MATH 4
INTERMEDIATE ALGEBRA
SUMMER TERM  2001

REAL NUMBER SYSTEM

Naturals, Integers, Fractions, Decimals, Rationals, Irrationals, Reals, Formulas, Scientific Notation, Graphing

Square Roots - The square root of a number is that number which when multiplied by itself produces the given number.

• The square root of 36 is 6 since 6² = 36.

• The square root of 2 is approximately 1.414 since 1.414² = 1.999396.

Square Root Algorithm - There are several methods in use for finding the square root of a number. Following is a technique for taking the square root of an arbitrarily large number that used to be part of the math curriculum. The technique is quite similar to long division. Here's how it works for decimal square roots.

1. Starting at the decimal point, of the number under the square root symbol, separate the digits into groups of two, first to the left and then to the right of the decimal point.

• If there is an odd number of digits to the left of the decimal point, there will be on group containing a single digit.
• If there is an odd number of digits to the right of the decimal point, add a zero so that each group contains two digits.

2. The decimal point of the answer is directly above the decimal point of the original number.

3. Find the largest square that subtracts from the left-most pair and still yields a positive result. Write it under the first group. The square root of this largest square is the first digit of the square root of the whole number.

4. Write the square root of this largest square above the first group, as the first figure of the square root.

5. Subtract the square number from the first group. This is the remainder that will be used in the next step.

6. Annex the next pair of digits with the remainder to form a dividend.

7. Form a trial divisor by multiplying the square root developed so far by 20. Note that the least significant digit is a zero.

8. Divide the dividend (Step 6) by the trial divisor (Step 7). Add the result to the root already found. Also, add the result to the trial divisor to form the complete divisor.

9. Multiply the complete divisor by the new figure of the root.

10. Subtract this product (Step 9) from the dividend (Step 4). This is a new remainder that will be used in the next step.

• If the result is zero or positive and less than the divisor continue to Step 11.
• If the result is negative decrease the result in Step 8 by one, and repeat Steps 9 & 10.

11. Return to Step 6 and repeat the process until all groups have been used, or the desired number of decimal points has been obtained.

12. Check by squaring the root (multiplying the answer by itself) to obtain the original number.

Complete the following problems on a separate sheet of paper. Be sure to show all of the work necessary to complete the problems. (Calculators may be used to check the answer, but should not be used to solve the problem originally.)

Square Root

1. Find the square root of 676.

• Step 1 - Starting at the decimal point, of the number under the square root symbol, separate the digits into groups of two.
• Step 2 - Locate the decimal point of the answer.
• Step 3 - Find the largest square (4) that subtracts from the left-most pair (6) and still yields a positive result. Write it under the first group.
• Step 4 - Write the square root (2) of this largest square (4) above the first group, as the first figure of the square root.
• Step 5 - Subtract the square number from the first group. This is the remainder (2) that will be used in the next step.
• Step 6 - Annex the next pair of digits with the remainder to form a dividend (276).
• Step 7 - Multiply the square root developed so far (2) by 20 to form a trial divisor (40).
• Step 8 - Divide the dividend (276) by the trial divisor (40). Add the result to the root already found. Also, add the result to the trial divisor to form the complete divisor (46).
• Step 9 - Multiply the complete divisor (46) by the new figure of the root (6).
• Step 10 - Subtract this product (276) from the dividend (276).
• Step 11 - Show that 26 squared equals 676.

Example 1:

2.  Find the square root of 7 to the nearest hundredth.

• Step 1 - Add three groups of 2 zeros each after the decimal point for 7. This will allow us to find the square root out to the thousandths position, and then round to the nearest hundredth.
• Step 2 - Locate the decimal point of the answer.
• Step 3 - Find the largest square (4) that subtracts from the left-most pair (7) and still yields a positive result. Write it under the first group.
• Step 4 - Write the square root (2) of this largest square (4) above the first group, as the first figure of the square root.
• Step 5 - Subtract the square number from the first group. This is the remainder (3) that will be used in the next step.
• Step 6 - Annex the next pair of digits with the remainder to form a dividend (300).
• Step 7 - Multiply the square root developed so far (2) by 20 to form a trial divisor (40).
• Step 8 - Divide the dividend (300) by the trial divisor (40). Add the result (6) to the root already found. Also, add the result to the trial divisor to form the complete divisor (46).
• Step 9 - Multiply the complete divisor (46) by the new figure of the root (6).
• Step 10 - Subtract this product (276) from the dividend (300). This is the remainder (24) that will be used in the next step.
• Step 11 - Annex the next pair of digits with the remainder to form a dividend (2400).
• Step 12 - Multiply the square root developed so far (26) by 20 to form a trial divisor (520).
• Step 13 - Divide the dividend (2400) by the trial divisor (520). Add the result (4) to the root already found. Also, add the result to the trial divisor to form the complete divisor (524).
• Step 14 - Multiply the complete divisor (524) by the new figure of the root (4).
• Step 15 - Subtract this product (2096) from the dividend (2400). This is the remainder (304) that will be used in the next step.
• Step 16- Annex the next pair of digits with the remainder to form a dividend (30400).
• Step 17 - Multiply the square root developed so far (264) by 20 to form a trial divisor (5280).
• Step 18 - Divide the dividend (30400) by the trial divisor (5280). Add the result (5) to the root already found. Also, add the result to the trial divisor to form the complete divisor (5285).
• Step 19 - Multiply the complete divisor (5285) by the new figure of the root (5).
• Step 20 - Subtract this product (26425) from the dividend (30400). This is the remainder (3975).
• Step 21 - Round the answer up to 2.65 since the third decimal position is 5.
• Step 22 - Show that 2.65 squared equals 7.0225.

Example 2:

HOMEWORK PROBLEMS :

3. Find the square root of 5184.

4. Find the square root of 8649.

5. Find the square root of 0.1225

6. Find the square root of 1.2996

7. Find the square root of 0.024964

8. Find the square root of 103041.

9. Find the square root of 31.561924

10. Find the square root of 451584.

11. Find the square root of 0.8836 correct to the nearest hundredth.

12. Find the square root of 42 correct to the nearest hundredth.

13. Find the square root of 768.4 correct to the nearest hundredth.

14. Find the square root of 9876 correct to the nearest hundredth.

15. Find the square root of 184325 correct to the nearest hundredth.

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