تقسیم عددی بر مضرب 10
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Multiples of 10
In this lesson, we will talk about ways to multiply and divide by 10 quickly. And also to multiply and divide by powers of 10. Now for some people everything in this video is going to be review. You don’t have to watch the video if that’s the case. But if you’re at all rusty with these skills, these are really important skills to know for the test.
The first idea is that when a multiple of 10 greater than 1 is written in standard form, the number of 0safter the 1 is equal to the factors of 10 in this multiple. So for example. 10,000 that has four 0s. So that’s 10 to the 4th or 4 factors of 10. 1 million has six 0s, so that’s 10 to the 6th or 6 factors of 10.
For powers of 10 less than 1, it get’s a little tricky here. The number of factors of 10 in the denominator equals the number of places to the right of the decimal point. So, for these we’re not longer counting number of 0s, we’re counting places to the right of the decimal point. So for example 0.01, that is 100th.
That has two decimal places to the right of the decimal point. So that’s 10 to the negative 2, or 1 over 10 times 10. 0.0001, that is 1 10,000th, that has four decimal places, so that’s 10 to the negative 4th, or 1 over 1 factors of 10. Of course in any number each place value is ten times more than the one to the right and ten times less than the one to the left.
This is a huge idea, this is a very important idea. It actually underlies everything in this video. This leads to a very quick trick when multiplying or dividing a number by some power of 10. First of all, when we multiply any number by 10, just plain, ordinary, regular 10, we multiply, we move the decimal point one place to the right.
So, 24 times 10. That decimal place right now is next to the 4. It moves one place to the right, so we have to stick in a place holding 0. One way to think about this, at least for numbers bigger than 1, is when we multiply by 10 it’s just like adding a 0 on the end. 2.53 times 10 to the, times 10.
Or for this we can’t just use the stick on a 0 anymore, we have to remember multiplying by 10 means move the decimal place one place to the right. So it starts out between the 2 and the 5, it winds up between the 5 and the 3, so we get 25.3. 6400 x 10, we move the decimal point 1 place to the right, we get a place holding 0, again it’s like adding 0 to the end.
And 0.00045, that would be 4,500 thousandths, that has three place holding 0s. When I multiply by 10, the decimal point moves one place to the right. And so I wind up with two place holding 0s. When we divide any number by 10 or multiply by 0.1, we move the decimal point one place to the left.
So now 24 divided by 10, the decimal point next to 4 moves one place to the left and it winds up between the 2 and the 4, 2.4. 0.00, 0.02 divided by 10, the decimal point moves one place to the left I wind up with two place holding 0s 0.002. 2,000ths. 39.85 times 0.1.
The decimal points moves one place to the left. It starts out between the 9 and the 8. It winds up between the 3 and the 9. And then we get 3.985. Finally, 0.00072.
That is 7,200 hundred thousandths. That has three placeholding 0s, when I multiply by 0.1 we move the decimal point one place to the left, we wind up with four placeholding 0s, 0.000072. When we multiply any number by any positive power of 10, we move the decimal point a number of spaces to the right equal to the power, that is, equal to the number of factors of 10.
So I multiply 350 times 100, 100 of course is two factors of 10, that’s 10 to the 2. That means the decimal point moves two places to the right which has the effect of adding on two more 0s at the end. Again that’s a trick you can use if the number is bigger then 1. 0.01728 times a 1,000, of course 1,000 is 10 to the 3rd.
There are three factors of 10. So multiplying by 1,000 will move the decimal place three places to the right. It will wind up between the 7 and the 2, 17.28. 8.3 times 10 to the 6, this is of course a scientific notation. We’re gonna move the decimal point six places to the right. The first place, it will be next to the 3, and then the next five places, we’ll need five place holding 0s.
And we’ll get 8,300,000. When we divide any number by any positive power of 10, or multiply by a negative power of 10, we move the decimal point a number of spaces to the left equal to the absolute value of the power. That is, equal to the number of factors of 10.
So if I divide by 100, again 100 is 10 to the 2, so there are 2 factors of 10. When I divide by 100, that moves the decimal point two places to the left. Starts out next to the 5, it winds up between the 2 and the 3, 12.35. 0.064 times 10 to the negative 2, I move the decimal point two places to the left, I start out with one place holding 0, and I’ll wind up with three place holding 0’s, 0.00064.
37.5 divided by 10,000. Well now I’m gonna have to move the decimal point four places to the left. Because there are four factors of 10 there. So the first two places it winds up next to the 3, and then I have to move it two more places, which means I’ll need two place holding 0s at the right of the decimal point, 0.00375.
64,000 times 0.0001. That is times 1 10,000th. That has four factors of 10 also because that is four decimal places. This means that the decimal point has to move four places to the left. It starts out next to that last 0 and winds up between the 6 and the 4. 5.4 times 10 to the negative 5th.
This again is scientific notation, it starts out between the 5 and the 4. It moves five places to the left, so we wind up with four placeholding 0s. 20.25 divided by 1 million divided by 10 to the 6th. This will move six places to the left. So the first two places get it in front of that, that first two.
The then it has to move four more places beyond that. And so we get four place holding 0s to the right of the decimal point. Notice that dividing by a negative power of 10, a number smaller than 1, has the same effect of making the product bigger. It is the equivalent to multiplying by a positive power of 10. So if I divide by 0.01, divide by a thous-, by a 100th, that’s the same as multiplying by 100.
And then I just follow the ordinary rule. If I divide by 10 to the negative 4th, that’s the same as multiplying by 10 to the positive 4th. And I follow the ordinary rule. One common mistake in these problems is, people sometimes get confused on which way to move the decimal point.
Always think about whether this multiplication or division will make the number bigger or smaller. Make sure that your answer is bigger or smaller as it should be. One thing that can be very helpful for building intuition on this, if this is something unfamiliar to you, just sit down with a calculator. Type in a number, and then multiply it by some power of 10, or divide it by some power of 10, and see if you can predict where the decimal place will wind up before you hit Enter.
Very good to practice this with a calculator to help you build intuition. And eventually your predictions will get so good that you wouldn’t need to do it on the calculator at all. In this video we learned how to multiply or divide numbers by powers of 10 simply by sliding the decimal point one way or the other.
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