The base-2 (binary) number system is particularly important in computing. It is also helpful to be familiar with number systems that are powers of 2, such as base 8 (octal) and base 16 (hexadecimal). Recall that the base value specifies the number of digits in the number system. Base 10 has ten digits (0–9), base 2 has two digits (0–1), and base 8 has eight digits (0–7).

Therefore, the number 943 could not represent a value in any base less than base 10, because the digit 9 doesn’t exist in those bases. It is, however, a valid number in base 10 or any base higher than that. Likewise, the number 2074 is a valid number in base 8 or higher, but it simply does not exist (because it uses the digit 7) in any base lower than that.

What are the digits in bases higher than 10? We need symbols to represent the digits that correspond to the decimal values 10 and beyond. In bases higher than 10, we use letters as digits. We use the letter A to represent the number 10, B to represent 11, C to represent 12, and so forth. Therefore, the 16 digits in base 16 are:

• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Let’s look at values in octal, hexadecimal, and binary to see what they represent in base 10. For example, let’s calculate the decimal equivalent of 754 in octal (base 8). As before, we just expand the number in its polynomial form and add up the numbers.

Let’s convert the hexadecimal number ABC to decimal:

Note that we perform the exact same calculation to convert the number to base 10. We just use a base value of 16 this time, and we have to remember what the letter digits represent. After a little practice you won’t find the use of letters as digits that strange.

Finally, let’s convert a binary (base-2) number 1010110 to decimal. Once again, we perform the same steps—only the base value changes:

Recall that the digits in any number system go up to one less than the base value. To represent the base value in any base, you need two digits. A 0 in the rightmost position and a 1 in the second position represent the value of the base itself. Thus 10 is ten in base 10, 10 is eight in base 8, and 10 is sixteen in base 16. Think about it. The consistency of number systems is actually quite elegant.

Addition and subtraction of numbers in other bases are performed exactly like they are on decimal numbers.

Recall the basic idea of arithmetic in decimal: 0 + 1 is 1, 1 + 1 is 2, 2 + 1 is 3, and so on. Things get interesting when you try to add two numbers whose sum is equal to or larger than the base value—for example, 1 + 9. Because there isn’t a symbol for 10, we reuse the same digits and rely on position. The rightmost digit reverts to 0, and there is a carry into the next position to the left. Thus 1 + 9 equals 10 in base 10.

The rules of binary arithmetic are analogous, but we run out of digits much sooner. That is, 0 + 1 is 1, and 1 + 1 is 0 with a carry. Then the same rule is applied to every column in a larger number, and the process continues until we have no more digits to add. The example below adds the binary values 101110 and 11011. The carry value is marked above each column in color.

We can convince ourselves that this answer is correct by converting both operands to base 10, adding them, and comparing the result: 101110 is 46, 11011 is 27, and the sum is 73. Of course, 1001001 is 73 in base 10.

The subtraction facts that you learned in grade school were that 9 – 1 is 8, 8 – 1 is 7, and so on, until you try to subtract a larger digit from a smaller one, such as 0 – 1. To accomplish this feat, you have to “borrow one” from the next left digit of the number from which you are subtracting. More precisely, you borrow one power of the base. So, in base 10, when you borrow, you borrow 10. The same logic applies to binary subtraction. Every time you borrow in a binary subtraction, you borrow 2. Here are two examples with the borrowed values marked above.

Once again, you can check the calculation by converting all values to base 10 and subtracting to see if the answers correspond.

Binary and octal numbers share a very special relationship: Given a number in binary, you can read it off in octal; given a number in octal, you can read it off in binary. For example, take the octal number 754. If you replace each digit with the binary representation of that digit, you have 754 in binary. That is, 7 in octal is 111 in binary, 5 in octal is 101 in binary, and 4 in octal is 100 in binary, so 754 in octal is 111101100 in binary.

To facilitate this type of conversion, the table below shows counting in binary from 0 through 10 with their octal and decimal equivalents.

To convert from binary to octal, you start at the rightmost binary digit and mark the digits in groups of threes. Then you convert each group of three to its octal value.

Let’s convert the binary number 1010110 to octal, and then convert that octal value to decimal. The answer should be the equivalent of 1010110 in decimal, or 86.

The reason that binary can be immediately converted to octal and octal to binary is that 8 is a power of 2. There is a similar relationship between binary and hexadecimal. Every hexadecimal digit can be represented in four binary digits. Let’s take the binary number 1010110 and convert it to hexadecimal by marking the digits from right to left in groups of four.

Now let’s convert ABC in hexadecimal to binary. It takes four binary digits to represent each hex digit. A in hexadecimal is 10 in decimal and therefore is 1010 in binary. Likewise, B in hexadecimal is 1011 in binary, and C in hexadecimal is 1100 in binary. Therefore, ABC in hexadecimal is 101010111100 in binary.

Rather than confirming that 101010111100 is 2748 in decimal directly, let’s mark it off in octal and convert the octal.

Thus 5274 in octal is 2748 in decimal.

In the next section, we show how to convert base-10 numbers to the equivalent number in another base.

The rules for converting base-10 numbers involve dividing by the base into which you are converting the number. From this division, you get a quotient and a remainder. The remainder becomes the next digit in the new number (going from right to left), and the quotient replaces the number to be converted. The process continues until the quotient is zero. Let’s write the rules in a different form.

These rules form an algorithm for converting from base 10 to another base. An algorithm is a logical sequence of steps that solves a problem. We have much more to say about algorithms in later chapters. Here we show one way of describing an algorithm and then apply it to perform the conversions.

The first line of the algorithm tells us to repeat the next three lines until the quotient from our division becomes zero. Let’s convert the decimal number 2748 to hexadecimal. As we’ve seen in previous examples, the answer should be ABC.

The remainder (12) is the first digit in the hexadecimal answer, represented by the digit C. So the answer so far is C. Since the quotient is not zero, we divide it (171) by the new base.

The remainder (11) is the next digit to the left in the answer, which is represented by the digit B. Now the answer so far is BC. Since the quotient is not zero, we divide it (10) by the new base.

The remainder (10) is the next digit to the left in the answer, which is represented by the digit A. Now the answer is ABC. The quotient is zero, so we are finished, and the final answer is ABC.

Although some of the early computers were decimal machines, modern computers are binary machines. That is, numbers within the computer are represented in binary form. In fact, all information is somehow represented using binary values. The reason is that each storage location within a computer contains either a low-voltage signal or a high-voltage signal. Because each location can have only one of two states, it is logical to equate those states to 0 and 1. A low-voltage signal is equated with a 0, and a high-voltage signal is equated with a 1. In fact, you can forget about voltages and think of each storage location as containing either a 0 or a 1. Note that a storage location cannot be empty: It must contain either a 0 or a 1.

Each storage unit is called a binary digit, or bit for short. Bits are grouped together into bytes (8 bits), and bytes are grouped together into units called words. The number of bits in a word is known as the word length of the computer. For example, IBM 370 architecture in the late 1970s had half words (2 bytes or 16 bits), full words (4 bytes), and double words (8 bytes). Modern computers are often 32-bit machines or 64-bit machines.

We have much more to explore about the relationship between computers and binary numbers. In the next chapter, we examine many kinds of data and see how they are represented in a computer. In Chapter 4, we see how to control electrical signals that represent binary values. In Chapter 6, we see how binary numbers are used to represent program commands that the computer executes.