He (Dase) multiplied together mentally two 8-figure
numbers in 54 seconds, two 20-figure numbers in 6
minutes, two 40-figure numbers in 40 minutes, and
two 100-figure numbers in 8 hours; he could extract
the square root of a 60-figure number in an "incredibly
short time," and the square root of a 100-figure
number in 52 minutes.


Johann Dase (1824-1861) was born in Hamburg and began public
exhibitions of his talents when he was 15. Dase could also count
objects with the greatest rapidity. With a single glance he could
give the number (up to thirty or thereabouts) of peas in a handful
scattered on a table; and the ease and speed with which he could
count the number of sheep in a herd, of books in a case, or the
like, never failed to amaze the beholder.

           

6

CHAPTER 6

VERTICALLY AND CROSSWISE


We come now to the general method of multiplication by which any two numbers, no matter how long, can be multiplied together by means of a simple pattern.
We will see a number of variations of this including squaring and the reverse process of division.

GENERAL MULTIPLICATION

This Vertically and Crosswise formula develops the left to right method of multiplication described in chapter 1 for multiplication by a single figure, and is
surely the most efficient general method of multiplication possible,

In this chapter we will put the numbers being multiplied one below the other.

Example 1
                                       2 1
                                       2 3 ×
                                    4 8 3

There are 3 steps:                                          2    1
A. multiply vertically in the left-hand          |
column: 2 × 2 = 4,                                        2    3 ×
so 4 is the first figure of the answer.           4      

B. multiply crosswise and add:                    2    1
2 × 3 = 6,                                                        ×
1 × 2 = 2, 6 + 2 = 8,                                     2    3 ×
so 8 is the middle figure of the answer.      4 8   

C. multiply vertically in the right-hand       2    1
column: 1 × 3 = 3,                                              |
3 is the last figure of the answer.                2    3 ×
                                                                      4 8 3

Example 2
                                       1    4
                                       2    1 ×
                                       2 9 4

A. vertically on the left: 1×2 = 2,
B. crosswise: 1×1 = 1, 4×2 = 8 and 1+8 = 9,
C. vertically on the right: 4×1 = 4.

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The previous examples involved no carry figures, so let us consider this next.

Example 3
                                       2 3
                                       4 1 ×
                                    9 4 3

The 3 steps give us: 2×4 = 8,
2×1 + 3×4 = 14,
3×1 = 3.

In building up the answer mentally from the left we combine these numbers as we did in chapter 1.

The mental steps are: 8
                                    8,14= 94 ( the 1 is carried over to the left)
                                    94,3 = 943

Example 4
                 2    3                   The steps are: 6
                 3    4 ×                                       6,17= 77
                 7 8 2                                          77,12= 782

Example 5
                 3   3                    The steps are: 12
                 4   4                                           12,24 = 144
             1 4 5 2                                           144,12= 1452

We can now multiply any two 2-figure numbers together in one line. With practice we will find that we can multiply the numbers mentally.

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Challenge Question: Can you see how this method simplifies when
                 a) both numbers end in a 1
                 b) the last figures of the numbers, or the first figures, or both figures
                      of one number, are the same?

You may have found in this exercise that you prefer to start with the crosswise multiplications, and put the left and right vertical multiplications on afterwards.

This method also works perfectly for algebraic multiplications (and divisions).

Explanation  It is easy to understand how this method works. The vertical product on the right multiplies units by units and so gives the number of units in the answer. The crosswise operation multiplies tens by units and units by tens and so gives the number of tens in the answer. And the vertical product on the left multiplies tens by tens and gives the number of hundreds in the answer.

USING THE VINCULUM

We now consider again the helpful vinculum. This was described in the last chapter and simplifies calculations by removing large digits like 6, 7, 8 and 9.

Example 6
29 × 34:                               3
                                             3 4 ×
                                           99 = 9 8 6

We write 29 as 3 in order to remove the large digit, 9.
Then the 3 steps are: 3 × 3 = 9, 3 × 4 +  ×3 = 12-3 = 9,  ×4 = .
The  may then be removed as shown in the previous chapter (99 = 990 - 4 = 986).

Of course the use of the vinculum is optional but it does remove the large digits, and the plus and minus numbers tend to cancel each other out.

Example 7
49 × 58                              5  
                                           6   ×           The steps are 30
                                   2  8  4  2                                     30,-16 = 3 = 284
                                                                                       284,2 = 2842

Example 8
28 × 42:                           3                     The steps are 12
                                         4  2 ×                                        12,  = 12
                                   1 1 7 6                                            12, , = 12= 1176

At which point to remove the vinculums is a matter of judgment, the answer being the same of course whatever method is decided upon.

