Question

Find two rational and two irrational numbers between root 2 and root 3
.
.
.
I want step by step explanation​

Answer 1

Step-by-step explanation:

• Write down the decimal representations of both square roots to as many digits as you want to write. You can get these values from a calculator if you have not memorized them as I did 50 years ago:
• √2 ≈ 1.4142136…
• √3 ≈ 1.7320508… (the dots remind us they keep going on)
• Play it safe:
• √2 ≈ 1.414213 7 (increase smaller number’s last digit by one)
• √3 ≈ 1.732050 7 (reduce larger number’s last digit by one)
• Any decimal number between these two values
• is rational and is between √2 and √3
• Did you notice that you already have two rational numbers between √2 and √3?
• 1.4142137 is definitely greater than √2
• 1.7320507 is definitely less than √3.
• You could do the same thing with a different number of digits:
• √2 ≈ 1.414213 → 1.414214
• √3 ≈ 1.732050 → 1.732049
• √2 ≈ 1.41421 → 1.41422
• √3 ≈ 1.73205 → 1.73204
• √2 ≈ 1.4142 → 1.4143
• √3 ≈ 1.7320 → 1.7319
• Another method:
• Look for the position at which the two numbers are at least 2 apart.
• 1.4 and 1.7 are three of the smallest units apart.
• Any decimal that starts with a decimal value BETWEEN these two values will fit your bill:
• 1.5 anything … such as 1.531415926
• 1.6 anything … such as 1.61620508
• The average method:
• Add your two values, write the answer as a decimal:
• √2 + √3 ≈ 1.414 + 1.732 = 3.146
• Divide that answer by two: 3.146 ÷ 2 = 1.573
• The average of two numbers will always be between those two numbers.
• A number written as a terminating decimal (like 1.573 or 1.573132185) will always be rational because, if you read them out loud, you are reading them as fractions.