The Power of One
Oct. 19th, 2007 01:08 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
rabidsamfanOkay, so you now know that dividing fractions works the way your teacher said it did, but you still don't understand WHY it works.
It works because of one. Or rather, it works because of the special thing which the number one does when you multiply or divide. That is to say, nothing... *screech of brakes*
Er, hmm. In our previous episode I was pretty punny with the word "nothing", but here I'm going to have to use it more carefully. Let's agree that when I mean the quantity I'll just write zero, okay? When I mean the symbol I'll put the word into brackets like so: [zero], and I'll reserve the word "nothing" for the non-technical sense.
In addition (and subtraction), zero is the special number that lets you do something and do nothing at the same time. If you add a zero to a number -- whatever number -- your total is equal to the number which wasn't the zero. (Well, actually it could have been a zero, and your total would be zero too, but still you end up with a total which is just the same as one of the numbers you started out with, right?) Subtract a zero, and you haven't changed a thing.
There's a special name for this. (Who says mathematicians don't like words?) Zero is the identity element for addition. (And subtraction.)
For multiplication and division the identity element is one. Take any number, multiply or divide it by one, and you're back where you started. The quantity in the answer is the same as the number you started out with.
So we've got a zero and we've got a one, and they're both identity elements, (you can think of them as mirrors) and identity elements are very useful. Here's why:
One can be written as a symbol: 1, or as a pattern. And the pattern for writing one is a fraction where the numerator and the denominator are equal. X over X, six over six, two halves, ten tenths, or a gazillion gazillionths, it's all one. And because multiplying a quantity by one doesn't change the quantity, multiplying it by a pattern that means one doesn't change the quantity either -- but it can change the symbols we use to write the quantity into a more useful form. That's what we did when we added fractions. Changed the forms of the fractions until the denominators matched so that we could add.
You spend an awful lot of time in algebra figuring out the most useful way to change something without changing it. One, written as a fraction, is the tool you use to change indivual terms into new forms. You can change the size of an entire equation by multiplying both sides too. But sometimes you have to change an equation into a more useful shape (without changing what it means!) by moving things around so that only one term is left on one side of the equation and that's where zero comes in handy.
Zero can be also written in a couple of ways. The symbol [zero]: 0, or a subtraction pattern that takes a number and subtracts the same number: X minus X, six take away six, one minus one, they all come out to zero. (WARNING: This pattern can either be helpful, or it can trip you up. You can write zero as (a-b) if a is equal to b, but if (a-b) ends up in the denominator of your fraction you've now tried to divide by zero and that's against the rules.)
When you have an equation you can add (or subtract) the same number on both sides and you haven't changed the meaning of the equation. So, if you want to get rid of a number on one side of the equal sign, you can do something that makes it be zero on that side of the equation as long as you do the same thing on the other side of the equation.
Which leads us to the idea of inverses.
The additive inverse of a number is the number that brings you back to the additive identity when you add it and the multiplicative inverse of a number is the number that brings you back to the multiplicative identity when you multiply with it. That is to say, for example, if you want to add something to five in order to get zero you have to add negative five, and if you want to multiply something times five in order to get one you have to multiply by one fifth. Adding a negative number is equivalent to subtracting. And multiplying the inverse of a number is equivalent to dividing. (Which means that zero hasn't got a multiplicative inverse.)
Wait a minute. What did I just say?
Multiplying the inverse of a number is equivalent to dividing.
The multiplicative inverse is so important an idea it has its own name. We call it a reciprocal. The reciprocal of a fraction is the number which we have to multiply the fraction by in order to get an answer of 1. And joy of joys, it's the easiest thing in the world to create. All you have to do is turn the fraction on its head.
That's right. The numerator of a fraction is the denominator of its reciprocal and vice versa. 2/3 x 3/2 = 1. Google/Yahoo x Yahoo/Google = 1. And that's why, when you want to divide with a fraction, you turn it upside down and multiply instead.
*looks up from notes*
Hellooooo, is anyone still there?
It works because of one. Or rather, it works because of the special thing which the number one does when you multiply or divide. That is to say, nothing... *screech of brakes*
Er, hmm. In our previous episode I was pretty punny with the word "nothing", but here I'm going to have to use it more carefully. Let's agree that when I mean the quantity I'll just write zero, okay? When I mean the symbol I'll put the word into brackets like so: [zero], and I'll reserve the word "nothing" for the non-technical sense.
