# Number Bases

Robert P. Webber and Don Blaheta, Longwood University

## Place-value notation

Why are we working on this? In our ultimate goal of representing all data, we need to take everything we've figured out how to represent as numbers—and things that were numbers in the first place—and put it in a form the computer can use. For pure human use, decimal is convenient, but binary form will work best for computers. Hexadecimal is a compromise: easily converted to binary for computer use but somewhat easier for humans to work with. Skills in this section: Translate positive integers freely among bases 2, 10, and 16 Concepts: Data representation, Classic algorithms, Limitations of representation

Our civilization uses the base 10 or decimal place-value system. There are ten different digits, 0-9, so a one-digit number in this system can represent ten possible different values. But by putting those same ten digits in different positions in a multi-digit number, we can represent arbitrarily large and small numbers, limited only by the number of digits we're willing to use for the number.

We call it a place-value system because the exact value that a digit represents is determined in part by where in a number it is found. The digit 7 represents exactly 7 in a number like 347, but it represents 70 in a number like 374, it represents 70,000 in a number like 3871524, and it represents
 7 10
in a number like 32.74. In a multi-digit number, each place, or position, is associated with a different power of ten. If there is no decimal point (we'll deal with those in detail a bit later), the rightmost digit is the "ones" place, to its left the "tens" place, then "hundreds" and so on.
That means that any time we write a number in the usual way, we are implying that each digit is multiplied by the appropriate power of ten, and the results are added together. So a number like 3574 means
3574 = 3×1000 + 5×100 + 7×10 + 4×1
We can make the "powers of ten" aspect of this more explicit by writing them with exponents:
3574 = 3×103 + 5×102 + 7×101 + 4×100
This general pattern holds no matter how large the number gets; we can just keep moving to higher powers of ten.
It is worth taking a moment to highlight the importance of the digit zero here. While 0 does in some sense mean "nothing", we still have to write it when it's in the middle of a number like 305. Why? Because our place-value system means that
305 = 3×102 + 0×101 + 5×100
and in this kind of a notation we need a zero in the tens place to explicitly say "no tens". In a place-value system there's really no way to directly abbreviate
3×100 + 5×1
and the number 35 is certainly not the same thing. (As an aside: There are number systems that do not use place-value notation; you've probably seen Roman numerals, which write 305 as CCCV:
305 = 100 + 100 + 100 + 5
so they don't need an explicit zero, but Roman numerals turn out to be inconvenient for a lot of other reasons.)

What we've seen so far is mostly review of what you learned about numbers in elementary school, but with an added layer of notation that may seem needlessly complex. The reason for that extra notation is to set us up to understand that 10 may be important for historical reasons (and is probably used in part due to the physiology of our hands), but there's nothing mathematically inevitable about using powers of ten for our place-value system. Now we'll see what happens when we use other bases to represent numbers.

## Binary numbers

One of the most important number bases other than 10 is base 2, also called binary. Just as base 10 uses ten digits (0-9) to represent numbers, base 2 uses two digits: 0 and 1. Mathematically, it's important because 2 is the smallest possible base for a place-value system, and mathematics is often about reducing complex systems and problems to more minimal systems. From an electrical engineering standpoint, binary numbers are a good foundation for computers because it's easier to design and build systems where a wire is either "on" or "off" like a light switch—two possible values—rather than having to distinguish three or four or ten different values on the same wire.

So let's apply our understanding of decimal notation to see how binary notation would work. We know we only have the digits 0 and 1 to work with, so what would a binary number like 1101 represent? Following our pattern from before:
1101 = 1×23 + 1×22 + 0×21 + 1×20
If we fill in more conventional representations for the powers of two we have
1101 = 1×8 + 1×4 + 0×2 + 1×1
And in our standard decimal notation, 8+4+1 = 13.

How do we even know that a number like 1101 is in binary? In the preceding paragraph, it was explicitly identified as such, but without some context, it could just as well be decimal (representing one thousand, one hundred and one). As a matter of convention, when working with numbers that may be in different numeric bases, we will often specify the base of a particular number as a subscript after the number. So 11012 is a base 2 number (as analysed above), while 110110 is a base 10 number (one thousand, one hundred and one).

The binary representation of a number is no better or worse than the decimal representation, and while it's correct to say that 11012 represents the number 1310, it is equally true to say that 1310 represents the number 11012. Both are just different ways of writing the same number, although the decimal representation will be more familiar and intuitive to most of us, since we've been working with base ten a lot longer! And because humans will generally find decimal more useful even when some computer application may call for binary, it's important to be able to convert between the two representations.

