Comparing decimal place values
We have the number 7,346,521.032. And what I want to think about is if I look at the same digit in two different places, in particular, I'm going to look at the digit 3 here and the digit 3 here, how much more value does this left 3 represent than this right 3? In order to think about that, we have to think about place value. So let's write down all the place values. So this right over here, this is the ones place. Now, we could move to the right. And as we move to the right in place values, each place represents 1/10 of the place before it. Or you could divide by 10 as we're moving to the right. So this is the ones place. This is divide by 10. This is the 1/10 place, or the tenths place. Divide by 10 again, this is the hundredths place. Divide by 10 again, this is the thousandths place. And that "s" I'm just saying to be plural-- hundredths, thousandths. Now, if we go to the left, now each place represents a factor of 10 more. So if this is ones, multiply by 10, this is the tens place. This is the hundreds place. This is the thousands place. This is the ten thousands place. I'm going to have to write a little bit smaller. This is the hundred thousands place. And then the 7 is in the millions place. So what does this number, what does this 3 represent? Well, it's in the hundred thousands place. It literally represents 3 hundred thousands, or you could say 300,000, 3 followed by five zeroes. Now, what does this 3 represent? It's in the hundredths place. It literally represents 3 hundredths. It represents 3 times 1/100, which is the same thing as 3, which is equal to 3 over-- let me do the 3 in that purple color. Which is the same thing as 3/100, which is the same thing as 0.03. These are all equivalent statements. Now let's try to answer our original question. How much larger is this 3 than that 3 there? Well, one way to think about it is how much would you have to multiply this 3 by to get to this 3 over here? Well, one way to think about is to look directly at place value. So we got to multiply by 10. Every time we multiply by 10, that's equivalent to thinking about shifting it to one place to the left. So we would have to multiply by 10 one, two, three, four, five, six, seven times. So multiplying by 10 seven times. Let me write this down. So this multiplied by 10 seven times should be equal to this. Let me rewrite this. 300,000 should be equal to 3/100-- let me write it the same way. 3/100 multiplied by 10 seven times, so times 10 times 10 times 10 times 10 times 10-- let's see, that's five times-- times 10 times 10. Now, multiplying by 10 seven times is the same thing as multiplying by 1 followed by seven zeroes. Every time you multiply by 10, you're going to get another zero here. So this is the same thing as 3/100 times 1 followed by one, two, three, four, five, six, seven zeroes. So this is literally 3/100 times 10 million. So let's see if this actually is the case. Does this actually equal 300,000? Well, if you divide 10 million by 100, dividing 10 million by 100, or I guess you'd say in the numerator, you have 10 million and the denominator you have 100, if you were to just multiply it like this, if you view this as 3 over 100 times 10 million over 1. Well, you divide the numerator by 100, you're going to get rid of two of these zeroes. Divide the denominator by 100, you're going to get rid of this 100 here. And so you're going to be left with 3 times-- now we got to be careful with the commas here, because since I removed two zeroes, the commas are going to be different. It's going to be 3 times-- we put our commas in the right place, so just like that. So this simplified to 3 times 100,000, which is indeed 300,000. So it did work out. Shifting the 3 one, two, three, four, five, six, seven decimal places makes that 3 worth 1 followed by seven zeroes more, or it essentially makes that 3 worth 10 million more. So this 3 represents 10 million times the value of this 3. Let me write down the numbers. So this 3 is 10 million times the value of that 3.