Last updated 6/2/97 - 5/8/2000.
Square root is a form of division where the divisor equals the quotient but the divsor and quotient are otherwise unknown. Adding a few steps to the standard division process reveals both the divisor and the quotient, one digit at a time.
3
Example: 32=9 _ ,---
\/ 9
1) Find the decimal point. (If not shown, mark it to the
right of the number. Even if the answer is a whole
number, the decimal point is needed for step 2.)2) Pair the digits. Work outward from the decimal point in both directions. There may be a single digit left at the beginning of the number. If a digit is left over at the end of the number, add a zero after it.
3) Mark a placeholder to the left of the problem but on the next line. An underline works well. This will hold the first estimated digit of the divisor.
4) Estimate the square root of the first group (What number times itself will fit into the first group of digits in the dividend?) Write that digit in both the placeholder and in the quotient above the first group in the dividend. Hint: It helps to have memorized the perfect squares of the one-digit numbers (see the table under Hints, below).
5) Multiply the new digit of the quotient by the complete divisor. Place the product below the first group in the divisor or the remainder from the previous division cycle.
6) Subtract the product from the digits above it.
7) Bring down the next group from the dividend.
8) Add the placeholder to the full value of the divisor. This has the effect of doubling the the placeholder and also takes care of carries. (Compare with the way the middle term doubles in producing a perfect-square trinomial when using FOIL.)
9) Put a new placeholder at the end of the new divisor.
10) Estimate the next digit of the divisor and write it in both the placeholder and in the quotient above the next group in the dividend. Allow for the fact that adding or changing the estimate changes both the divisor and the quotient which literally multiplies the effect of the change. If the estimate is large (7, 8 or 9), consider reducing it by at least one before testing it.
11) Repeat from step 5 until the quotient has enough
digits.
Hints:
When estimating the rounding of the last digit for both square root and long division, ask whether the remainder is clearly more or less than half the divisor and round accordingly. This practice can save a lot of time since the divisors grow longer from cycle to cycle. In the relatively rare cases where it is difficult to tell which way to round, continue the work one more decimal place before rounding.
If a remainder is zero, just put a zero into the quotient and another into the placeholder, add on another placeholder and bring down the next pair of digits.
If a remainder is greater than the corresponding divisor, check to make sure that the digit of the quotient and the placeholder could not be one or even two greater in value (they must match). It is barely possible that they could be greater since both the divisor and the quotient vary together, therefore the remainder varies faster than the divisor.
If the decimal point in the
dividend
is moved an even number of places to make an otherwise-identical problem, the decimal point in the quotient will change by half that many places. There will be no other changes either to the answer or the work. However, if the decimal point is moved an
odd number of places, there will be no resemblance between the two problems.
Estimating first and last digits of square root
Digit Square Answer Ends In
----- ------ --------------
02 = 0 -- Itself If a dividend ends in 2, 3, 7, 8
.-> 12 = 1 ---------------------. or an odd number of zeros, the
| 22 = 4 ----------------. | square root cannot come out
| 32 = 9 -----------. | | exact in the calculation (cannot
| 42 = 16 ------.4 |3 |2 |1 be a rational number) since
| 52 = 25 -- 5 |or |or |or |or there is no way to multiply any
| 62 = 36 ------'6 |7 |8 |9 number times itself to get a
| 72 = 49 -----------' | | product which ends in one of
| 82 = 64 ----------------' | these digits. The answer will
| 92 = 81 ---------------------' never end and never repeat
`- 102 = 100 -- 0 (irrational number).
Applications:Square root is used anywhere distance on a diagonal is calculated from lengths of sides or coordinates, where area ratios are used, for many curved shapes and for any calculation involving a rate of change.
Examples: Building construction, scale models and cartography (map-making), quilting, chemistry (energy and equilibrium calculations), physics, electricity and electronics, heat and power, automotive, aircraft, boat and engine design, architecture, surveying, landscaping, navigation and fire control, profit projections.
Why learn to do manual square root? Many, if not most, of the environments where one uses square root on the job are hostile to calculators (heat, dirt and moisture). For example: roof framing, surveying, landscaping, small-boat navigation, water-plant valve stations, artillery fire-control.
