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9.1 Binary Numbers

Sometimes it is said that engineering takes abstract science and makes it concrete. It’s no exaggeration to say that the logic you learn in this textbook changed the course of history when Alan Turing, Claude Shannon, and other logicians realized how to make it concrete with electricity running through circuits.

In order to understand how that’s possible, you first have to know how binary numbers work.

The base-ten numerical system that we use has ten numerals, 0-9. Since it has no numeral for ten, it represents ten by cycling the ones’ place back to 0 and putting a 1 in the tens’ place: 10.

In base-four, 10 = four.

That makes it feel like ten is a special “round” number. Similarly for 100, 1,000, etc. But really there is nothing special about the base-ten system. We can represent the same numbers with any system.

For example, a base-four system works like this: it has four numerals, 0-3, and to count to four you cycle the ones’ place back to 0 and put a 1 in the next place: 10.

It’s tempting to call that next place the “tens’ place”, but since this is base-four, it is the fours’ place.

In base-four, four and sixteen feel like special “round” numbers, because
10 = four
100 = sixteen

In base-ten, “23” means 2 tens and 3 ones, or twenty-three. But in base-four, “23” means 2 fours and three 1s, or eleven.

Numbers: objects studied by math.
Numerals: symbols we use to refer to numbers.

When talking about different base systems, it is important to distinguish between numbers and numerals. Numbers are the objects or entities “out there” in the world of mathematics. Numerals are symbols we use to refer to those numbers.

For example, there is only one number five, but we can have many different numerals that refer to five. The Arabic numeral “5” is not identical to the number five any more than the Roman numeral “V” is.

To understand different base systems, you just have to remember that “10” is a numeral; it isn’t literally the number ten. And “10” always refers to your base number. So in base-four, 10 means four. In base-nine, 10 means nine, and in base-two, 10 means two.

Binary: the base-two system

Binary is just the base-two system. It has two numerals, 0 and 1, which can represent every number.

Some helpful binary numbers to know:
10 = 2
100 = 4
1000 = 8
10000 = 16

Figuring out larger numbers in binary just requires this idea: in base-ten, 100 is 10×10, or one hundred.

So in base-two (binary), 100 is 2×2, or 4.

In order to write a larger number in binary, just divide it into chunks based on the powers of 2.

For example, to write 9 in binary, we need 8 + 1, and hence 1001.