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B.2.4. Gottfried Wilhelm Leibniz (1646-1716) was a German mathematician and philosopher who
is credited with the discovery of binary numbers. Leibniz described his ideas in a topic titled
The
Dyadic System of Numbers
, published in 1679. His motivation was to develop a system of notation
and rules for describing philosophical reasoning. Leibniz was fascinated by binary numbers and
believed that the sequences of 0s and 1s revealed the mystery of creation of the universe and
that everything should be expressed by binary numbers. In 1697 he wrote a letter to the Duke of
Brunswick suggesting a design for a celebration of binary numbers to be minted in a silver coin.
The Latin text at the top of the coin reads “One is sufficient to produce out of nothing everything.”
On the left of the table there is an example of binary addition and on the right, an example of
multiplication. At the bottom the text in Latin says the “Image of Creation.”
1101 = (1 × 2
3
) + (1 × 2
2
) + (0 × 2
1
) + (1 × 2
0
) = 13 in decimal
We can add and multiply binary numbers in a similar but much simpler way as
in the decimal system. When adding in the decimal system, we automatically
align the powers of ten in the numbers and perform a “carry” to the next power
when needed, for example:
47
85
132
+
Fig. 2.4. A silver coin capturing the con-
cept of binary numbers.
For binary addition we have similar sum and carry operations. Adding the deci-
mal numbers 13 and 22 using binary notation leads to:
1101
10110
100011
+
We have just used the basic rules of binary addition:
000
011
101
110
+=
+=
+=
+=plus a carry
1
Let's take a look at multiplication in both the decimal and binary systems by
multiplying 37 by 5 in both decimal and binary notation:
In decimal:
37
5
185
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