Irrational Numbers on a Number Line

When placing irrational numbers on a number line, note that your placement will not be exact, but a very close estimation. For instance, when placing √15 (which is 3.87), it is best to place the dot on the number line at a place in between 3 and 4 (closer to 4), and then write √15 above it. Writing 3.87 would not be entirely accurate, and you can’t write out every decimal number because it ever ends. This is why √15 is the best thing to write.

Irrational Numbers on a Number Line

Irrational Numbers on a Number Line

I’m going to write some irrational numbers up here on the board, and we’re going to take a look at these rational numbers to get an idea of the value of each one of them. Then, we’ll place them in their corresponding places on the number line. These numbers right here are what we call irrational numbers. They’re going to have really long decimals. These decimals aren’t going to end and they’re not going to be repeating. Sometimes we are taking the square root of a number like four and we get a clean number like two.

That’s a rational number. These numbers are irrational. We need to find the square root of all these numbers. You may want to have a calculator with you, but I’ve already done the calculations. I’ll save you some time. The square root of 15 is 3 .87. This number actually goes on and on and on and on. I just rounded this to the hundreds place. After the decimal we have the tens place and then the hundreds place. Going to the hundreds place gives us a pretty good idea of the value of this number. We go to after three and we go almost to four because it’s at 3.87, or you could say almost 3.9. It’s going to be about right there. I’m just going to write √15 above it. We find the square root of three and round into the hundreds place giving us 1.73. The number goes on and on and on.

Now, we’re looking at one and two, somewhere in between there, and we could say 1.73 is really close to 1.75, which is between 1.5 and 2. We’ll go about right there. This number line isn’t really big, so we don’t have to be very exact. This dot is going to take up a big area. We’ll write √3. Now, for √58 we get 7.62 and we round to the hundreds place with many many more numbers after it. 7.62 is a little bit after 7.5, so we’ll go about right there and write √58. Then, we come to the √90. √90 is about 9.49. We’re looking at pretty much right in between 9 and 10. That’s where we put √90. Notice here that we have all these dots and we’re writing the number they represent above them. We’re doing that instead of just writing these numbers above, because these numbers aren’t exact. By putting the number with the square root sign over it, this is the most exact number you can get.

As soon as you find the square root of these numbers, you’re no longer exact, because the decimal goes on forever and you can’t write all those numbers. It’s more exact to keep the number in whole form. You’re more exact that way. Those are where all the numbers lie on the number line.

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