Lesson 2 · from §90 of the 1910 text
Square Root of Numbers
Square root of numbers
Since 1 = 1^2 and 81 = 9^2, a 1-digit or 2-digit square has one digit in its square root. Since 100 = 10^2 and 9801 = 99^2, a 3-digit or 4-digit square has two digits in its root; a 5- or 6-digit square has three.
So, beginning at the decimal point, point off periods of two digits each: 78'43'56, \;7'84'35, \;.98'01, \;.03'27'40 (an incomplete period on the right is completed by annexing a zero). The first digit of the root is the greatest integer whose square does not exceed the left-hand period: 8, 2, 9 and 1 in these four numbers. The number of digits in the root of a perfect square equals the number of periods.
The work rests on the same formula as for polynomials, (t + u)^2 = t^2 + 2tu + u^2, with t the tens and u the units. For 2809 = (50 + 3)^2 = 2500 + 300 + 9:
| Root: 53 | ||
|---|---|---|
| 28'09 | ||
| 5^2 = | 25 | |
| Trial divisor 2 \cdot 5 = 10 | 3\,09 | 30 \div 10 gives 3 |
| Complete divisor 103 | 3\,09 = 103 \cdot 3 |
With three digits, 17{,}424: the root begins with 1; the trial divisor 2 into 7 suggests 3, and 23 \cdot 3 = 69 leaves 524; the trial divisor 26 into 52 gives 2, and 262 \cdot 2 = 524. So \sqrt{17424} = 132.
Rule
- Beginning at the decimal point, point off periods of two digits each, to the left for an integer, to the right for a decimal, both ways for a mixed decimal.
- Find the greatest integer whose square is not more than the left-hand period; write it as the first digit of the root. Subtract its square and annex the next period.
- Double the root found so far for a trial divisor; divide it into the remainder (leaving off the right-hand digit) and write the integral part of the quotient as the next digit of the root.
- Annex that digit to the trial divisor to make the complete divisor; multiply by the digit, subtract, and annex the next period.
- Repeat until the desired number of digits is found. Point off one decimal place in the root for every decimal period in the number.
Check: square the root (adding any remainder); you should get the original number.
Too large a digit. For \sqrt{32301} the trial divisor 2 into 22 gives 11, which is impossible; 9 \cdot 29 = 261 and 8 \cdot 28 = 224 are both more than 223, so the second digit is 7 (7 \cdot 27 = 189).
A zero digit. In \sqrt{9'42.49} the trial divisor 6 does not go into 4, so the second digit is 0; bring down the next period: the trial divisor 60 gives 7, and the root is 30.7.
Roots that never end. Annexing periods of zeros to 2 and applying the rule gives 1.414\ldots without ever ending: 2 has no exact square root. We say 1.414 is the square root of 2 to three decimal places. A common fraction (or the fractional part of a mixed number) should be changed to a decimal before extracting the root, unless the root is seen to be exact.
Typing answers: type the positive root; for an inexact root give three decimal places, like
2.236. Answers within about 0.001 of the true value are accepted, whether you round the last digit or cut it off as the book does.
From the history of algebra
A method of extracting square roots not unlike ours was used by the Greek mathematician Theon of Alexandria about 350 A.D. In the Middle Ages square roots were found with fair accuracy from the approximation formulas \sqrt{a^2 + r} \approx a + \frac{r}{2a} \qquad\text{and}\qquad \sqrt{a^2 + r} \approx a + \frac{r}{2a + 1}. The true root lies between the two results. For \sqrt{65} = \sqrt{8^2 + 1} they give 8\tfrac{1}{16} and 8\tfrac{1}{17}. This method was known to the Arabs. Decimal fractions and the decimal point were not in common use until the eighteenth century, so the full method of the text is comparatively recent.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Extract the square root, correct to three decimal places, of:
6241
16129
223{,}729
2
5
7.135
.6279
.0451
.0035
\tfrac{11}{5}
1\tfrac67
\tfrac{2}{21}