Question

........ is a rational number between root 2 and root 3 . step by step
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Answer 1

Answer:

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.