Number Bases: Introduction & Binary Numbers

On the other hand, in the octal system (that is, in the base-8 system), there are only eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. There is no digit "8" in the octal system. And in the hexidecimal system (that is, in the base-16 system), there are sixteen digits: 0 through 9, followed by A, B, C, D, E, and F.

While the topic of different bases may seem somewhat pointless to you, the rise of computers and computer graphics has increased the need for knowledge of how to work with different, non-decimal base systems, particularly the binary system (consisting of just ones and zeroes) and the hexadecimal system.

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What are other number bases called?

We use the decimal number base, having ten digits; other number bases have their own names. For instance, the base-11 number base is called the "undecimal" base; base-12 is called "dozenal" (as in, "it has a dozen digits"). The base-8 system is called "octal"; the base-16 system is called "hexidecimal"; the base-2 system is called "binary". Other number-base systems have distinct names, too, and many of these other systems have been used, and some still to this day are still used, by various cultures and in various contexts.

Digits versus numbers

In our customary base-ten system, we have digits for the numbers zero through nine. We do not have a single-digit numeral for "ten". (The Romans did, in their character "X".) Yes, we can write the number "10", but this stands for "1 ten and 0 ones". This is two digits — a 1 and a 0. We have no single solitary digit that stands for "ten".

Instead, when we need to count to one more than nine, we zero out the ones column and add one to the tens column. When we get too big in the tens column — when we need one more than nine tens and nine ones (which is written as "99"), we zero out the tens and ones columns, and add one to the ten-times-ten, or hundreds, column. The next column is the ten-times-ten-times-ten, or thousands, column. And so forth, with each bigger column being ten times larger than the one before. We place digits in each column, telling us how many copies of that power of ten we need.

The only reason base-ten math seems "natural" to you — and the other bases don't — is that you've been doing base-ten since you were a child. And (nearly) every civilization has used base-ten math probably for the simple reason that we have ten fingers. If instead we lived in a cartoon world, where we would have only four fingers on each hand (count them next time you're watching TV or reading the comics), then the "natural" base system would likely have been base-eight, or "octal".

What are binary numbers?

Binary numbers are numbers which are written in the binary, or base-two, number system. The only digits in the binary system are 0 and 1. Binary numbers are, as you're probably aware, the lifeblood of computers; at the most basic level, computers operate by shuffling zeroes and ones.

Let's look at base-two, or binary, numbers. How would you write, for instance, 12₁₀ ("twelve, base ten") as a binary number? You would have to convert to base-two columns, the analogue of base-ten columns. In base ten, you have columns or "places" for 10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1000, and so forth. Similarly in base two, you have columns or "places" for 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, and so forth.

The first column in base-two math is the units column. But only 0 or 1 can go in the units column. When you get to "two", you find that there is no single solitary digit that stands for "two" in base-two math. Instead, you put a 1 in the twos column and a 0 in the units column, indicating "1 two and 0 ones". The base-ten "two" (2₁₀) is written in binary as 10₂.

The number "three" in base two is actually "1 two and 1 one", so it is written in binary as 11₂. "Four" is actually two-times-two, so we zero out the twos column and the units column, and put a 1 in the fours column; 4₁₀ is written in binary form as 100₂. Here is a listing of the first few numbers in their binary form:

Converting between binary and decimal numbers is fairly simple, as long as you remember that each digit in the binary number represents a power of two.

• Convert 101100101₂ to the corresponding base-ten number.

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Converting decimal numbers to binaries is nearly as simple: just divide by 2.

• Convert 357₁₀ to the corresponding binary number.

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This method of conversion will work for converting to any non-decimal base. Just don't forget to include that first digit on the top, before the list of remainders. If you're interested, an explanation of why this method works is available here.

You can convert from base-ten (decimal) to any other base. When you study this topic in class, you will probably be expected to convert numbers to various other bases, so let's look at a few more examples...