Positive real number which when multiplied by itself gives 7
Square root of 7
The square root of 7 is the positive
real number that, when multiplied by itself, gives the
prime number7. It is more precisely called the principal square root of 7, to distinguish it from the negative number with the same property. This number appears in various geometric and number-theoretic contexts. It can be denoted in
surd form as:
which can be rounded up to 2.646 to within about 99.99% accuracy (about 1 part in 10000); that is, it differs from the correct value by about 1/4,000. The approximation 127/48 (≈ 2.645833...) is better: despite having a
denominator of only 48, it differs from the correct value by less than 1/12,000, or less than one part in 33,000.
More than a million decimal digits of the square root of seven have been published.
The extraction of decimal-fraction approximations to square roots by various methods has used the square root of 7 as an example or exercise in textbooks, for hundreds of years. Different numbers of digits after the decimal point are shown: 5 in 1773 and 1852, 3 in 1835, 6 in 1808, and 7 in 1797.
An extraction by
Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".
For a family of good rational approximations, the square root of 7 can be expressed as the
The successive partial evaluations of the continued fraction, which are called its
convergents, approach :
Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257…(sequence A041008 in the
OEIS) , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,…(sequence A041009 in the
Each convergent is a
best rational approximation of ; in other words, it is closer to than any rational with a smaller denominator. Approximate decimal equivalents improve linearly (number of digits proportional to convergent number) at a rate of less than one digit per step:
Every fourth convergent, starting with 8/3, expressed as x/y, satisfies the
When is approximated with the
Babylonian method, starting with x1 = 3 and using xn+1 = 1/2(xn + 7/xn), the nth approximant xn is equal to the 2nth convergent of the continued fraction:
All but the first of these satisfy the Pell's equation above.
The Babylonian method is equivalent to
Newton's method for root finding applied to the polynomial . The Newton's method update, is equal to when . The method therefore
converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).