About the Header Image

Header image thumbnail

This image is designed to remind you of space exploration (imagine a photo taken from the surface of the moon). The symbols “floating in space” are exemplary of the topics to be explored on this site. The black and white imagery is also symbolic of binary encoding, which is characterized by two opposing “states.”

(A reader proposed an alternate view of the image — that of a chalkboard full of mathematical symbols!)

What do the symbols mean?

Here is a brief explanation of each of the sixteen symbols in the image, from left to right:

  • The eight symbols on the left side
    • 15 = 1 + 2 + 4 + 8: This is an example showing that any whole number can be written as a sum of powers of two.
    • 11001 = 25: The binary number 11001 equals

      \mbox{\footnotesize{1\cdot2^4 + 1\cdot2^3 + 0\cdot2^2 + 0\cdot2^1 + 1\cdot2^0 = 25}}.

    • Binary search tree with 7 nodes.: This is a binary search tree with seven nodes. If you were searching for the value 5, for example, you would take the right branch from node 4 to node 6 (since 5>4) and then the left branch to node 5 (since 5<6).
    • \footnotesize{\frac{1}{2} + \frac{1}{4} + \frac{1}{16} = \frac{13}{16}} = \mbox{\footnotesize{0.1101}}:This is an example of a dyadic fraction (a sum of negative powers of two) written in binary.
    • ‘R’ == 0x52: Using C-language syntax, this says that capital ‘R’ has the ASCII value of hexadecimal 52.
    • \mbox{\footnotesize{10100100 \vee \ 01011100 = 11111100}}: This shows the bitwise “OR” of two bytes (a bit is set to ‘1’ in the answer if there is a 1-bit in the corresponding bit position in either operand).
    • The first five rows of Pascal's Triangle.: This is the first five rows of Pascal’s Triangle. Each row sums to \mbox{\footnotesize{2^n}}, where n is the row number (row numbers start from 0 at the top).
    • AND gate with inputs 1 and 0 and output 0.: This shows a logic gate performing the logical AND of 1 and 0, which is 0.
  • The eight symbols on the right side
    • \mbox{\footnotesize{\displaystyle\sum_{i=2}^\infty 2^{-i} = \frac{1}{2}}}: This is an infinite geometric series summing all the negative powers of two except for 1/2. This series converges to 1/2.
    • 101 + 101 = 1010.This is an example of binary addition. You can verify this by noting that the binary number 101 equals five and the binary number 1010 equals ten.
    • Half, quarter, eighth and sixteenth notes.: This shows a half, quarter, eighth and sixteenth note. (They are dyadic).
    • \mbox{\footnotesize{2^4 \cdot 2^4 = 2^8}}:This is an example of how exponent arithmetic works. When you multiply two numbers with the same ‘base’ (2 in this case), the result is the base raised to the sum of their exponents. Such a computation might appear when using the binary exponentiation algorithm to computer powers of two efficiently.
    • I Ching hexagram for clustering.: This is not the letter `H’, but an example of an I Ching (pronounced ‘e-ching’) hexagram. This particular hexagram signifies “clustering” or “gathering” (as in this site being a gathering place for those interested in binary). Each hexagram is made up of 6 lines — each either solid or broken — giving \mbox{\footnotesize{2^6}}, or 64 hexagrams total.
    • \mbox{\footnotesize{\overline{x {\wedge} y} =  \overline{x} \vee \overline{y}}}: From De Morgan’s theorem, this states that the negation of “x ANDed with y” is equivalent to the negation of x ORed with the negation of y.
    • \mbox{\footnotesize{\log _2 \left( 32 \right) = 5}}: This says the logarithm in base 2 of 32 is 5. This is the same as saying that \mbox{\footnotesize{2^5}} = 32.
    • 07D8: The hexadecimal number 07D8 equals

      \mbox{\footnotesize{0\cdot16^3 + 7\cdot16^2 + 13\cdot16^1 + 8\cdot16^0 = 2,008}}

      (Not coincidentally, this site was launched in 2008.)

If you want the details behind these explanations, stay tuned to this site!

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2 comments

  1. @Paresh,

    There are many octal/hex/any base calculators and converters on the Web that work with integer values. My decimal/binary converters are unique in that they work (accurately) with fractional values as well. I haven’t perceived much need for a similarly designed octal or hex calculator. Could you tell me what uses you’d have for it? Thanks.

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