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Challenge Question: Can you prove the By One More than the One Before method from Chapter 3 for finding products like 34 × 36 and 452, using Vertically and Crosswise?

MULTIPLYING THREE-FIGURE NUMBERS

Example 9
                                     1 2 3               The Vertically and Crosswise formula can be
                                     1 3 2 ×            extended to deal with this, but in fact the
                              1 6 2 3 6                previous vertical/crosswise/vertical pattern
                                                             can be used on this sum also.

       We can split the numbers up into 12/3 and 13/2, treating the 12 and 13
       as if they were single figures.:

                                    12     3              vertically 12×13 = 156,
                                    13     2              crosswise 12×2 + 3×13 = 63,
                                  162 3 6               vertically 3×2 = 6.

       Combining these mentally we get: 156
                                                                156,63= 1623
                                                                1623,6 = 16236

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Example 10
304 × 412

Here we may decide to partition after the first figure: 3/04 × 4/12.

          3       04       When we split the numbers so that there are pairs of
          4       12       digits on the right the answer appears two digits at a
        12 52 48       time.

The 3 steps of the pattern are: 3×4 = 12,
                                                  3×12 + 4×4 = 52,
                                                  4×12 = 48.

These give the 3 pairs of figures in the answer.

Example 11
1201 × 1312

Here we can split the numbers halfway:

             12       01                           The pattern gives: 12×13 = 156
             13       12                           12×12 + 1×13 = 157,
           157 57 12                           1×12 = 12

The mental steps are:            156,157= 15757
                                               15757,12 = 1575712
                                               (working 2 digits at a time)

Example 12
312 × 1011

             3       12                     We have:   30,153 = 3153
           10       11 ×                                    3153,132 = 315432
           31 54 32


Example 13
198 × 303          Here we can apply the vinculum.

                            2                                treating the sum as 2/× 3/03
                            3        03
                            6 00   = 59994

                                       20        
                           or        30         3 ×               treating the sum as 20/× 30/3
                                       600  0   = 59994

Given a choice about how to split the numbers, as in the last example, it is generally best to mark off two figures on the right and then work with pairs of figures. This way we tend to avoid the carry figures.

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MOVING MULTIPLIER

In multiplying a long number by a single figure, for example 4321 × 2, we multiply each of the figures in the long number by the single figure. We may think of the 2 moving along the row, multiplying each figure vertically by 2 as it goes.

Example 14
4321 × 32

4 3 2 1         Similarly here we put 32 first of all at the extreme left.
3 2               Then vertically on the left, 4 × 3 = 12.
                     And crosswise, 4×2 + 3×3 = 17.

4 3 2 1         Then move the 32 along and multiply crosswise:
   3 2            3×2 + 2×3 = 12.

4 3 2 1         Moving the 32 once again:
      3 2         multiply crosswise, 2×2 + 1×3 = 7.
                    Finally the vertical product on the right is 1×2 = 2.

These 5 (in bold above) results: 12,17,12,7,2 are combined mentally, as
they are obtained, in the usual way:
1217 = 137
13712 = 1382
1382,7,2 = 138272

So we multiply crosswise in every position, but we multiply vertically also at the very beginning and at the very end.

Example 15
31013 × 21

Here the 21 takes the positions:

The 6 mental steps give: 6,5,1,2,7,3
so the answer is 651273.

You may find with practice that your mind simplifies this procedure further by adding, in this example, twice each digit to the digit before. And this indicates a very easy way of multiplying by 11 and by 9:

Example 16
3213 × 11 = 35343

Suppose that there is a zero on each end of the number, and starting on the left, add to each figure the figure before it.

Example 17
3927 × 9 = 365 = 35343

Put a zero on each end of the number, and starting on the left subtract
from each figure the figure before it: 3-0, 9-3, 2-9, 7-2, 0-7.

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In fact this vertical/crosswise/vertical pattern can be used to find the product of any two numbers. However, we now move on to see how the pattern itself can be extended.

THREE AND FOUR-FIGURE MULTIPLICATIONS

Example 18
  

The pattern now becomes:

A    Vertically on the left, 5×3 = 15.

               
B    Then crosswise on the left,  
       5×2 + 0×3 = 10.
       Combining the 15 and 10:
       1510= 160.

C    Next we take 3 products and add them up,   
       5×1 + 0×2 + 4×3 = 17. And 16017= 1617.
       (actually we are gathering up the hundreds
       by multiplying hundreds by units, tens by
       tens and units by hundreds)

D    Next we multiply crosswise
       on the right,
       0×1 + 4×2 = 8: 1617,8 = 16178.

