Find square of 312 and 135 by duplex method
Answers
Answer:
620,944
Step-by-step explanation:
First, what is a duplex?
If a, b, c, d, e, f, etc. are digits, then
d(a) = a^2
d(ab) = 2ab
d(abc) = 2ac + b^2
d(abcd) = 2ad + 2bc
d(abcde) = 2ae + 2bd + c^2
etc.
In other words, from the outside working inwards, take the outermost digits, multiply them, double them, then take the next adjoining digits, multiply then, double them, then the next digits. etc.
If you are left with one single middle digit, square it.
Then add all these numbers.
If we want to square a 2 digit number: ab then the square is:
d(a) | d(ab) | d(b)
If we want to square a 3 digit number: abc then the square is:
d(a) | d(ab) | d(ac) | d(bc) | d( c )
If we want to square a 4 digit number: abcd then the square is:
d(a) | d(ab) | d(abc) | d(abcd) | d(bcd) | d(cd) | d(d)
Let’s start simple:
11^2 =
a=1
b=1
d(1) | d(11) | d(1) =
1 | 2 | 1 =
121
17^2 =
a=1
b=7
d(1) | d(17) | d(7) =
1 | 14 | 49 =
carry the 1 and the 4:
1+1 | 4+4 | 9 =
289
78^2 =
a=7
b=8
d(7) | d(78) | d(8)
49 | 2*56 | 64 =
49 | 112 | 64 =
carry 11 into the hundreds and 6 into the tens:
60 | 8 | 4 =
6084
alternatively:
78^2 = (80-2)^2
a=8
b=-2
d(8) | d(8, -2) | d(-2)
64 | 28-2 | 4 =
64 | -32 | 4 =
you want to end up with positive digits, so I turn 32 into -40+8 and carry the -40:
64 | -40 +8 | 4 =
carry the -40 into the thousands:
60 | 8 | 4 =
6084
Let’s square 345:
345^2 =
d(3) | d(34) | d(345) | d(45) | d(5) =
9 | 2 * 12 | 2 * 3 * 5 + 4^2 | 2 * 4 * 5 | 5^2 =
9 | 24 | 30 + 16 | 40 | 25 =
9 | 24 | 46 | 40 | 25 = (now handle the carries)
11 | 8 | 10 | 0 | 25 =
11 | 9 | 0 | 2 | 5 = (no carries left, we are done)
119025
A four digit number:
1234^2 =
d(1) | d(12) | d(123) | d(1234) | d(234)| d(34) | d(4) =
1 | 4 | 10 | 20 | 25 | 24 | 16 =
1522756
We can also do this two digits at a time:
12 34^2:
a=12
b=34
12^2 | 2 * 12 * 34 | 34^2 =
144 | 816 | 1156 =
keep in mind that when working with 2 digits at a time, we work with hundreds between two ‘|’'s. So in 1156, we carry 11 and not 115:
144 + 8 | 16 + 11 | 56 =
152 | 27 | 56
1522756
Or with negative numbers:
788^2:
for 788 use 800 - 12, so:
a = 8 and b = -12:
d(8) | d(8 , -12) | d(-12)
64 | 2*-12*8 | 144 =
64 | -192 | 144 =
192 = 200-8, so -192 = -200+8.
64 | -200 + 8| 144 =
Do the carry negatively:
62 | 08 | 144 =
62 | 09 | 44 =
620,944
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