Radix point

In mathematics and computing, a radix point (or radix character) is the symbol used in numerical representations to separate the integer part of a number (to the left of the radix point) from its fractional part (to the right of the radix point).[1] "Radix point" applies to all number bases. In base 10 notation, the radix point is more commonly called the decimal point, where the prefix deci- implies base 10. Similarly, the term "binary point" is used for base 2.

In English-speaking countries, the radix point is usually a small dot (.) placed either on the baseline or halfway between the baseline and the top of the numerals.[2] In many other countries, the radix point is a comma (,) placed on the baseline. It is important to know which notation is being used when working in different software programs. The respective ISO standard defines both the comma and the small dot as decimal markers, but does not explicitly define universal radix marks for bases other than 10.

The negative powers of a base or the right side of the raddix can be viewed as similar to the tenths hundredths thousands columns of decimal notation. Whereas the ones column is x to the zero power the powers become negative to the right of the point. The negative powers are the inverse of the positive powers in that they are effectively 1 divided by base to nth power. So in the example there is the 1 halfs column the 1 quarters column and the 1 eights column. In base three the columns to right of raddix start with 1/3s, 1/9s, 1/27s, 1/(3^4)s columns and so forth. See the correlation with x to the negative power of y and 1 divided by x^y. Also see x^(1/3) as opposed to the third root of x.

Examples

• In base 10 (decimal): 13.625

In this example, 13 is the integer to the left of the radix point, and 625 (i.e. 625/1000) is the fractional part to the right.

• In base 2 (binary): 1101.101

The binary number 1101.101 has the following digits:

Power of 2	3	2	1	0		−1	−2	−3
Binary digit	1	1	0	1	.	1	0	1

Hence, its decimal value can be calculated as:

${\displaystyle {\begin{aligned}1101.101_{2}&=1\times 2^{3}+1\times 2^{2}+0\times 2^{1}+1\times 2^{0}+1\times 2^{-1}+0\times 2^{-2}+1\times 2^{-3}\\&=1\times 8+1\times 4+0\times 2+1\times 1+1\times 0.5+0\times 0.25+1\times 0.125\\&=8+4+0+1+0.5+0+0.125\\&=13.625_{10}\end{aligned}}}$

It is now seen that 1101, which is to the left of the radix point, is the binary representation of the decimal number 13. To the right of the radix point is 101, which is the binary representation of the decimal fraction 625/1000 (or 5/8).

See also

• Floating point
• Fixed point
• Positional notation

References

1. Van Verth, James M.; Bishop, Lars M. (2008), Essential Mathematics for Games and Interactive Applications: A Programmer's Guide (2nd ed.), CRC Press, p. 7, ISBN 9780123742971.
2. "International Language Environments Guide". Oracle Corporation Docs. Archived from the original (html) on 15 January 2012. Retrieved 19 December 2018. Great Britain and the United States are two of the few places in the world that use a period to indicate the decimal place. Many other countries use a comma instead. The decimal separator is also called the radix character. Likewise, while the U.K. and U.S. use a comma to separate groups of thousands, many other countries use a period instead, and some countries separate thousands groups with a thin space.