E     Finally, vertically on the right,
       4×1 = 4: 16178,4 = 161784.

Note the symmetry in the 5 steps:
first there is 1 product, then 2, then 3, then 2, then 1.

We may summarise these steps as follows:

Note also the dot which moves through the middle of the sum from left to right.

Example 19
    3    2    1
    3    2    1 ×              The 5 results are 9,12,10,4,1
1 0 3 0 4 1                  The mental steps are       9
                                                                             912= 102
                                                                             10210= 1030
                                                                             1030,4,1 = 103041

Example 20
123 × 45

This can be done with the moving multiplier (see Example 14) or by the smaller vertical and crosswise pattern, treating 12 in 123 as a single digit (see Example 9). Alternatively, we can put 045 for 45 and use the latest method:

   1 2 3
   0 4 5                   For the 5 steps we get 0,4,13,22,15.
5 5 3 5                   Mentally we think 4; 53; 552; 5535

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You may find that you prefer to build up the answer as each product is found (rather than at each step), so that in number 16, for example, where the products are 3,5,6,6,10,12,12,20,24 the answer is built up: 3;35;41;416;426;438;4392;
4412;44144. (The order in which the products are taken at each step can be chosen to give the simplest possible route to the answer.)

SQUARING

The Vertically and Crosswise formula simplifies nicely when the numbers being multiplied are the same, and gives us a very easy method for squaring numbers.
We will use the term "Duplex", D, to denote:

for 1 figure D is its square, e.g. D(4) = 4²=16;
for 2 figures D is twice their product, e.g. D(43) = 2×4×3 = 24;
for 3 figures D is twice the product of the outer pair + the square of the middle digit,
e.g. D(137) = 2×1×7 + 3² = 23;
for 4 figures D is twice the product of the outer pair + twice the product of the inner pair,
e.g. D(1034) = 2×1×4 + 2×0×3 = 8;
D(10345) = 2×1×5 + 2×0×4 + 3² = 19;
and so on.

The square of any number is just the total of its Duplexes, combined in the way we have been doing in Chapter 1 and in this chapter.

Example 22
43² = 1849

D(4) = 16, D(43) = 24, D(3) = 9,
combining these three results in the usual way we get: 16
                                                                                          1624 = 184
                                                                                          184,9 = 1849

Example 23
64² = 4096

D(6) = 36, D(64) = 48, D(4) = 16.      So mentally 3648 = 408
                                                                                    40816 = 4096

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Example 24
341² = 116281

Here we have a 3-figure number:

D(3) = 9, D(34) = 24, D(341) = 22, D(41) = 8, D(1) = 1.

Mentally  924 = 114
                11422 = 1162
                1162,8,1 = 116281


Example 25
4332² = 18766224

D(4) = 16, D(43) = 24, D(433) = 33, D(4332) = 34,
D(332) = 21, D(32) = 12, D(2) = 4.

Mentally 1624 = 184
                18433 = 1873
                187334 = 18764
                1876421 = 187661
                18766112 = 1876622
                1876622,4 = 18766224


Example 26
21034² = 442429156

D(2) = 4, D(21) = 4, D(210) = 1, D(2103) = 12, D(21034) = 22,
D(1034) = 8, D(034) = 9, D(34) = 24, D(4) = 16.

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GROUPING

The grouping of digits is also available for squaring.

Example 27
123² = 15129

Here we may think of 123 as 12/3, as if it were a 2-figure number:

D(12) = 144, D(12/3) = 72, D(3) = 9.

Combining these: 14472 = 1512
                              1512,9 = 15129


Alternatively, had we known that 23² = 529 we might have preferred to see 123 as 1/23.
Then D(1) = 1, D(1/23) = 46, D(23) = 529 so that we combine: 1,46,529, in groups of two.
This gives 15129 again.

Example 28
312² = 97344

If we think of 312 as 3/12 we must work with pairs of digits:

D(3) = 9, D(3/12) = 72, D(12) = 144.

Combining: 9,72 = 972
                     972144 = 97344

Example 29
235² = 55225

There are several ways of doing this, but suppose we decide to split
the number into 2/35:

D(2) = 4, D(2/35) = 140, D(35) = 1225:   4140 = 540
                                                                       5401225 = 55225

Example 30
192² = 22² = 4, ,9, ,4 = 494 = 4/9/4 = 36864

Seeing the large digit, 9, suggests using the vinculum.

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