complicated equations,

在這裡，你會得到紛繁複雜的方程式、

and it's your job to simplify them and rearrange them

而你的工作就是簡化它們，重新排列它們

until it is a nice simple equation (like x = 3)

直到它成為一個漂亮的簡單方程（如 x = 3）

where it's easy to tell what the unknown values are.

在這裡，很容易看出未知值是什麼。

We're going to learn a lot more about

我們將進一步瞭解

how you actually do that (how you solve equations)

如何實際操作（如何解方程）

in the next several videos,

在接下來的幾個視頻中、

but for now, let's learn some important rules about

但現在，讓我們學習一些重要規則，瞭解

how symbols can and can't be used in algebraic equations.

代數方程中如何使用和不能使用符號。

The first rule you need to know is that the same symbol (or letter)

您需要知道的第一條規則是，相同的符號（或字母）

can be used in different algebra problems to stand for different unknown values.

可用於不同的代數問題，代表不同的未知值。

For example, in the problem we just solved,

例如，在我們剛剛解決的問題中、

the letter 'x' was used to stand for the number 3, right?

字母'x'被用來代表數字 3，對嗎？

But 'x' could stand for a different number in a different problem.

但是，'x'可以代表不同問題中的不同數字。

Like, if someone asks us to solve the equation, 5 + x = 10.

比如，有人讓我們解方程 5 + x = 10。

In order for the two sides of this equation to be equal,

為了使等式的兩邊相等、

'x' must have the value '5' in this problem, because 5 + 5 = 10.

在這個問題中，'x'的值一定是'5'，因為 5 + 5 = 10。

So 'x' (or any other symbol) can stand for different values in different problems.

是以，'x'（或任何其他符號）在不同的問題中可以代表不同的值。

That's okay,

沒關係

but what's NOT okay is for a symbol

但不允許的是一個符號

to stand for different values in the same problem at the same time!

同時代表同一問題中的不同數值！

For example, what if you had the equation: x + x = 10?

例如，如果你有這樣一個方程：x + x = 10？

This equation says that if we add 'x' to 'x' we will get 10.

這個等式表示，如果我們將 'x' 加到 'x'，就會得到 10。

And there are a lot of different numbers we could add together to get 10, like 6 and 4.

有很多不同的數字可以相加得到 10，比如 6 和 4。

But, if we had the first 'x' stand for 6 and the second 'x' stand for 4,

但是，如果我們讓第一個'x'代表 6，第二個'x'代表 4、

then 'x' would stand for two different value at the same time

那麼 'x' 將同時代表兩個不同的值

and things could get really confusing!

事情可能會變得非常混亂！

If you wanted symbols to stand for two different numbers at the same time,

如果您想讓符號同時代表兩個不同的數字、

you would need to use two different symbols, like 'x' and 'y'.

您需要使用兩個不同的符號，如 'x'和 'y'。

So in Algebra, whenever you see the same symbol repeated more than once in an equation,

是以，在代數中，只要你看到同一個符號在等式中重複出現不止一次、

it's representing the same unknown value.

它代表的是同一個未知值。

Like if you see a really complicated Algebraic equation (like this),

比如你看到一個非常複雜的代數方程（像這樣）、

where 'x' is repeated a lot of different times,

其中，'x'重複了很多次、

all of those 'x's stand for the same value,

所有這些 'x'都代表相同的值、

and it will be your job to figure out what that value is.

你的工作就是找出這個值是多少。

Okay, so for any particular equation,

好吧，那麼對於任何一個特定的等式、

we can't use the same letter to represent two different numbers at the same time,

我們不能同時用同一個字母來表示兩個不同的數字、

but what about the other way around?

但反過來呢？

Could we use two different letters to represent the same number?

我們可以用兩個不同的字母來表示同一個數字嗎？

Yes! - Here's an example of that.

這就是一個例子。

Let's say you have the equation: a + b = 2

假設方程為：a + b = 2

What could 'a' and 'b' stand for so that the equation is true?

