Martian Math

Whenever I failed to find a lesson plan for whatever teacher, whenever I arrived at school to substitute for the real teacher, who was absent, I would turn to my trusty notebook and teach the unruly class how to do what I called Martian Math (which was really binary math, '1's and '0's, like a computer does.) This was just after I got out of college and couldn’t find a real job. So I'd substitute-teach Martian Math to High School kids.

On those sick-days-for-somebody-else, I thanked the gods that I paid attention to 1^st Class Petty Officer Odom from Albany Georgia (rhymes with “rainy”, Albany that is, not Georgia,) I thanked the gods that I paid attention to 1^st Class Petty Officer Odom while he was teaching Boolean Math & various gate circuits, like And gates, Nand, Or and Nor gates to our sleepy class of would-be electronic technician strikers back in Fleet Ballistic Missile school in 1963 Dam Neck Virginia. (An unlikely scientist, Petty Officer Odom. When he spoke, you might judge him to be the redneck he resembled in outward appearance.)

(These days, if I were constructing a lesson plan about troubleshooting electronic computer circuits, instead of Boolean logic and various individual electronic gating circuits, I’d teach about microchip inputs, and their wholesale substitution. Forget about troubleshooting an individual transistor circuit when microchips can contain thousands of transistors. Back then computers were big. Those thousands of transistors would easily fill up your hands and spill onto the deck below your feet.)

So, back in those bad old uncertain times when I substituted for ailing teachers and the lesson plan was also absent, I’d turn to my notes on Boolean stuff so I could teach Martian Math to an unruly room full of girly, wannabe porn starlets, and manly, wannabe hoods, general knob polishers (who polished their own knobs in the absence of their feminine classmates. For it seemed to be true, that old navy saying, that 95% of all men used to masturbate, while the other 5% still do—not that there’s anything wrong with that. I guess that those kids weren’t as bad as I thought back then; appearances, like Odom’s appearance, may not be real. My mind was slowly changing about appearances. Maybe the Navy was just worried about controlling a crowd of guys alone on a ship, at sea…)

Anyway, what I called “Martian Math” was actually binary math, dressed up in a spacesuit to attract the geeks in the room, who recognized it as something they might like to know, and so, would take a front row seat to get away from the noisy kids at the back of the room, the inevitable crowd of slackers any class of forced students contains.

To seal the deal, so to speak, I’d begin by talking about Martians as funny looking individuals who had, for example, only two fingers where our hands have ten. And there, the science fiction would start. “Imagine the mathematics of such a civilization,” I’d say, “Instead of being based on ten numbers, on 0-9, like ours, Martians would base their math on only two. Oh, they’d still use something like the powers of ten to calculate stuff.“ I’d say. “Only it would be the powers of two.” We can use the powers of ten to convert decimal numbers to binary numbers and Martians can use the powers of two to convert binary numbers to decimal numbers.

And, if no one asked what are the powers of ten, or two? If no one asked how do you use the powers of ten to convert binary numbers to decimal numbers, or how do you use the powers of two to convert decimal numbers to binary numbers I’d just pretend someone had asked. I’d just forge on: “Some ask, what’re the powers of ten and how do you use them to convert binary numbers to decimal numbers?”

Let me show you the powers of ten. The decimal number 8765 contains 8 numbers worth a thousand each, 7 numbers worth a hundred each, 6 numbers worth ten each, and 5 numbers worth one each. From another viewpoint, starting at the right and moving through a decimal number like 8765, a decimal number contains units, tens, hundreds, thousands… From yet another viewpoint, to assemble a decimal number with the powers of ten you multiply the number in the far right column (“5” in “8765”) by 10^0. Multiply “6” by 10¹, “7 by 10² and “8” by 10³. (Notice that each column of any decimal number is one power of ten more than the column on your right and one power of ten less than the column on your left--…10³, 10², 10¹, 10⁰, or, thousands, hundreds, tens, units.

Working backward let me show you the powers of two. To assemble a decimal number with the powers of two you multiply the number in the far right column by 2⁰, Moving left, multiply the digit in the next column by 2¹, 2² next, 2³next…(Notice that—as in the powers of ten above--each column of any binary number is one power of two more than the column on your right and one power of two less than the column on your left--…2³, 2², 2¹, 2⁰. Of course, 2³, 2², 2¹, 2⁰ equals 8, 4, 2, and 1 respectively.

Anyone can reconstruct the powers of ten or the powers of two by remembering that any number raised to the zero power equals one. 10⁰ equals 1. 2⁰ equals1.

Now someone might ask: "how do you use the powers of ten to convert binary numbers to decimal numbers, or how do you use the powers of two to convert decimal numbers to binary numbers?”

“Starting with easy stuff, let me show you how to convert a binary number, say, the number 1000₂, convert it to a decimal number. (Remember, any binary number has only two digits: a “0” and a “1”. Thus 101010₂ is a binary number. 1001001₂ is also a binary number.)"

To convert the binary number 1000₂ to a decimal number multiply each column by its power of two for that column. Thus, 0 times 1 equals 0. 0 times 2 equals 0. 0 times 4 equals 0. And 1 times 8 equals 8. 1000₂ equals 8₁₀.

To convert the decimal number 500₁₀ to a binary number list out the powers of two and build the binary number by inspection. (I recommend this method because the binary number will be a long string of '1's and '0's. This method makes the conversion manageable.)

The powers of two, ascending, are: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024… 500₁₀ contains a binary 256 or 100000000 (the binary representation of a decimal 256, a 256₁₀). That leaves us with 244₁₀ (500 – 256). 244₁₀ contains a binary 128 or 10000000. 116 (244 – 128) contains a binary 64 or 1000000. 52 contains a 32 or 100000. 20 contains a 16 or 10000. But 4 contains no 8 or 0000. 4 contains a 4 or 100, but 2 and 1 are, in this example, 0 and 0. In sum, we express all the binary numbers with this combination:111110100₂.

111110100₂ equals 500₁₀.

If I were lucky, someone who followed the above reasoning would ask, “but what about the Martians who have 16 fingers?”

“Their math,” I’d reply, “is based on the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. But remember, any number to the zero power equals 1; any hexidecimal number to the zero power equals 1; 16⁰ equals 1. You can figure out the rest.” And if my luck held, the bell would ring:-)