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Mathematics Magazine for Grades 1-12 |
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1/2004 |
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They are never alone
that are accompanies by noble thoughts. Grade 7
Theory:Calculate Square Root Manually To
find a square root by the "longhand" method, we proceed as follows. I
intersperse numbered steps with an example. We will find the square root of 113
to three decimal places. 1.Draw a square root symbol, or radical, with the number whose root you are seeking underneath. Start with the decimal point and mark off digits in both directions in groups of two. Put a decimal point above the radical, and directly above the other decimal point. . /-------------
\/ 1 13.00 00 00 2. Start with the first group of 1 or 2 digits. Find the largest square of a single-digit integer less than it. Write the single digit above the radical, and its square under the first group. Draw a line under that square, and subtract it from the first group. 1 . /------------- \/ 1 13.00 00 00 1 ----
0 3. Bring down the next group below the last line drawn. This forms the current remainder. Draw a vertical line to the left of the resulting number, and to the left of that line put twenty times the number above the radical, a plus sign, a blank space (to be filled in during step 4), an equals sign, and some blank space for the answer. 1 . /------------- \/ 1 13.00 00 00 1 ----
20 +_= ?? | 0 13 4. Pick the biggest digit D that would fit into the underscore place, and give a number such that D times it is less than the current remainder. (If you guess too large a D, the remainder will be negative. If you guess too small a D, the remainder will be greater than the number to the left of the vertical line.) Put it above the radical above the last group of digits brought down, and put it in the blank space you left in step 3. Compute the number given by the expression, and put it after the equals sign. Multiply D times that number, and put that below the current remainder, draw a horizontal line below that, and subtract, to give a new current remainder. 1 0. /------------- \/ 1 13.00 00 00 1 ---- 20 + 0 = 20 | 0 13 0 -----
13 5.
If the current answer, above the radical, has the desired accuracy, stop.
Otherwise, go back to step 3. Step 3: 1 0. /------------- \/ 1 13.00 00 00 1 ---- 20 + 0 = 20 | 0 13 0 -----
200 +_= ??? | 13 00 Step 4: 1 0. 6 /------------- \/ 1 13.00 00 00 1 ---- 20 + 0 = 20 | 0 13 0 ----- 200 + 6 = 206 | 13 00 12 36 --------- 64 Step 3: 1 0. 6 /------------- \/ 1 13.00 00 00 1 ---- 20 + 0 = 20 | 0 13 0 ----- 200 + 6 = 206 | 13 00 12 36 --------- 2120 +_= ???? |
64 00 Step 4: 1 0. 6 3 /------------- \/ 1 13.00 00 00 1 ---- 20 + 0 = 20 | 0 13 0 ----- 200 + 6 = 206 | 13 00 12 36 -------- 2120 + 3 = 2123 | 64 00 63 69 --------
31 Step 3: 1 0. 6 3 /------------- \/ 1 13.00 00 00 1 ---- 20 + 0 = 20 | 0 13 0 ----- 200 + 6 = 206 | 13 00 12 36 -------- 2120 + 3 = 2123 | 64 00 63 69 -------- 21260 + _ = ????? |
31 00 Step 4: 1 0. 6 3 0 /------------- \/ 1 13.00 00 00 1 ---- 20 + 0= 20 | 0 13 0 ----- 200 + 6 = 206 | 13 00 12 36 --------- 2120 + 3 = 2123 | 64 00 63 69 -------- 21260 + 3 = 21263 | 31 00 0 -----
31 00
Step 5: Stop. Thus
the square root of 113 to three decimal places is 10.630.Checking, 10.6302 =
112.9969, and 10.6312= 113.0182, so the answer is correct. The
underlying principle is (10
or
even better, (100
Here
is a more detailed explanation of what is going on: N is the integer made up of
first groups of digits, and a is the integer part of the square root of N
already calculated. N - a2 is the current integer remainder. When you
bring down the next two-digit group n, and append it to the integer remainder,
you are multiplying the integer remainder by 100 and adding n to give 100
The
20
The
old square root was a, and the new one is 10
Solutions from the Previous Issue:Solve:
Solution:
|-27
– 7 | = |-34 | = 34
Solution:
|7| =
7
Solution:
|49 – (-36)|= |49 + 36|= 85
Solution:
|-15| = 15 Proposed Exercises:
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