Comparing Decimal Numbers Math Antics Extras

Is comparing decimal numbers harder than comparing multi-digit whole numbers? Nope! In fact, it’s basically the same process. You line up the number places and then, starting with the biggest number places to the left, you look for digits that you can compare to tell which number has the greatest value. But it will help if you really understand how decimal places work, so be sure to watch our lesson about decimal place value if you haven’t already. Okay, let’s start by comparing these two decimal numbers: 3.015 and 2.49. Which has the greatest value? First we line up the number places, but instead of lining up the number place on the right hand side like we did with whole numbers, we need to line up the decimal points. That’s really important because the decimals you're comparing might have different number of decimal digits like these do. And if you just line up the right-most digits, the number places won't be aligned correctly. But when you align the decimal points, no matter how many digits are on either side of it, the number places are always lined up too. Next, just like with whole numbers, we compare the digits in the largest number place which is always the farthest to the left. In this case, that’s the ones place which doesn’t seem very large, but it’s larger than either of the other decimal number places to the right. The first number has 3 in the ones place and the second number has 2. Since 3 is greater than 2, the decimal 3.015 must be greater than the decimal 2.49. Because we could tell which number was bigger just from the ones place digits, we didn’t even need to look at the decimal digits. But what if you have to compare two decimal numbers like these? …0.5 and 0.34 If we line up the decimal points, you’ll notice that they each have zero in the ones place. Since those digits are the same, we can’t use them to tell which is bigger. Instead we need to shift to the next biggest number place; the tenths place. The first number has 5 in the tenths place (representing five-tenths), while the second number has 3 in the tenths place (representing three-tenths). 5 is greater than 3 which means that five-tenths is greater than three-tenths. Therefore, 0.5 must be greater than 0.34 Now if you watched the last video about comparing multi-digit whole numbers, you might remember me saying that the number with the most digits is automatically bigger. But that doesn’t apply to decimal digits. 0.34 has more digits than 0.5, but its value is smaller. Here’s an extreme example of that. Which of these decimals has the largest value? …0.1 or 0.099999 Well, all those nines sure make the second number seem big, but looks can be deceiving. Let’s line the decimal points up to see what’s really going on here. Starting with the biggest number place, both have zero there, so we can’t tell from those digits. Moving to the tenths place, the first number has 1 (representing one-tenth) while the second number has 0 (representing zero-tenths). 1 is definitely greater than 0 which means 0.1 must be greater than 0.099999, even with all those extra digits. And here’s the difference… With multi-digit whole numbers, if one of the numbers you’re comparing has more digits, those extra digits are always to the left which means they’re always in the bigger number places. That’s why they automatically make the number bigger. But when number has more decimal digits, those extra digits are always to the right, which means they’re in smaller number places. So no matter how many of them there are, they can only make very small difference. 0.9, 0.999, and 0.99799 are all small numbers… they’re all less than 1. But which has the greatest value and which has the smallest? Well, we can’t tell from the first two digits on the left because they’re the same for all three numbers. But when we get to the hundredths place, we do notice difference. The first number doesn’t have digit there while the others do. That means the first number is the smallest because we could put zero in the hundredths place without changing its value. But what about the others? They both have 9 in the hundredths place, so you can’t tell which is bigger by comparing those digits. You need to look at the next place to the right; the thousandths place. This number has 9 in that place while the last number has 7. Since 9 is greater than 7, we know that 0.999 must be greater than 0.99799 Again, notice that the extra digits at the end of 0.99799 didn’t make it greater than 0.999 because the thousandths place is actually larger than the ten-thousandths or hundred-thousandths place. Even though the number places sound like they are getting bigger as they go to the right “thousandths place”, “millionths place”, “billionths place”. They are actually getting smaller and smaller because if you divide something up into billion pieces, each piece is going to be very small! Okay, the way to get really good at comparing decimal numbers is to practice lot and check your work to make sure you are getting the right answers. And be sure to check out www.mathantics.com to see more topics and to learn about our paid membership for access to print-based practice sheets.