1. Write three irrational numbers between 5.6 and 5.7 (Show your work).
We begin by squaring 5.6 and 5.7
5.6² = 31.36
5.7² = 32.49
This means that if we multiply both sides by 100, we get
5.6²(100) = (5.6)(5.6)(10)(10) = (5.6)(10)(5.6)(10) = (56)(56) = 56² = 3136
5.7²(100) = (5.7)(5.7)(10)(10) = (5.7)(10)(5.7)(10) = (57)(57) = 57² = 3249
We know that the square root of any integer between two perfect squares is
irrational. There are no perfect squares between perfect squares 56²=3136
and 57²=3249. So we pick any three integers between 3136 and 3249 and their
square roots will be irrational.
So let's pick, say, 3163, 3193, and 3223 at random between them.
Taking square roots:
√3163 which is approximately 56.24055476...
√3193 which is approximately 56.50663678...
√3223 which is approximately 56.77147171...
Dividing under the square root by 100 amounts to dividing by 10, so
√31.63 which is approximately 5.624055476...
√31.93 which is approximately 5.650663678...
√32.23 which is approximately 5.677147171...
All three are irrational.
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Now, do the same exact thing with 3.8 and 3.9.
Edwin