為了使等式成立，'a'和'b'分別代表什麼？

Well, If 'a' was 0 and 'b' was 2, then the equation would be true.

好吧，如果 'a'為 0，而 'b'為 2，那麼等式就成立了。

Or, we could switch them around.

或者，我們可以把它們調換一下。

If 'a' was 2 and 'b' was 0, the equation would also be true.

如果 'a'為 2，而 'b'為 0，等式也將成立。

But there's another possibility:

但還有另一種可能：

If 'a' was 1 and 'b' was also 1, that would make the equation true, right?

如果'a'為 1，'b'也為 1，那麼等式就成立了，對嗎？

So, even though 'a' and 'b' are different symbols,

是以，儘管 'a' 和 'b' 是不同的符號、

and would usually be used to represent different numbers,

通常用來表示不同的數字、

there are times when they might happen to represent the same number.

有時，它們可能恰好代表同一個數字。

Oh, and this problem can help us understand something

哦，這個問題可以幫助我們理解一些東西

very important about how symbols are used in Algebra.

在代數中如何使用符號非常重要。

Did you notice that there were different possible solution for this equation?

你注意到這個方程有不同的可能解嗎？

In other words, 'b' could have the value 0, 1, or 2 depending on what the value of 'a' was.

換句話說，'b'的值可能是 0、1 或 2，這取決於 'a'的值是多少。

If 'a' is 0 then 'b' must be 2

如果 'a' 為 0，那麼 'b' 必須為 2

If 'a' is 1 then 'b' must be 1

如果 'a' 是 1，那麼 'b' 一定是 1

If 'a' is 2 then 'b' must be 0

如果 'a' 是 2，那麼 'b' 一定是 0

'b' can't have two different values at the same time,

'b'can't have two different values at the same time、

but it's value can change over time if the value of 'a' changes.

但如果 &#39；a&#39；的值發生變化，它的值也會隨之變化。

In Algebra, 'b' is what's called a "variable" because it's value can 'vary' or change.

在代數中，'b'被稱為'變量'，因為它的值可以'變化'或改變。

In fact, in this equation, both 'a' and 'b' are variables

事實上，在這個等式中，'a'和'b'都是變量

because their values will change depending on the value of each other.

因為它們的值會隨著彼此的值變化而變化。

Actually, it's really common in Algebra to refer to any letter as a variable,

實際上，在代數中，將任何字母稱為變量都很常見、

since letters can stand for different values in different problems.

因為在不同的問題中，字母可以代表不同的值。

But at Math Antics, we'll usually just use the word "variable"

但在數學趣談中，我們通常只使用 "變量 "一詞；

when we're talking about values that can change or vary in the same problem.

當我們討論同一問題中可能變化或不同的值時。

Alright, so far we've learned that Algebra is a lot like arithmetic,

好了，到目前為止，我們已經瞭解到代數與算術有很多相似之處、

but that in includes unknown values and variables that we can solve for in equations.

但其中包括我們可以在方程中求解的未知值和變量。

There's one other really important thing that I want to teach you

我還想教你一件非常重要的事

that will help you understand what's going on in a lot of Algebra problems,

它將幫助你理解許多代數問題的來龍去脈、

and it has to do with multiplication.

和乘法有關。

Here are the four basic arithmetic operations:

以下是四種基本算術運算：

addition, subtraction, multiplication and division.

加法、減法、乘法和除法。

Although in Algebra, you'll usually see division written in fraction form, like this.

雖然在代數中，你通常會看到用分數形式寫的除法，就像這樣。

In Arithmetic, all four operations have the same status,

在算術運算中，所有四種運算都具有相同的狀態、

but in Algebra, multiplication get's some special treatment.

但在代數學中，乘法得到了一些特殊處理。

In Algebra, multiplication is the 'default' operation.

