Take some guesses and try them out.
Whole numbers:
Is the lower number 2 ?
If it is, then 2² is less than 3.
2² is 4. That's more than 3. So 2 is too big to be the lower number.
2 can be the upper number.
Is the lower number 1 ?
If it is, then 1² is less than 3.
1² is 1. 1 is less than 3.
So 1 can be the lower number.
√3 is between the two rational numbers 2 and 3.
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Numbers with 1 decimal place:
Can the lower number be 1.9 ?
If it can, then 1.9² is less than 3.
1.9² is 3.61. That's more than 3.
So 1.9 is too big to be the lower number.
Can the lower number be 1.8 ?
If it can, then 1.8² is less than 3.
1.8² is 3.24. That's more than 3.
So 1.8 is too big to be the lower number.
Can the lower number be 1.7 ?
If it can, then 1.7² is less than 3.
1.7² is 2.89. Yay ! That's less than 3.
So 1.7 can be the lower number.
√3 is between the rational numbers 1.7 and 1.8 .
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Numbers with 3 decimal places:
Can the lower number be 1.741 ?
If it can, then 1.741² is less than 3.
1.741² is 3.031... . That's more than 3.
So 1.741 is too big to be the lower number.
Can the lower number be 1.735 ?
If it can, then 1.735² is less than 3.
1.735² is 3.0102... . That's more than 3.
So 1.735 is too big to be the lower number.
Can the lower number be 1.734 ?
If it can, then 1.734² is less than 3.
1.734² is 3.0067... . That's more than 3.
So 1.734 is too big to be the lower number.
Can the lower number be 1.733 ?
If it can, then 1.733² is less than 3.
1.733² is 3.00328... . That's more than 3.
So 1.733 is too big to be the lower number.
Can the lower number be 1.732 ?
If it can, then 1.732² is less than 3.
1.732² is 2.9998... . Yay! That's less than 3.
√3 is between the rational numbers 1.732 and 1.733 .
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Of course, if you already KNOW the square root of 3,
or if you go to a book or a calculator and find out what
it is, or if you know how to figure out what it is and you
take a pencil and paper and you do that, then you can
just pick any old rational number that's bigger and any
old rational number that's smaller, and say "THERE !
Here are two rational numbers that √3 is between."
On my calculator, √3 starts out with
1.7320508 ...
So it must be between
the two rational numbers 1.7320507
and 1.7320509 .
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In the question you posted, they GAVE you some pairs
of rational numbers. You can test each pair and see
if it works.
If the √3 is between them, then
(the smaller one)² must be less than 3, and
(the bigger one)² must be more than 3.
Let's checkum out:
A). 1.2 and 1.3
(1.2)² = 1.44 . Could be ! Let's check the other one.
(1.3)² = 1.69 . Aw shucks.
The √3 is bigger than both of them, not between them.
B). 1.7 and 1.8
(1.7)² = 2.89 . Could be! Let's check the other one.
(1.8)² = 3.24. Yay ! The √3 is between these.
C). 2.6 and 2.7
(2.6)² = 6.76. Phooey! The √3 can't be 'between' these, because
it's even smaller than the smaller number.
D). 2.7 and 2.8
(2.7)² = 7.29. Phooey! The √3 can't be 'between' these, because
it's even smaller than the smaller number.
