ریاضی هفتم. جذر و مجذور

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Finding square roots by guess & check method

One simple way to find a decimal approximation to, say √2 is to make an initial guess, square the guess, and depending how close you got, improve your guess. Since this method involves squaring the guess (multiplying the number times itself), it actually uses the definition of square root, and so can be very helpful in teaching the concept of square root.

Example: what is √20 ?

Children first learn to find the easy square roots that are whole numbers, but quickly the question arises as to what are the square roots of all these other numbers. You can start out by noting that (dealing here only with the positive roots) since √16 = 4 and √25 = 5, then √20 should be between 4 and 5 somewhere.

Then is the time to make a guess, for example 4.5. Square that, and see if the result is over or under 20, and improve your guess based on that. Repeat the process until you have the desired accuracy (amount of decimals). It's that simple and can be a nice experiment for children.

Example: Find √6 to 4 decimal places

Since 22 = 4 and 32 = 9, we know that √6 is between 2 and 3. Let's just make a guess of it being 2.5. Squaring that we get 2.52= 6.25. That's too high, so make the guess a little less. Let's try 2.4 next. To find approximation to four decimal places we need to do this till we have five decimal places, and then round the result.

Guess	Square of guess	High/low
2.4	5.76	Too low
2.45	6.0025	Too high but real close
2.449	5.997601	Too low
2.4495	6.00005025	Too high, so between 2.449 and 2.4495
2.4493	5.99907049	Too low
2.4494	5.99956036	Too low, so between 2.4494 and 2.4495
2.44945	5.9998053025	Too low, so between 2.44945 and 2.4495.

This is enough since we now know it would be rounded to 2.4495 (and not to 2.4494).

Finding square roots using an algorithm

There is also an algorithm that resembles the long division algorithm, and was taught in schools in days before calculators. See the example below to learn it. While learning this algorithm may not be necessary in today's world with calculators, working out some examples can be used as an exercise in basic operations for middle school students, and studying the logic behind it can be a good thinking exercise for high school students.

Example: Find √645 to one decimal place.

First group the numbers under the root in pairs from right to left, leaving either one or two digits on the left (6 in this case). For each pair of numbers you will get one digit in the square root. 

To start, find a number whose square is less than or equal to the first pair or first number, and write it above the square root line (2).

2
√6	.45

2 √6 .45 - 4 2 45	2 √6 .45 - 4 (4 _) 2 45	2 √6 .45 - 4 (45) 2 45
Square the 2, giving 4, write that underneath the 6, and subtract. Bring down the next pair of digits.	Then double the number above the square root symbol line (highlighted), and write it down in parenthesis with an empty line next to it as shown.	Next think what single digit number something could go on the empty line so that forty-something times somethingwould be less than or equal to 245. 45 x 5 = 225 46 x 6 = 276, so 5 works.
2 5 √6 .45 .00 - 4 (45) 2 45 - 2 25 20 00	2 5 √6 .45 .00 - 4 (45) 2 45 - 2 25 (50_) 20 00	2 5 . 3 √6 .45 .00 - 4 (45) 2 45 - 2 25 (503) 20 00
Write 5 on top of line.  Calculate 5 x 45, write that  below 245, subtract, bring down the next pair of digits (in this case the decimal digits 00).	Then double the number above the line (25), and write the doubled number (50) in parenthesis with an empty line next to it as indicated:	Think what single digit number somethingcould go on the empty line so that five hundred-something  times something would be less than or equal to 2000. 503 x 3 = 1509 504 x 4 = 2016, so 3 works.
2 5 . 3 √6 .45 .00 .00 - 4 (45) 2 45 - 2 25 (503) 20 00 - 15 09 4 91 00	2 5 . 3 √6 .45 .00 .00 - 4 (45) 2 45 - 2 25 (503) 20 00 - 15 09 (506_) 4 91 00	2 5 . 3 9 √6 .45 .00 .00 - 4 (45) 2 45 - 2 25 (503) 20 00 - 15 09 (506_) 4 91 00
Calculate 3 x 503, write that  below 2000, subtract, bring down the next digits.	Then double the 'number' 253 which is above the line (ignoring the decimal point), and write the doubled number 506 in parenthesis with an empty line next to it as indicated:	5068 x 8 = 40544 5069 x 9 = 45621, which is less than 49100, so 9 works.

Thus to one decimal place, √645 = 25.4

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