在代數中，乘法是默認運算。

That means, if no other arithmetic operation is shown between two symbols,

也就是說，如果兩個符號之間沒有其他算術運算、

then you can just assume they're being multiplied. The multiplication is 'implied'.

那麼你就可以認為它們是相乘的。乘法是隱含的。

For example, instead of writing 'a' times 'b',

例如，不要寫 'a'乘以 'b'、

you can leave out the times symbol and just write 'ab'.

可以省略次數符號，只寫 'ab'。

Since no operation is shown between these two symbols,

因為這兩個符號之間沒有運算、

you know that you're supposed to multiply 'a' and 'b'.

你知道你應該乘以&#39；a&#39；和&#39；b&#39；。

Of course, you can't actually multiply 'a' and 'b'

當然，你不可能把 &#39；a&#39；和 &#39；b&#39；相乘；

until you figure out what numbers they stand for.

直到你弄清楚它們代表什麼數字。

The advantage of this rule about multiplication is that

這條乘法規則的優點是

it makes many algebraic equations less cluttered and easier to write down.

它使許多代數方程不再雜亂無章，更易於書寫。

For example, instead of this: a * b + c * d = 10

例如：a * b + c * d = 10

You could just write: ab + cd = 10.

你可以直接寫出：ab + cd = 10。

You can also use this shorthand when you are multiplying

在進行乘法運算時，也可以使用這種速記方法

a variable and a known number... like 2x, which means the same thing as 2 times 'x'

一個變量和一個已知數......比如 2x，它的意思和 2 乘以'x&#39 是一樣的；

or 3y which means the same thing as 3 times 'y'

或 3y，意思與 3 次 'y&#39 相同；

Since the symbol and the number are right next to each other,

因為符號和數字緊挨著、

the multiplication is implied.

乘法是隱含的。

You don't have to write it down.

你不必寫下來。

Finally some good news!

終於有好消息了

Now I never have to write down that pesky multiplication symbol again!

現在，我再也不用寫下那討厭的乘法符號了！

Oh yeah!!

是的

Ah, not so fast...

啊，沒那麼快......

there are some cases in Algebra where still need to use a multiplication symbol.

在代數中，有些情況下仍然需要使用乘法符號。

For example, what if you want to show 2 x 5?

例如，如果您想顯示 2 x 5 呢？

If you just get rid of the times symbol and put the '2' right next to the '5',

只要去掉時間符號，把 '2' 放在 '5' 旁邊即可、

it's going to look like the two-digit number twenty-five,

它看起來就像兩位數二十五、

which is NOT the same as 2 x 5.

這與 2 x 5 不一樣。

So, whenever you need to show multiplication between two known numbers,

是以，無論何時需要顯示兩個已知數之間的乘法關係、

you still have to use the times symbol, unless...

您仍然必須使用時間符號，除非...

you use parentheses instead.

而不是使用括號。

But aren't parentheses used to show grouping in math?

但是，數學中的括號不是用來表示分組的嗎？

How can you use that to show multiplication?

如何用它來表示乘法？

Ah, that's a good question.

啊，這個問題問得好。

Parentheses are used to group things,

括號用於分組、

but whenever you put two groups right next to each other,

但每當你把兩組人放在一起時、

with no operation between them,

之間沒有任何操作、

guess what operation is implied.

猜猜這意味著什麼操作。

Yep! Multiplication!

沒錯！ 乘法

For example, if you see the this,

例如，如果您看到以下內容

it means that the group (a+b) is being multiplied by the group (x+y).

這意味著 (a+b) 組與 (x+y) 組相乘。

We could put a times symbol between the groups, but we don't have to

我們可以在各組之間加一個時間符號，但不必這樣做

because it's the default operation in Algebra.

因為它是代數中的默認操作。

The multiplication is just implied.

乘法只是一種暗示。

So, going back to our problem: 2 x 5

那麼，回到我們的問題：2 x 5

If you wanted to, you could put each of the numbers inside parentheses like this,

如果你願意，可以像這樣把每個數字都放在括號裡、

and then you could get rid of the multiplication sign.