It is a fundamental skill which can often be done faster than it
takes to locate a calculator.
Examples:
3 3 3
_ ,--- _ ,--- _ ,--- _ ,--- _ ,---
\/ 9 \/ 9. 3 \/ 9. 3 \/ 9. 3 \/ 9.
- - - 9 - -9
---
0
.....................................................................................
4 4 4
_ ,---- _ ,---- _ ,---- _ ,---- _ ,----
\/ 16 \/ 16. 4 \/ 16. 4 \/ 16. 4 \/ 16.
- - - 16 - -16
----
0
.....................................................................................
1 1 1
_ ,----- _ ,------ _ ,------ _ ,------ _ ,------
\/ 169 \/ 1'69. 1 \/ 1'69. 1 \/ 1'69. 1 \/ 1'69.
- - - 1 - -1
---
0
1 1 1 3 1 3 1 3
_ ,------ _ ,------ _ ,------ _ ,------ _ ,------
1 \/ 1'69. 1 \/ 1'69. 1 \/ 1'69. 1 \/ 1'69. 1 \/ 1'69.
- -1 +1 -1 +1 -1 +1 -1 +1 -1
------ --- ------ --- ------ --- ------ --- ------
69 2 ) 69 23 ) 69 23 ) 69 23 ) 69
- - - 69 - -69
----
0
.....................................................................................
6 6 6
_ ,------- _ ,------- _ ,------- _ ,------- _ ,-------
\/ 4,096 \/ 40'96. 6 \/ 40'96. 6 \/ 40'96. 6 \/ 40'96.
- - - 36 - -36
----
4
6 6 6 4 6 4 6 4
_ ,------- _ ,------- _ ,------- _ ,------- _ ,-------
6 \/ 40'96. 6 \/ 40'96. 6 \/ 40'96. 6 \/ 40'96. 6 \/ 40'96.
- -36 +6 -36 +6 -36 +6 -36 +6 -36
------- --- ------- --- ------- --- ------- --- --------
4 96 12 ) 4 96 124 ) 4 96 124 ) 4 96 124 ) 4 96
- - - 4 96 - -4 96
------
0
2 2
_ ,-------- _ ,--------- _ ,--------- _ ,---------
\/ 65,536 \/ 6'55'36. 2 \/ 6'55'36. 2 \/ 6'55'36.
- - - 4
2 2 2 2 5
_ ,--------- _ ,--------- _ ,--------- _ ,---------
2 \/ 6'55'36. 2 \/ 6'55'36. 2 \/ 6'55'36. 2 \/ 6'55'36.
- -4 - -4 +2 -4 +2 -4
--- ------- --- ------- --- -------
2 2 55 4 ) 2 55 45 ) 2 55
- -
2 5 2 5 2 5 2 5
_ ,--------- _ ,--------- _ ,--------- _ ,---------
2 \/ 6'55'36. 2 \/ 6'55'36. 2 \/ 6'55'36. 2 \/ 6'55'36.
+2 -4 +2 -4 +2 -4 +2 -4
--- ------- --- -------- --- -------- --- --------
45 ) 2 55 45 ) 2 55 45 ) 2 55 45 ) 2 55
- 2 25 - -2 25 - -2 25 +5 -2 25
------ --------- ---- ---------
30 30 36 50 ) 30 36
-
2 5 6 2 5 6 2 5 6
_ ,--------- _ ,--------- _ ,---------
2 \/ 6'55'36. 2 \/ 6'55'36. 2 \/ 6'55'36.
+2 -4 +2 -4 +2 -4
--- -------- --- -------- --- --------
45 ) 2 55 45 ) 2 55 45 ) 2 55
+5 -2 25 +5 -2 25 +5 -2 25
---- --------- ---- --------- ---- ---------
506 ) 30 36 506 ) 30 36 506 ) 30 36
- - 30 36 - -30 36
-------
0
.....................................................................................
3
_ ,---------------- _ ,------------------ _ ,------------------
\/ 90,000,000,000 \/ 9'00'00'00'00'00. 3 \/ 9'00'00'00'00'00.