In our explanation of what binary notation is, we've already seen how to convert from binary to decimal, but here is another example:
Example: Convert 10110102 to base ten.
Solution: Write out the full place-value interpretation of each digit, and solve from there.
 10110102 = 1×26 + 0×25 + 1×24 + 1×23 + 0×22 + 1×21 + 0×20 = 1×64 + 0×32 + 1×16 + 1×8 + 0×4 + 1×2 + 0×1 = 64 + 16 + 8 + 2 = 90
So 10110102 = 9010.
Converting in the other direction is a matter of reversing this process, filling in blanks in the binary place value chart, starting as far left as possible. That is, knowing that a binary number deals with powers of two, you can set up the place-value notation with blanks for the unknown parts, and fill in from there. Reversing the order of the above example:
 9010 = ?×64 + ?×32 + ?×16 + ?×8 + ?×4 + ?×2 + ?×1 = 1×64 + 0×32 + 1×16 + 1×8 + 0×4 + 1×2 + 0×1 = 1×26 + 0×25 + 1×24 + 1×23 + 0×22 + 1×21 + 0×20 = 10110102
How do you know how to fill in the blanks? Start with out the largest power of 2 that "fits" in the original number. Subtract it from the original number, and repeat with the remainder. Continue until the remainder is 0.
Example: Convert 10210 to binary.
Solution: 64 ( = 26) is the largest power of 2 that is less than or equal to 102. Put 1 in the 64’s place and subtract it off.
 10210 = 1×64 + ?×32 + ?×16 + ?×8 + ?×4 + ?×2 + ?×1 (102−64 = 38 remaining)
Repeat, this time using 38. The largest power of 2 less than or equal to 38 is 32 (=25). Put 1 in the 32’s position and subtract it off.
 10210 = 1×64 + 1×32 + ?×16 + ?×8 + ?×4 + ?×2 + ?×1 (38−32 = 6 remaining)
Continuing, 6 – 4 = 2. Put 1 in the 4’s position.
 10210 = 1×64 + 1×32 + ?×16 + ?×8 + 1×4 + ?×2 + ?×1 (6−4 = 2 remaining)
Because we have bypassed the 16's place and the 8's place, the rules of place-value notation require us to fill them in with zeroes:
 10210 = 1×64 + 1×32 + 0×16 + 0×8 + 1×4 + ?×2 + ?×1 (still 2 remaining)
The remaining 2 lets us put a 1 in the 2's place, and then zeroes in any further positions (in this case just the 1's place):
 10210 = 1×64 + 1×32 + 0×16 + 0×8 + 1×4 + 1×2 + 0×1 = 11001102

Try counting in binary, starting with 0. Do you get 0, 1, 10, 11, 100, 101, 110, 111, 1000, …? What is the next number after 1111 in binary? Why?

Just as we use base 10 and base 2 in place-value notation, we could use other bases. In particular, base 16, also called hexadecimal, is used frequently in computer science.

Since 162 = 256, 163 = 4096, and so on, hexadecimal place values increase rapidly!

Let's jump right in and apply the principles we learned earlier, with binary, to convert a base 16 number into base 10.
Example: Convert 15816 to decimal.

Solution:
 15816 = 1×162 + 5×161 + 8×160 = 1×256 + 5×16 + 8×1 = 256 + 80 + 8 = 34410
It's important to remember that just because 158 (without subscript or context) looks just like how we write one hundred and fifty-eight, since we have said it is a hexadecimal representation of a number, the places mean different things and we need to use some conversion process to interpret its value into our more familiar decimal notation. 15816 ≠ 15810.

We're still missing an important piece of hexadecimal notation, though. We've already seen that base 10 has ten distinct digits (0-9) that it can use in each position, and base 2 has two (0 and 1). We can then infer—correctly—that base 16 needs to be able to use sixteen different digits to work; because if 1016 is 1×16+0 or 1610, we need to have something to represent the numbers that are greater than 9 but less than 1016.

The standard convention in this case is to use the first six letters of the alphabet as digits representing the numbers 10-15. So when you add one to 9, in hexadecimal, you end up with A. Add another one and you get B. If you add one to F16, which corresponds to 1510, you get 1016, which corresponds to 1610.

So let's work out an example that makes use of one of these new digits.
Example: Convert 10A316 to base 10.

Solution:
 10A316 = 1×163 + 0×162 + A×161 + 3×160 = 1×4096 + 0×256 + 10×16 + 3×1 = 4096 + 160 + 3 = 425910
This example was carefully selected to involve easy arithmetic (like multiplying by ten), but you might notice that these numbers are in danger of becoming very unwieldy if we wanted to convert a number like 7C916. Hold that thought; a little later we'll see another way to convert from hexadecimal to decimal that might be a little easier.