然後就可以去掉乘號了。

Now this can't be confused with the number twenty-five,

現在，這不能與數字 25 混淆、

and since the groups are right next to each other,

而且這兩個小組緊挨著、

you know that you need to multiply them.

你知道你需要把它們加倍。

Of course, it might seem strange to have just one thing inside 'group' symbols like this,

當然，在這樣的'組'符號中只有一件事可能會顯得很奇怪、

but it's okay to do that in math.

但在數學課上這樣做是可以的。

An alternate way that you could do the same thing would be

做同樣事情的另一種方法是

to put just one of the numbers in parentheses, like this.

在括號中只寫一個數字，就像這樣。

Again, you won't confuse this with a two-digit number

同樣，您不會將其與兩位數混淆

and you know that multiplication is implied.

就知道乘法是隱含的。

Okay, so we've learned that Algebra is a lot like arithmetic,

好了，我們已經瞭解到代數與算術有很多相似之處、

but it involves unknown values or variables that we need to solve for.

但它涉及我們需要求解的未知值或變量。

And we learned that in Algebra, the multiplication sign is usually not shown,

我們還學到，在代數中，乘號通常是不顯示的、

because it's the default operation.

因為這是默認操作。

You can just assume that two things right next to each other are being multiplied.

你可以假定相鄰的兩樣東西正在相乘。

But why do we even care about Algebra?

但我們為什麼還要關心代數呢？

Is it good for anything in the 'real world'?

在'現實世界'中，它有什麼好處嗎？

or is it just a bunch of tricky problems that keep students busy in school?

還是隻是一堆讓學生在學校裡忙得不可開交的難題？

Actually, Algebra is very useful for describing or "modeling" things in the real world.

事實上，代數對於描述或模擬現實世界中的事物非常有用。

It's a little hard to see that when you are just looking

當你只是看的時候，很難發現這一點

at all these numbers and symbols on the page of a math book.

數學書上的這些數字和符號。

But, it's a lot easier to see

但是，它更容易看到

when you start taking algebraic equations and graphing their solutions.

當你開始計算代數方程並繪製解的圖形時。

Graphing an equation is like using its different solutions

繪製方程圖就像使用方程的不同解法

to draw simple lines and curves that can be used to describe and predict things in real life.

繪製簡單的直線和曲線，用於描述和預測現實生活中的事物。

For example, there's a whole class of equations in Algebra

例如，代數學中有一類方程

called "linear" equations because they form straight lines when you graph them.

稱為"線性方程"是因為它們在作圖時會形成直線。

Those sorts of equations could help you describe the slope of a roof

這些等式可以幫助你描述屋頂的坡度

or tell you how long it will take to get somewhere.

或告訴你到達某個地方需要多長時間。

Another class of algebraic equations, called "quadratic" equations

另一類代數方程稱為二次方程

can be used to design telescope lenses,

可用於設計望遠鏡鏡片、

or describe how a ball flies through the air,

或描述球是如何在空中飛行的、

or predict the growth of a population.

或預測人口增長。

So Algebra is used all the time in fields like

是以，代數一直被用於以下領域

science, engineering, economics and computer programming.

科學、工程學、經濟學和計算機編程。

And even though you might not need Algebra to get by in your day to day life,...

儘管在日常生活中您可能不需要代數來維持生計，但您仍然需要代數。

...so divide both sides by 3. That means x = 42.

......所以兩邊都除以 3，即 x = 42。

So... in three... two... one...

所以，三、二、一

YES!!

是的！

Alright... now to see how much butter I need.

好吧......現在看看我需要多少黃油。

It's still a very useful part of math.

它仍然是數學中非常有用的一部分。

Thanks for watching Math Antics, and I'll see ya next time!

感謝您收看《數學趣事》，我們下次再見！

Learn more at www.mathantics.com

瞭解更多資訊，請訪問 www.mathantics.com