- -
3 3 3 0 0, 0 0 0
_ ,------------------ _ ,------------------ _ ,------------------
3 \/ 9'00'00'00'00'00 3 \/ 9'00'00'00'00'00. 3 \/ 9'00'00'00'00'00.
_ 9 - -9 - -9
--- ---
0 0
. . 0 0 0 0
_ ,------------- _ ,----------------- _ ,-----------------
\/ .0000000004 \/ .00'00'00'00'04 \/ .00'00'00'00'04
- -
. 0 0 0 0 2 . 0 0 0 0 2 . 0 0 0 0 2
_ ,----------------- _ ,----------------- _ ,-----------------
\/ .00'00'00'00'04 \/ .00'00'00'00'04 \/ .00'00'00'00'04
2 2 4 2 -4
- - - ---
0
.....................................................................................
. 6. 6. 6.
_ ,------ _ ,------- _ ,------- _ ,------- _ ,-------
\/ 37.1 \/ 37.10 6 \/ 37.10 6 \/ 37.10 6 \/ 37.10
- - - 36 - -36
----
1
6. 6. 6. 0 6. 0
_ ,------- _ ,------- _ ,------- _ ,----------
6 \/ 37.10 6 \/ 37.10 6 \/ 37.10 6 \/ 37.10'00
- -36 +6 -36 +6 -36 +6 -36
------- ---- ------- ---- ------- ---- ----------
1 10 12 ) 1 10 120 ) 1 10 120 ) 1 10 00
- - -
6. 0 9 6. 0 9 6. 0 9 When next digit
_ ,---------- _ ,---------- _ ,---------- of the quotient
6 \/ 37.10'00 6 \/ 37.10'00 6 \/ 36.10'00 (and the divisor)
+6 -36 +6 -36 +6 -36 is zero, as above,
--- ----------- --- ----------- --- ------------ just bring down
1209 ) 1 10 00 1209 ) 1 10 00 1209 ) 1 10 00 the next group
- - 1 08 81 - -1 08 81 in the dividend
--------- and add another
1 19 placeholder.
The answer, rounded to the nearest tenth, is closer to 6.09 than to 6.1
since 119 is clearly less than half of 1209.
. . 1 . 1
_ ,------ _ ,-------- _ ,-------- _ ,--------
\/ .036 \/ .03'60 1 \/ .03'60 1 \/ .03'60
- - 1
. 1 . 1 . 1 . 1 8
_ ,-------- _ ,-------- _ ,-------- _ ,--------
1 \/ .03'60 1 \/ .03'60 1 \/ .03'60 1 \/ .03'60
- -1 - -1 +1 -1 +1 -1
--- ------ --- ------- --- -------
2 2 60 2 ) 2 60 28 ) 2 60
- -
. 1 8 . 1 8 . 1 8 . 1 8
_ ,-------- _ ,-------- _ ,----------- _ ,-----------
1 \/ .03'60 1 \/ .03'60 1 \/ .03'60'00 1 \/ .03'60'00
+1 -1 +1 -1 +1 -1 +1 -1
--- ------- --- -------- --- -------- --- --------
28 ) 2 60 28 ) 2 60 28 ) 2 60 28 ) 2 60
- 2 24 - -2 24 - -2 24 +8 -2 24
------ --------- ---- ---------
36 36 00 36 ) 36 00
-
. 1 8 9 . 1 8 9 . 1 8 9 The remainder
_ ,----------- _ ,----------- _ ,----------- is greater than
1 \/ .03'60'00 1 \/ .03'60'00 1 \/ .03'60'00 the divisor in
+1 -1 +1 -1 +1 -1 the step above.
--- -------- --- -------- --- -------- This is just
28 ) 2 60 28 ) 2 60 28 ) 2 60 possible since
+8 -2 24 +8 -2 24 +8 -2 24 the divisor
---- --------- ---- --------- ---- --------- and quotient
369 ) 36 00 369 ) 36 00 369 ) 36 00 vary together.
- 33 21 -33 21
-------
2 79
The answer, rounded to the nearest thousandth, is closer to .190 than to .189
since 279 is clearly more than half of 369.
Last updated 6/2/97 - 5/8/2000.
