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Basic Binary Board Book

The Basic Binary Board Book is for small people who aspire to one day count in binary from 0 to F. I am not yet selling physical copies, but the PDF is cheap and Pint Size Productions can print one out for you on demand and ships within the United States. We produced a monochrome 1st edition for our first child. This colorful 10nd edition with more information commemorates our second. It looks like this interactive version.

full page

What follows is an explanation for parents and lifelong learners.

Digits #

The word digit has accumulated a lot of meanings. On the one hand, digit means finger. With the other hand, there are 10 digits. The hands of a clock point to numbers from 1 to 12. The hand of a compass points to numbers from 0 to 359. Although computers don’t have hands, nothing could be more digital. So, a robot is all kinds of handy.

In a decimal number, each digit is a number from 0 to 9. The last digit counts for one, and each digit represents 10 times as much as the one to its right. So, if you multiply each digit by its place value and add those numbers up, that is the value of the whole number.

123
1⨉100 + 2⨉10 + 3⨉1
300 + 20 + 1

All of the wonderous utility and complexity of expression and communication facilitated by computers emerges from the stark reality that a computer is a very long list of switches that can flip themselves very, very quickly. And each of those switches is a digit in the binary number system.

The computer above has has four binary digits or bits. Whenever you flip a switch, you change the number. You can touch or click a bit on this computer to toggle.

So, if you take the number zero and click or touch the center of the circle or the last bit of 0b0000, it will turn to page 1. If you then click or touch the next ring, with two segments, the number will turn to three, which has both of the least significant bits. If you turn off the center bit again, you go back down to two.

When a switch is off, it represents the binary digit zero. When a switch is on, it represents the binary digit one. These are the only digits in binary.

Concentric rings #

On each of the pages of this binary book, there are four concentric rings, each representing a bit of a four digit number. The color of the ring and the number of segments in the ring represents its place value. So, the outer ring has eight segments. The next ring has four. The next ring has two. The innermost ring is simply a circle with one segment. Each segment has the same area, so should give you a visual sense for the doubling in value from inside to out and the total value of each number.

Binary numerals #

Numbering systems like binary and decimal encode the number of expressible digits in their name. The bi- in binary means two. The dec- in decimal means ten. The hexadec- in hexadecimal means sixteen. (The dodec- in dodecimal means twelve, but so does the dozen in dozenal, and that's more fun. The sexages- in sexagesimal means sixty.)

As with decimal, the last bit of a binary number stands for the value of one. The place value of ever other bit is twice as much as the bit to its right.

So, each switch, when on, stands for eight, four, two, and one respectively. Adding them up, as illustrated on top of the computer, produces the decimal sum on the top-left corner.

So, with four bits, we can represent the sixteen numbers: 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, and 1111. To help distinguish these from decimal numbers, we add a 0b. The initial 0 indicates that the word that follows is a number and not a name. The b stands for binary.

When people discuss numbers that fit nicely in computers, writing all the bits out is sometimes illustrative, but writing out four digits for numbers from 0 to 15 is tedious, and writing the same numbers in decimal sometimes requires one digit (like 7) and sometimes two (like 12).

Every four-bit number in binary corresponds to a single digit in hexadecimal. For the numbers from zero to nine, we reuse the decimal digits 0 to 9. But for the numbers ten to fifteen, we use the letters from A to F.

In the same way we use the 0b prefix for binary numbers, in hexadecimal notation, we use the prefix 0x. So, hexadecimal 0xA means binary 0b1010 and decimal 10.

0xA in hexadecimal
0b1010 in binary
10 in decimal

Hexadecimal numerals #

The hexadecimal notation captures an even number of binary digits in a much-compressed form, but not so compressed that humans must memorize the values of many more than the traditional base-ten digits.

The memory of a computer is a long, long list of metaphorical switches. The computer has various tiers of memory and can move chunks of memory between those tiers. The smallest and fastest tier has registers and the smallest addressable registers on modern computers converged on having eight bits, which we call a byte. So, when we draw a representation of a byte in hexadecimal, it has two hexadecimal digits: the high nibble and the low nibble. I would have spelled that nybble, but nobody asked me. In fairness, I didn't exist yet.

The hexadecimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. So, the largest expressible number in a byte is 255 and we write that as 0xFF.

Of bucks and bytes #

By dint of a strange coïncidence, the word “bit” means an eighth of a byte and also an eighth of a dollar. The word dollar comes from Spanish, and the Spanish got the word from a specific silver mine in a town of modern Czechia called Joachimsthal, or Saint Joachim’s Dale. So, the German “thal” means valley, is also seen in “neanderthal”, and gave English both the word “dale” and “dollar”.

