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.
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
We will see a number of variations of this including squaring and the reverse process of division.
This Vertically and Crosswise formula develops the left to right method
of multiplication described in chapter 1 for multiplication by a single figure,
surely the most efficient general method of multiplication possible,
In this chapter we will put the numbers being multiplied one below the other.
There are 3 steps: 2
B. multiply crosswise and add: 2
The previous examples involved no carry figures, so let us consider this next.
The 3 steps give us: 2×4 = 8,
In building up the answer mentally from the left we combine these numbers as we did in chapter 1.
The mental steps are: 8
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.
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.
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.
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.
At which point to remove the vinculums is a matter of judgment, the answer being the same of course whatever method is decided upon.
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?
We can split the numbers
up into 12/3 and 13/2, treating the 12 and 13
12×13 = 156,
Combining these mentally
we get: 156
Here we may decide to partition after the first figure: 3/04 × 4/12.
When we split the numbers so that
there are pairs of
The 3 steps of the pattern are: 3×4 = 12,
These give the 3 pairs of figures in the answer.
Here we can split the numbers halfway:
pattern gives: 12×13 = 156
have: 30,153 = 3153
the sum as 2/×
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.
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.
4 3 2 1 Then move
the 32 along and multiply crosswise:
4 3 2 1 Moving the 32 once
These 5 (in bold above) results: 12,17,12,7,2 are combined mentally,
So we multiply crosswise in every position, but we multiply vertically also at the very beginning and at the very end.
Here the 21 takes the positions:
The 6 mental steps give: 6,5,1,2,7,3
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:
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.
Put a zero on each end of the number, and starting on the left subtract
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.
The pattern now becomes:
C Next we take 3 products and add them up,
D Next we multiply crosswise
E Finally, vertically on the right,
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.
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
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.)
The Vertically and Crosswise formula simplifies nicely when the numbers
being multiplied are the same, and gives us a very easy method for squaring
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.
D(4) = 16, D(43) = 24, D(3) = 9,
D(6) = 36, D(64) = 48, D(4) = 16. So
mentally 3648 = 408
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
D(4) = 16, D(43) = 24, D(433) = 33, D(4332) = 34,
Mentally 1624 = 184
D(2) = 4, D(21) = 4, D(210) = 1, D(2103) = 12, D(21034) = 22,
The grouping of digits is also available for squaring.
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
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
There are several ways of doing this, but suppose we decide to split
D(2) = 4, D(2/35) = 140, D(35) = 1225: 4140 =
Seeing the large digit, 9, suggests using the vinculum.
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