Every society has had to create a method of counting. Over the centuries there have been many such systems. The most common number system today uses ten symbols called digits—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—and combinations of these symbols.

Kinds of numbers

Numbers can be classified in many ways. The simplest class is the counting, or natural, numbers (1, 2, 3, …). With the addition of 0 these are known as the whole numbers. For each of the natural numbers there is a corresponding negative number (-1, -2, -3, …). The natural numbers, their negative equivalents, and 0 constitute the integers. The natural numbers are also called positive numbers because they are greater than 0, while the negative numbers are less than 0. The set of integers may be pictured as the set of points on a line that continues forever in either direction:

The set of real numbers consists of all the positive numbers, all the negative numbers, and 0. In addition to the integers, these include the fractions, which represent parts of a whole. They are written using the same symbols used to write integers, but the symbols are used in a different way. Common fractions are written as digits separated by a line, as in ²/₃ or ²⁴⁵/₃₅₆. In a common fraction, the number below the line is called the denominator, and the number above the line is the numerator. In reading a common fraction, the numerator is stated first. Thus, ²/₃ is read as two thirds. Fractions are very helpful because they make possible measurements in other than whole numbers. They can also be indicated on a number line:

Fractions can also be written as decimals. To change a common fraction into a decimal, one must divide the numerator by the denominator. In this way, ³/₄ can be changed into the decimal 0.75. Not all common fractions, however, can be changed into such precise decimals: ²/₃ as a decimal is an endless series of sixes to the right of the decimal point: 0.6666….

Ancient number systems

The first system of numbers was probably the tally system, which involved making a separate mark for every item being counted. This system is practical for dealing with small numbers only. In more complicated numbering systems, ten is often a key number. This is because people have ten fingers and often use them in counting. In fact, the word digit comes from digitus, which means “finger” in the Latin language. However, not all number systems used ten as a base. The ancient Babylonians used 60 as their base, and the Maya used 20.

The ancient Egyptians developed a complex system for writing large numbers in hieroglyphics. There was a single hieroglyphic symbol for the number 1,000. To write the number 999, however, it was necessary to write the symbol for 100 nine times, then the symbol for 10 nine times, and finally the symbol for 1 nine times—27 symbols in all.

Later the ancient Romans used letters to represent numbers—I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, and M for 1,000. In this system, known as Roman numerals, CCLVI means 256. The Romans used what is called the subtraction principle to shorten their numbers. For example, CMXCIX means 999 because the first C subtracts 100 from the M, the first X subtracts 10 from the second C, and the I subtracts 1 from the second X.

Base-ten and other systems

Our number system today is what is called a base-ten system. This means that there are ten digits (0–9) that can be arranged in various combinations to write any number. Further, each “place” or column in a number stands for a different factor of ten—the “ones place,” the “tens place,” the “hundreds place,” and so forth. In the number 456, for example, the 4 is in the hundreds place, the 5 is in the tens place, and the 6 is in the ones place. Written in another way, the number 456 actually represents (4 × 100) + (5 × 10) + (6 × 1). This system dates back to the Arabs of the Middle Ages. For this reason, the digits 0–9 are called Arabic numerals.

For some purposes, other number systems are more useful than base-ten. Computers, for example, use a base-two (or binary) number system. Instead of ten digits, this system uses only two—0 and 1. In a computer these numbers stand for “off” and “on,” the only two possible states of the computer's electric circuits. As in the base-ten system, the digits in the base-two system can be arranged to represent any number. The base-two system, however, is based on multiples of two (2, 4, 8, 16, 32, and so forth) instead of multiples of ten (10; 100; 1,000; 10,000; 100,000; and so forth).

The use of fewer digits in the base-two system than in the base-ten system means that more places are required to write numbers. In base-two notation, as in base-ten, the number one is represented by the digit 1. Because only 0 and 1 are used in base-two, however, every number above one requires more than one place. The first place is 1. Each place after that represents a multiple of 2. The number two, for example, requires two places and is written as 10. Written in another way using base-ten digits, this means (1 × 2) + (0 × 1). Similarly, three is written as 11, meaning (1 × 2) + (1 × 1); four is written as 100, meaning (1 × 4) + (0 × 2) + (0 × 1); five is written as 101, meaning (1 × 4) + (0 × 2) + (1 × 1); six is written as 110, meaning (1 × 4) + (1 × 2) + (0 × 1); seven is written as 111, meaning (1 × 4) + (1 × 2) + (1 × 1); eight is written as 1000, meaning (1 × 8) + (0 × 4) + (0 × 2) + (0 × 1); and so forth. The number 999 is written as 1111100111, meaning (1 × 512) + (1 × 256) + (1 × 128) + (1 × 64) + (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (1 × 1).