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• Hi! This is Rob. Welcome to Math Antics.

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

• about a whole branch of math called Algebra.

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

• It follows all the rules of arithmetic

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

• addition, subtraction, multiplication and division.

• But Algebra introduces a new element...

• the element of the unknown.

• When you were learning arithmetic,

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

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

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

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

• we use a symbol in its place.

• that symbol is usually just any letter of the alphabet.

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

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

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

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

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

• What we have here is a very basic algebraic equation.

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

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

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

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

• are equal to what's on the other side,

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

• One of the main goals in Algebra

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

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

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

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

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

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

• That almost seems too easy, doesn’t it?

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

• like this: x - 2 = 1.

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

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

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

• where you are given mixed-up, complicated equations,

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

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

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

• We're going to learn a lot more about

• how you actually do that (how you solve equations)

• in the next several videos,

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

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

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

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

• For example, in the problem we just solved,

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

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

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

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

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

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

• That's okay,

• but what's NOT okay is for a symbol

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

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

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

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

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

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

• and things could get really confusing!

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

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

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

• it's representing the same unknown value.

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

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

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

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

• Okay, so for any particular equation,

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

• but what about the other way around?

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

• Yes! - Here's an example of that.

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

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

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

• Or, we could switch them around.

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

• But there's another possibility:

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

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

• and would usually be used to represent different numbers,

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

• Oh, and this problem can help us understand something

• very important about how symbols are used in Algebra.

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

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

• If 'a' is 0 then 'b' must be 2

• If 'a' is 1 then 'b' must be 1

• If 'a' is 2 then 'b' must be 0

• '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.

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

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

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

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

• since letters can stand for different values in different problems.

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

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

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

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

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

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

• and it has to do with multiplication.

• Here are the four basic arithmetic operations:

• addition, subtraction, multiplication and division.

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

• In Arithmetic, all four operations have the same status,

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

• In Algebra, multiplication is the 'default' operation.

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

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

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

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

• Since no operation is shown between these two symbols,

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

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

• until you figure out what numbers they stand for.

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

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

• You could just write: ab + cd = 10.

• You can also use this shorthand when you are multiplying

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

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

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

• the multiplication is implied.

• You don't have to write it down.

• Finally some good news!

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

• Oh yeah!!

• Ah, not so fast...

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

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

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

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

• which is NOT the same as 2 x 5.

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

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

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

• How can you use that to show multiplication?

• Ah, that's a good question.

• Parentheses are used to group things,

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

• with no operation between them,

• guess what operation is implied.

• Yep! Multiplication!

• For example, if you see the this,

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

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

• because it's the default operation in Algebra.

• The multiplication is just implied.

• So, going back to our problem: 2 x 5

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

• and then you could get rid of the multiplication sign.

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

• and since the groups are right next to each other,

• you know that you need to multiply them.

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

• but it's okay to do that in math.

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

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

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

• and you know that multiplication is implied.

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

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

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

• because it's the default operation.

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

• But why do we even care about Algebra?

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

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

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

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

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

• But, it's a lot easier to see

• when you start taking algebraic equations and graphing their solutions.

• Graphing an equation is like using its different solutions

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

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

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

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

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

• Another class of algebraic equations, called "quadratic" equations

• can be used to design telescope lenses,

• or describe how a ball flies through the air,

• or predict the growth of a population.

• So Algebra is used all the time in fields like

• science, engineering, economics and computer programming.

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

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

• So... in three... two... one...

• YES!!

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

• It's still a very useful part of math.

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

Hi! This is Rob. Welcome to Math Antics.

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# 代数学の基礎代数とは何か？- 数学アンチックス (Algebra Basics: What Is Algebra? - Math Antics)

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Yassion Liu に公開 2021 年 01 月 14 日