The importance of hexadecimal to computer science is that although it may be less familiar than decimal, it is much easier for humans to work with than binary; and unlike decimal, it is very easy to convert hexadecimal into binary (and back). Because 16 is itself a power of 2, conversion between base 16 and base 2 can use an algorithm that doesn't work for bases that don't have that relationship.

Each digit in a hexadecimal number corresponds exactly to four digits (bits) in a binary number. Thus to convert from hexadecimal into binary, you can proceed digit-by-digit rather than having to do any division or subtraction or anything like that.
Example: Convert C60516 to base 2.
Solution: Expand each hexadecimal digit to its four-bit binary form. First, C16 is 1210, which (since it is 8+4) we can turn into the four-bit binary sequence 11002:
 C60516 = 1100 ... 2 (605 remaining)
6 is 4+2, so it is 110 in binary, but because each hexadecimal digit corresponds to four binary digits, we must write it as 0110:
 C60516 = 1100 0110 ... 2 (05 remaining)
We process the digits 0 (can't skip it!) and 5 in the same way:
 C60516 = 1100 0110 0000 01012
It is also correct to write the answer without spaces, as 11000110000001012, which is a good reminder that inside the computer there are no "spaces" in this sense. However, when binary numbers are to be processed by humans, they're just a lot easier to work with if they're broken up a little bit (for the same reason we often write a comma to separate groups of three digits in a decimal number). Because of the importance of groups of four, the binary digits are nearly always grouped that way; indeed, one way to think of hexadecimal (and the reason it's so important in computer science) is as a more compact way of writing groups of four bits. That is, "7" in this context is just a fancy way of writing "0111", and "D" just a fancy way of writing "1101".
Converting in the other direction essentially reversing this process with the only distinction being that the grouping should be done right-to-left so that the incomplete group of four, if any, comes first.

Solution: Group the bits by fours, starting on the right. For emphasis, write two leading zeros to round out the first group of bits.
 10110110012 = 10 1101 10012 = 0010 1101 10012 = 2D916

Earlier we mentioned an easier way to convert from hexadecimal to decimal (and we never even showed converting in the other direction). Basically, because the conversion between base 16 and base 2 is so easy, in all but the simplest cases it's almost always easier to do a decimal-hexadecimal conversion (in either direction) by first converting to binary as an intermediate step. That way you never have to multiply by anything but zero and one, and you don't have to separately keep track of powers of sixteen.

The process works in both directions, but first let's use it for the direction we've already seen.
Example: Convert 3C516 into decimal notation.
Solution:
 3C516 = 0011 1100 01012 = 1×29 + 1×28 + 1×27 + 1×26 + 0×25 + 0×24 + 0×23 + 1×22 + 0×21 + 1×20 = 1×512 + 1×256 + 1×128 + 1×64 + 0×32 + 0×16 + 0×8 + 1×4 + 0×2 + 1×1 = 512 + 256 + 128 + 64 + 4 + 1 = 96510
Again, working in the other direction is a matter of reversing the direction of the process.
Example: Convert 64310 into hexadecimal notation.
Solution:
 64310 = 1×512 + ... (643−512 = 131 remaining) = 1×512 + 1×128 + ... (131−128 = 3 remaining) = 1×512 + 1×128 + 1×2 + ... (3−2 = 1 remaining) = 1×512 + 1×128 + 1×2 + 1×1 = 1×29 + 1×27 + 1×21 + 1×20 = 1×29 + 0×28 + 1×27 + 0×26 + 0×25 + 0×24 + 0×23 + 0×22 + 1×21 + 1×20 = 10 1000 00112 = 28316
Don't forget that since hexadecimal numbers don't have to contain the letter-digits A-F, a number written as 283 can be a perfectly good hexadecimal number (though it does not represent two hundred and eighty three). It's especially important in cases like this to indicate clearly that the number is hexadecimal, through a subscript or some other notation.

## Exercises

1. Convert the binary numbers to decimal.
1. 10110
2. 11100111
2. Convert the decimal numbers to binary.
1. 86
2. 131
3. Convert the base 10 numbers to hexadecimal.
1. 68
2. 543
3. 127
4. Convert the hexadecimal numbers to decimal.
1. 10D
2. 345
3. BEEF
5. Convert the base 16 numbers to binary.
1. 53
2. 94B0
3. 3ED
6. Convert the binary numbers to hexadecimal.
1. 1010 0101
2. 0011 0000 1101 1111
3. 1001 0111 0110 1000
7. Convert the base 10 number 36 to
1. binary