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. ____________________________________
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 . ____________________________
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 . ____________________________________
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 . ____________________________
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.
