Numeration systems

Numeration systems, or number systems, method of arranging and representing numbers. The most familiar and widely used system today is the decimal system. Based on innovations of Hindu mathematicians of the 4th and 3rd centuries B.C., and introduced to Europe by the Arabs, it uses 10 numerals or digits, 0 through 9, which stand for different values according to whether they represent 1s, 10s, 100s, or other powers of 10. For this reason the decimal system is also called “numbers to the base 10.” Originally, numerals 1–9 were combined with words to accomplish this. The number 263 might be read as “2 hundreds, 6 tens, and 3 ones.” With the Hindu invention of the zero c.A.D. 600, a more efficient method became possible. Here the position of the digit determines its value. In the number 200, 2 zeros are needed to hold the positions of the tens and ones, so that the digit 2 can be correctly interpreted as 2 hundreds. This system, disseminated by Arab mathematicians, reached Europe in the 12th century. The decimal system was later extended to include a representation of fractions. Positions of digits representing fractions are separated by those representing integers (whole numbers) by a decimal point. Thus, while values to the left of the decimal point represent 10⁰ (ones), 10¹ (tens), 10² (hundreds), and so on, values to the right of the decimal point represent 10⁻¹ (1/10 or tenths), 10⁻² (1/100 or hundredths), 10⁻³ (1/1,000 or thousandths), and so on. Numeration systems can be based on numbers other than 10 while applying the same principle of positional value. The binary system or base 2, for example, uses only 2 digits, 0 and 1; positional values are based on powers of 2. For instance: in the binary system 1011 means (1 × 2⁰) + (1 × 2¹) + (0 × 2²) + (1 × 2³) = 1 + 2 + 0 + 8 = 11. Invented by the German mathematician Gottfried Wilhelm Leibniz (1646–1716), the binary system became important in the development of computers, because “on” and “off switches or electrical circuits could be used to represent 0 and 1. The hexadecimal system, base 16, uses 16 digits (commonly, 0–9 and A-F), with place values equal to powers of 16. It is also of importance in computers. The number system of the ancient Babylonians, which had a base 60, survives in the division of the hour into 60 minutes, each divided into 60 seconds. It is also reflected in the degrees, minutes, and seconds in which angles may be measured.