05 Comparison of Irrational Numbers
Comparing Irrational Numbers
Like comparing whole numbers or fractions or decimals, you can also compare irrational numbers. Approximating them as decimals is sometimes necessary to make comparing easier.
For example:
Out of the following numbers, which is the largest?
√17, 4, 3.03
4 and 3.03 are easy enough to compare since they are in decimal and whole number form. 4 is clearly greater than 3.03.
What we do not know is where √17 lies among the three. So, we need to find what whole number √17 is closest to. For this, we sqaure it, giving us √172 = 17. We then need to find which two sqaure numbers this liws between. 17 lies between 16 and 25. We can then write 16 and 25 as 42 and 52.
So, root of 17 lies between root of 16 and root of 25, which are 4 and 5. So √17 lies between 4 and 5.
Let’s write this step wise to understand it better.
(√17)2
16 < 17 < 25
√16 < √17 < √25
4 < √17 < 5
This tells us that √17 is bigger than both 4 and 3.03.
So this is how you comapre irrational numbers are order them.
Estimating Expressions
When You have expressions with whole number, farctions, decimals, we can easily solve them and get one number answers. This makes it easier to comapre the expressions. But say you have an expression with a root term. You cannot operate on the root term, so how do you decide its value? And how do you comapre it with other expressions? Let’s use an example to see how this is done. Which is bigger? √3 + 5 or √2 + 5?
In both cases, we have two different root terms, √3 and √2. To decide which is bigger, we can simply look at the root term since 5 is common in both. We know that √2 is smaller than √3, and when the same number (5) is added to both, √2 + 5 is smaller. Easy right?
Let’s look at one more example when the added term is not the same.
Are 4√2 and 2√4 the same? (Here 4 and 2 are not part of the root but multiplied -> 4 x √2 and 2 x √4)
For this, let’s see what √2 and √4 mean.
√2 lies between 1 and 2 since √22 lies between 12 and 22.
(√2)2
1 < 2 < 4
√1 < √2 < √4
1 < √2 < 2
If we multiply by 4, we get:
4 x 1 < 4 x √2 < 4 x 2
4 < 4√2 < 8
So 4 x √2 will be between 4 and 8.
Now for √4, which we know is equal to 2. So 2√4 is simply 4. So the two are not the same.
The magnitude of the whole term depends on the whole number being multiplied outside the root - it has more weight towards the overall product. So we see that using approximate of irrational roots, we can estimate the value of expressions as well.