# MATH MAGIC - PERFORMING THE     MULTIPLICATION OF LARGE     NUMBERS EASILY AND QUICKLY

Mental Math before Math Magic

That knowledge is a prerequisite here.

There we applied the "the Urdhva Tiryak Sutra"
(meaning : Vertically and cross-wise) to perform
multiplication of three digit numbers mentally,
easily and speedily.

Here, we extend the method to perform
multiplication of four digit numbers.

There and Here we deal with the principles
appled to Multiplication. For the application
to other topics of math magic, go to
Vedic Maths eBook.
for more details, see
near the bottom of the page.

Multiply vertically and crosswise
to get the digits of the product.

Examples will clarify the method.

Before seeing the examples, let us see
the formula for finding the product of
four digit numbers.

Let us say the four digits of the first number be
'a'(Thousands' digit) and 'b'(Hundreds' digit),
'c'(tens' digit) and 'd'(units' digit).

And those of the second number be
'p'(Thousands' digit) and 'q'(Hundreds' digit),
'r'(tens' digit) and 's'(units' digit).

Write the digits of the two numbers
one below the other as follows.

```          a  b  c  d
p  q  r  s
```

The product of these numbers has seven parts
which are given below seperated by '/'.

```          a  b  c  d
p  q  r  s
-----------
ap/(aq+pb)/(ar+pc+bq)/(as+pd+br+qc)/(bs+qd+cr)/(cs+rd)/ds
-----------
```

'ap' is the millions part which is the
vertical product of the first column.

(aq+pb) is the hundred thousands' part
which is the sum of the cross-wise
product of the first and third columns.

(ar+pc+bq) is the ten thousands' part which is the
sum of the cross-wise product of the first and third
columns and the vertical product of
the second column.

(as+pd+br+qc) is the Thousands' part which
is the sum of the cross-wise products of
first and last columns and the cross-wise
products of second and third columns.

(bs+qd+cr) is the Hundreds' part which is
the sum of the cross-wise products
of second and last columns and the vertical
product of third column.

(cs+rd) is the Tens' part which is the sum of
the cross-wise products of the last two columns.

'ds' is the units' part which is
the vertical product of the last column.

Let us see the method by examples.

Example 1 of Math Magic

To find 2011 x 1402

```          2011
1402
-----
2x1/2x4+1x0/2x0+1x1+0x4/2x2+1x1+0x0+4x1/
0x2+4x1+1x0/1x2+0x1/1x2
= 2/8/1/9/4/2/2
2011 x 1402 = 2819422
```

Having seen the steps for two and three
digit numbers, following here for the
four digit numbers should not be difficult.

The first and last terms are vertical
products of the first and last columns.

```          2
1
---
2x1
---
```
```          1
2
---
1x2
---
```

The second term is the sum of the cross
product of the first two columns.

```          2 0
1 4
----
2x4+1x0
----
```

Similarly the second term from last, is the sum
of the cross product of the last two columns.

```          1 1
0 2
----
1x2+0x1
----
```

The third term is the sum of the cross product
of the first and third columns and the
vertical product of the second column.

```          2 0 1
1 4 0
-----
2x0+1x1+0x4
-----
```

Similarly the third term from last, is the sum of
the cross product of the second and last columns
and the vertical product of the third column.

```          0 1 1
4 0 2
-----
0x2+4x1+1x0
-----
```

The fourth term is the sum of the cross product
of the first and last columns and the cross
product of the second and third columns.

```          2 0 1 1
1 4 0 2
-------
2x2+1x1 + 0x0+4x1
-------
```

We can do the calculations mentally and write
the seven parts directly as shown below.

```
2011
1402
-----
2/8/1/9/4/2/2
-----

2011 x 1402 = 2819422

```

Thus 2011 x 1402 = 2,819,422

Example 2 of Math Magic

To find 4201 x 5130

```          4201
5130
----
4x5/4x1+5x2/4x3+5x0+2x1/4x0+5x1+2x3+1x0/
2x0+1x1+0x3/0x0+3x1/1x0
=20/14/14/11/1/3/0
=20/14/14+1/1/1/3/0
=20/14+1/5/1/1/3/0
=20+1/5/5/1/1/3/0
=21/5/5/1/1/3/0

4201 x 5130 = 21551130
```

Here, the seven parts are obtained as in previous
example. But some parts have more than one digit. In
such cases, units's digit is retained and the other
digits are carried over to the immediate left place.

We can do the first step mentally and write
the second step directly. Also, we can do the
carry over from right mentally and write it
as the next step as shown below.

```
4201
5130
----
20/14/14/11/1/3/0
= 21/5/5/1/1/3/0  (worked out from right)
4201x5130 = 21551130

```

Thus, 4201 x 5130 = 21,551,130

Example 3 of Math Magic

To find 1234 x 5678

```          1234
5678
----
1x5/1x6+5x2/1x7+5x3+2x6/1x8+5x4+2x7+6x3/
2x8+6x4+3x7/3x8+7x4/4x8
=5/16/34/60/61/52/32
=5/16/34/60/61/52+3/2
=5/16/34/60/61+5/5/2
=5/16/34/60+6/6/5/2
=5/16/34+6/6/6/5/2
=5/16+4/0/6/6/5/2
=5+2/0/0/6/6/5/2
=7/0/0/6/6/5/2

1234 x 5678 = 7006652

```

We can do the process mentally and write
the answer in two steps as shown below.

```
1234
5678
----
5/16/34/60/61/52/32
=7/0/0/6/6/5/2 (worked out from right)
1234x5678 = 7006652

```

Thus 1234 x 5678 = 7,006,652

Example 4 of Math Magic

To find 3456 x 789

```
3456
0789
----
0/21/52/94/118/93/54
=2/7/2/6/7/8/4 (worked out from right)
3456x789 = 2726784

```

Thus, 3456 x 789 = 2,726,784

See the math magic of arriving
at the answer in two steps.

Example 5 of Math Magic

To find 9876 x 543

```
9876
0543
----
0/45/76/94/82/45/18
=5/3/6/2/6/6/8 (worked out from right)
9876x543 = 5362668

```

Thus, 9876 x 543 = 5,362,668

See the math magic of arriving
at the answer in two steps.

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## Proof of the method adopted : Math Magic

We know
(ay3 + by2 + cy + d)(py3 + qy2 + ry + s)
= y6(ap) + y5(aq + pb) + y4(ar + pc + bq)
+ y3(as + pd + br + qc) + y2(bs + qd + cr) + y(cs + rd) + ds

 y6 term y5 term y4 term y3 term y2 term y term constant term ap (aq+pb) (ar+pc+bq) (as+pd+br+qc) (bs+qd+cr) (cs+rd) ds

Multiplying four digit numbers is similar with y = 10

106term and units term are vertical products
and 105 term, 104 term,
103 term, 102 term,
10 term are cross-wise products' sums.

[a(10)3 + b(10)2 + c(10) + d][p(10)3 + q(10)2 + 10y + s]
= 106(ap) + 105(aq + pb) + 104(ar + pc + bq)
+ 103(as + pd + br + qc) + 102(bs + qd + cr) + 10(cs + rd) + ds

 Millions' Hundred Thousands' Ten Thousands' Thousands' Hundreds' Tens' Units' ap (aq+pb) (ar+pc+bq) (as+pd+br+qc) (bs+qd+cr) (cs+rd) ds

The digits of the two numbers :

```          a  b  c  d
p  q  r  s
```

The seven parts of the product :

ap/(aq+pb)/(ar+pc+bq)/(as+pd+br+qc)/(bs+qd+cr)/(cs+rd)/ds (proved.)

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