An Etymological Dictionary of Astronomy and Astrophysics
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A → numeral system that has 2 as its base and uses only two digits, 0 and 1. The positional value of each digit in a binary number is twice the place value of the digit of its right side. Each binary digit is known as a bit. The decimal numbers from 0 to 10 are thus in binary 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, and 1010. And, for example, the binary number 11101₂ represents the decimal number (1 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰), or 29. In electronics, binary numbers are the flow of information in the form of zeros and ones used by computers. Computers use it to manipulate and store all of their data including numbers, words, videos, graphics, and music.

→ binary; → number; → system.

A system of numerals for representing real numbers that uses the → base 10. It includes the digits from 0 through 9.

→ decimal; → system.

Same as → numeral system.

→ number; → system.

The conversion of a → number system with a given → base to another system with a different base; such as the conversion of a → decimal number (base 10) to a → binary number system (base 2). In order to convert a number into its representation in a different number base, we have to express the number in terms of powers of the other base. For example, to convert the decimal number 100 to base 3, we must figure out how to express 100 as the sum of powers of 3. We proceed as follows:
1: Divide the decimal number to be converted (100) by the value of the new base (3).
2: Get the remainder from Step 1 (that is 1) as the rightmost digit (least significant digit) of new base number.
3: Divide the quotient of the previous divide (33) by the new base.
4: Record the remainder from Step 3 (0) as the next digit (to the left) of the new base number.
Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes zero in Step 3 (2 and 0).
The last remainder thus obtained (1) will be the most significant digit of the new base number.
Therefore, 100₁₀ = 10201₃.
Conversely, to convert from another base to decimal we must:
1: Determine the column (positional) value of each digit.
2: Multiply the obtained column values (in Step 1) by the digits in the corresponding columns.
3: Sum the products calculated in Step 2. The total is the equivalent value in decimal.
For example, the binary number 1100100 is determined by computing the place value of each of the digits of the number:
(1 × 2⁶) + (1 × 2⁵) + (0 × 2⁴) + (0 × 2³) + (1 × 2²) + (0 × 2¹) + (0 × 2⁰) = 64 + 32 + 0 + 0 + 4 + 0 + 0 = 100.

→ number; → system; → conversion.

A → number system in which the value of each digit is determined by which place it appears in the full number. The lowest place value is the rightmost position, and each successive position to the left has a higher place value. In the → number system conversion, the rightmost position represents the "ones" column, the next position represents the "tens" column, the next position represents "hundreds", etc. The values of each position correspond to powers of the → base of the number system. For example, in the usual decimal number system, which uses base 10, the place values correspond to powers of 10. Same as → place-value notation and → positional notation. See also → number system conversion.

→ positional; → number; → system.