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.
Yes. because In mathematics, the irrational numbers are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers
Exponential functions are patterns that get continuously multiplied by some number. It's exponential growth when the base of our exponential is bigger than 1, which means those numbers get bigger. It's exponential decay when the base of our exponential is in between 1 and 0 and those numbers get smaller.