The silver dollar also gave Spanish pesos de ocho, or pieces of eight, which were slices of the dollar coin. Americans of the Victorian era called these bits, so a “two-bit saloon” costed twenty-five cents.

Thus, there are eight bits to the dollar. The other coining of bit was for parts of a byte, but long before there was agreement that a byte should have eight bits. Many other word lengths existed.

Counting #

In the binary book, you can add one to a number by turning the page. In the computer based on that book above, the “>” button increments and “<” decrements. If you count from 0 to F, you will notice that the inside circle, representing the least significant bit flips on and off. When a bit flips off, the value of the bit must be carried up to the next ring or next most significant bit. Carrying bits ripples up until it flips a switch from off to on.

A strange thing happens when the four bit number runs out of bits. When you go to the next page from fifteen, you return to zero. This phenomenon is called overflow, and although its a sharp departure from mathematics, it is eminently useful for computers because integer addition can use the exact same circuitry as subtraction.

Hexadecimal pronunciation #

In 2019, Liz Henry unearthed an article titled A Hexadecimal Pronunciation Guide from Datamation Magazine, Volume 14 by Robert A. Magnusen in 1968. This long-lost naming scheme was both sensible and rife with puns.

Magnusen proposed that the numbers 10 to 15 be named anna, bet, chris, dot, ernest, and frost. It naturally follows that 0xB0 is betty and 0x1C is christeen.

a page from Datamation magazine with a sample of Magnusen's pronunciation guide for hexadecimal

So, 0xB0B5_C0FFEE_FACADE is pronounced, “betty, betty-five, christy, frosty-frost, ernesty-ernest, frosty-anna, christy-anna, dotty-ernest”.

Latin and Greek #

This book also notes the Latin and Greek roots for English numbers, which you will see in the names of number systems and geometric shapes.

English uses Latin roots to construct all sorts of English words. We’ve already visited names of number systems, like binary. You can use these roots to name arbitrary number systems, like quaternary. They also make words like octagon and triangle

English uses the Greek roots for words like monochrome for one color, pentagon for a shape with five sides and tesseract, a shape with four dimensions.

English uses these roots in all manner of inconsistent and nonsensical ways that are never-the-less broadly accepted, like nonagon instead of enneagon, using a Latin root for 9-sided shapes although every other n-gon is named according to Greek roots. We have Greek double and triple, but Latin single and quadruple.

Etymology is the fossil record of confusion.

Roman numerals #

The pages of this book do not speak about Tally Marks, the only number system simpler than binary, especially since simple is related to single and literally based on one, one less than two.

For no other reason that to set up for the next joke, this book catalogues the Roman Numerals. This is a decimal numeral system whose only apparent virtue is that it can be etched with an economy of strokes using a stylus.

Binary Roman numerals #

The Roman numerals employ the letters I, V, and X. I propose new values for these glyphs, in ascending complexity, and introduce the next obvious symbol for the representation of binary numbers.

The last is sometimes called octothorpe, which has the virtue of never confusing British bean counters or American weights and measures. It also suggests that X is a quadrathorpe. V is arguably a bithorpe and I an unthorpe, but I will concede that this is a stretch. It can be much more convincingly argued that - is a bithorpe, ⨉ is a quadrathorpe, and asterisk * is a hexathorpe. A simple thorpe is a thing of myth and legend.

Regardless, it follows that Binary Roman numerals represent a four-bit binary number as a sequence of #XVI, including only the bits that are set, so their value is a mere counting of thorpes.

The binary numerals are:

This representation highlights property unique to binary numerals. Because the only possible values of a digit are 0 or 1, the ommission of a digit can imply 0. This is not a book about how computers represent real numbers, but computers use at least one practical application of this insight. If you were to create a scientific notation out of binary digits, the only number that doesn't start with 1 is 0. So, that first 1 can be implied and doesn't require a switch.

Basic binary board book #

If you have gotten this far, you probably already received a copy of Basic Binary Board Book, either because your child received it as a birthday present, or you lost at a White Elephant gift exchange.

But, I would be remiss if not to remind you that you have an opportunity to buy one for yourself, your child, and all their peers at the preschool or gradschool. I am not yet selling physical copies, but the PDF is cheap and Pint Size Productions can print one out for you on demand and ships within the United States.