In addition (and subtraction), zero is the special number that lets you do something and do nothing at the same time. If you add a zero to a number -- whatever number -- your total is equal to the number which wasn't the zero. (Well, actually it could have been a zero, and your total would be zero too, but still you end up with a total which is just the same as one of the numbers you started out with, right?) Subtract a zero, and you haven't changed a thing.
There's a special name for this. (Who says mathematicians don't like words?) Zero is the identity element for addition. (And subtraction.)
For multiplication and division the identity element is one. Take any number, multiply or divide it by one, and you're back where you started. The quantity in the answer is the same as the number you started out with.
So we've got a zero and we've got a one, and they're both identity elements, (you can think of them as mirrors) and identity elements are very useful. Here's why:
One can be written as a symbol: 1, or as a pattern. And the pattern for writing one is a fraction where the numerator and the denominator are equal. X over X, six over six, two halves, ten tenths, or a gazillion gazillionths, it's all one. And because multiplying a quantity by one doesn't change the quantity, multiplying it by a pattern that means one doesn't change the quantity either -- but it can change the symbols we use to write the quantity into a more useful form. That's what we did when we added fractions. Changed the forms of the fractions until the denominators matched so that we could add.
You spend an awful lot of time in algebra figuring out the most useful way to change something without changing it. One, written as a fraction, is the tool you use to change indivual terms into new forms. You can change the size of an entire equation by multiplying both sides too. But sometimes you have to change an equation into a more useful shape (without changing what it means!) by moving things around so that only one term is left on one side of the equation and that's where zero comes in handy.
Zero can be also written in a couple of ways. The symbol [zero]: 0, or a subtraction pattern that takes a number and subtracts the same number: X minus X, six take away six, one minus one, they all come out to zero. (WARNING: This pattern can either be helpful, or it can trip you up. You can write zero as (a-b) if a is equal to b, but if (a-b) ends up in the denominator of your fraction you've now tried to divide by zero and that's against the rules.)
When you have an equation you can add (or subtract) the same number on both sides and you haven't changed the meaning of the equation. So, if you want to get rid of a number on one side of the equal sign, you can do something that makes it be zero on that side of the equation as long as you do the same thing on the other side of the equation.
Which leads us to the idea of inverses.
The additive inverse of a number is the number that brings you back to the additive identity when you add it and the multiplicative inverse of a number is the number that brings you back to the multiplicative identity when you multiply with it. That is to say, for example, if you want to add something to five in order to get zero you have to add negative five, and if you want to multiply something times five in order to get one you have to multiply by one fifth. Adding a negative number is equivalent to subtracting. And multiplying the inverse of a number is equivalent to dividing. (Which means that zero hasn't got a multiplicative inverse.)
Wait a minute. What did I just say?
Multiplying the inverse of a number is equivalent to dividing.
The multiplicative inverse is so important an idea it has its own name. We call it a reciprocal. The reciprocal of a fraction is the number which we have to multiply the fraction by in order to get an answer of 1. And joy of joys, it's the easiest thing in the world to create. All you have to do is turn the fraction on its head.
That's right. The numerator of a fraction is the denominator of its reciprocal and vice versa. 2/3 x 3/2 = 1. Google/Yahoo x Yahoo/Google = 1. And that's why, when you want to divide with a fraction, you turn it upside down and multiply instead.
*looks up from notes*
Hellooooo, is anyone still there?
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
(no subject)
Date: 2007-10-19 06:43 pm (UTC)*applauds*
(no subject)
Date: 2007-10-19 07:19 pm (UTC)(no subject)
Date: 2007-10-19 07:59 pm (UTC)I will take your word for it! But it was a very fun read!
(no subject)
Date: 2007-10-19 08:04 pm (UTC)(Goes back to drawing board.)
(no subject)
Date: 2007-10-21 02:10 pm (UTC)But then when you brought in algebra--well, that's just me, I think. It just seems so counter-intuitive to me to substitute letters for numbers, that my brain just fogs up at the idea of A+A=X-B or whatever...
Of course, that's where I got lost in RL math as well. Once we were past plain arithmetic. And of course, the brief and completely confusing interlude of "New Math" in the mid 1960s did not help at all--first time in my entire life I got an "F" in anything--that was seventh grade, I think...
(no subject)
Date: 2007-10-21 03:05 pm (UTC)And if it's any comfort to you, if Pythagoras came into a modern classroom and saw them teaching his theory, he'd be completely confused by the a's and b's and c's too.
(no subject)
Date: 2007-10-20 01:31 am (UTC)(which generated a lot of snickering, because at first we thought he said "flip the sucker")
(no subject)
Date: 2007-10-20 02:04 am (UTC)