A magnifying glass to locate the roots

Compare √2, √4 and √5. Change the decimal places and watch the range get narrower.

Explore the controls and explain what changed in your own words.

Who belongs to the real line?

Number Classification Justification
−3 Whole and rational −3 = −3/1
0,5 Rational 0,5 = 1/2
0,333… periodic Rational It is equal to 1/3
√2 Irrational Cannot be written as a quotient of integers

They are all real. The decimal writing of a rational is finite or possibly periodic. That of an irrational is infinite and non-periodic.

Getting closer is not making the same

To find √2, compare squares: 1,4² = 1,96 and 1,5² = 2,25. Thus, 1,4 < √2 < 1,5. Refining: 1,41² = 1,9881 and 1,42² = 2,0164; therefore, 1,41 < √2 < 1,42.

Writing √2 is exact. 1,414 is a rational approximation of an irrational number: √2 ≈ 1,414, not √2 = 1,414.

Between two numbers there are more numbers

Between 1,4 and 1,5 we can choose 1,45. Between 1,4 and 1,45, we can choose 1,425. On the real line there is no next number immediately after a given number: between two different reais, their average is another real.

Roots challenge

Are √9 and √3 both irrational? Between which integers is √3?

Check out the investigation

√9 = 3 is rational. √3 is irrational and lies between 1 and 2, since 1² < 3 < 2². Not every root is irrational.