Comparing decimals 2

Let's compare 0.1 to 0.070. So this 1 right over here, it is in the tenths place. So it literally represents 1 times 1/10, which is obviously the same thing as 1/10. Now, when we look at this number right over here, it has nothing in the tenths place. It has 7 in the hundredths place. So this is the hundredths place right over here. And then it also has nothing in the thousandths place. So this number can be rewritten as 7 times 1/100, or 7/100. And now we could compare these two numbers. And there's two ways you could think about it. You could try to turn 1/10 into hundredths. And the best way to do that, if you want the denominator to be increased by a factor of 10, you need to do the same thing to the numerator. So all I did is I multiplied the numerator and denominator by 10. Ten 100's is the exact same thing as 1/10. And here it becomes very clear, 10/100 is definitely larger than 700. Another way you could think about this is, look, if you were to increment by hundredths here, you would start at 7/100, 8/100, 9/100, and then you would get to 10/100. So then you would get to that number. So this number, multiple ways you could think about it, is definitely larger. So let me write this down. This is definitely larger, greater than. This is greater than that. The greater than symbol opens to the larger value. So here we have 0.093 and here we have 0.01. So let's just think about this a little bit. So this 9-- get a new color here. This 9 is not in the tenths, the hundredths. It's in the thousandths place. It's in the thousands place. And this 3 is in the-- I'm running out of colors again. This 3 is in the ten thousandths place. So the 3 is in the ten thousandths place. So you could literally view this as 9/1,000 plus 3/10,000. And if you just wanted to write it in terms of ten thousandths, you can multiply the 9 and 1,000 by 0. And so it becomes 90/10,000. And if you want to add them together, you could, of course, write this as 93/10,000. Ten thousandths. I always have trouble with that "-ths" at the end. Now, let's think about this number right over here, 0.01. Well, this 1 right over here is in the hundredths place. It's in the hundredths place. So it literally represents 1/100. So how can we compare 1/100 to 93/10,000? So the best way to think about it is, well, what's 1/100 in terms of ten thousandths? Well, let's just multiply both the numerator and the denominator here by 10 twice. Or you could say, let's multiply them both by 100. If you multiply by 10 once, you get to ten thousandths. It's the same thing as 1/100. Multiply by 10 again, you get 100/10,000 is the same thing as 1/100. And we know that, 100 times 100 is 10,000. So here it becomes very clear, 100/10,000, or 1/100, is definitely a larger than 93/10,000. So this quantity right over here is less than this quantity there. Less than symbol, the small end points to the smaller number, larger end to the larger number. In fact, that's true with the less than and greater than. So let's see, this one right over here, 0.6 versus 0.06. So here, I have a 6 in the tenths place. So it literally represents 6/10. And in the second, I have a 6 in the hundredths place. Well, 6/100 is definitely smaller than 6/10. A hundredth is a tenth of a tenth. So this one is pretty straightforward. This is going to be the larger value. 0.6 is greater than 0.06. Now, let's think about 0.3 versus 0.06. So this 3 literally represents 3/10 while this 6 right over here represents 6/100. And if you wanted to compare them directly, you could multiply 3/10 times-- well, both the numerator and the denominator by 10 so you're not changing its value. 10/10 is essentially 1, or it is 1. So this becomes 30/300. 3/10 is the same thing as 30/100. And 30/100 is a lot larger than 6/100. So this is greater than.