Standard notation needed for binary numbers as exponents

In summary, this expert summarizer found that using hexidecimal notation along with arithmetic operations can make multiplication and division easier to understand. They also found that using 1011 for a (four-bit) binary number is very widely used and conveys the same information as the {3, 1, 0} notation.
  • #1
coolul007
Gold Member
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I work a lot in binary. I am organizing some of my work and need a way to write expressions. I can always create my own notation, but i would rather not invent something that already exists.
1011 is binary for 11 base 10. I use this {3,1,0} to represent the binary with just exponents. When I multiply this is the notation I use: {3,1,0}(1) ={4,2,1} This is ok , but division is awkward, {4,2,1}/(1) = {3,1,0} Any help will be appreciated.
 
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  • #3
Actually, I was looking for a better way to show the operation of multiplication and division. I didn't make that clear
 
  • #4
whats wrong with using the arithmetic operators and hex notation?

0xA * 0xB = 10 * 11 = 110 = 0x6E
 
  • #5
do you also have a problem when you multiply with a non power of two number

(3,1,0) * (3,1,0) = (3,1,0) * 3 + (3,1,0) * 1 + (3,1,0) * 0 = (6,4,3) + (4,2,1) + (3,1,0) = (6,4,3,2,1,0) ?

It seems like I'm trying to do things in Roman numerals with this notation so I'm sure folks are going to get real confused.

This is just my thought though you should still explore its pros and cons.
 
  • #6
coolul007 said:
I work a lot in binary. I am organizing some of my work and need a way to write expressions. I can always create my own notation, but i would rather not invent something that already exists.
1011 is binary for 11 base 10. I use this {3,1,0} to represent the binary with just exponents. When I multiply this is the notation I use: {3,1,0}(1) ={4,2,1} This is ok , but division is awkward, {4,2,1}/(1) = {3,1,0} Any help will be appreciated.
What's wrong with 1011 for a (four-bit) binary number? That notation is very widely used, and conveys exactly the same information as your {3, 1, 0} notation. Both forms of notation represent ##1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0##. The former notation writes the coefficients, and keeps track of the exponents by the position in the bitstring. Your notation keeps track of the exponents of the terms with nonzero coefficients.
 
  • #7
coolul007 said:
Actually, I was looking for a better way to show the operation of multiplication and division. I didn't make that clear
Using bitstrings (as in the usual notation for binary numbers) multiplication is performed analogously with how it is performed in decimal arithmetic when multidigit numbers are multiplied. The only facts that are needed though, are 0 * 0 = 0, 0 * 1 = 1, 1 * 0 = 0, and 1 * 1 = 1. Division also works analogously with how it is performed with decimal numbers.
 
  • #8
Time to end this thread. I am not interested in binary vs other bases,etc. I wanted to know if there existed notation analgous to set theory for strictly dealing with operations on exponents. I chose binary as there are no coefficients. This makes operating with numbers, quicker and more straight forward. I am also working with very large integers, thousands to millions of bits. On average eliminating the zero bits cuts the work in half.
 
  • #9
What if in your notation you used a -1 for the division?

What kind of work are you doing that requires arithmetic operations on very large binary numbers?
 
  • #10
I'm programming a PowerMod function and factoring large numbers. PowerMod requires squaring of numbers. Squaring is a "pig" operation with large numbers in a computer. The key is to make the operation as quick as possible. The notation question arises because to explain my squaring method I needed something to demonstrate the method in a paper. It is easy to program without notation, I was just wondering if such notation already existed. I can always continue using my own.
 

1. What is standard notation for binary numbers as exponents?

The standard notation for binary numbers as exponents is a way to express a binary number as a power of 2. For example, the binary number 101 can be written as 2^2 + 2^0, which is equivalent to 4 + 1 = 5.

2. How do you convert a binary number to standard notation?

To convert a binary number to standard notation, you need to first identify the positions of the 1s in the binary number. Then, you can write the binary number as a sum of 2 raised to the power of the positions of the 1s. For example, the binary number 1101 can be written as 2^3 + 2^2 + 2^0, which is equivalent to 8 + 4 + 1 = 13.

3. What is the significance of using standard notation for binary numbers as exponents?

Standard notation for binary numbers as exponents is important in computer science and digital electronics. It allows us to represent large binary numbers in a more concise and efficient way, making computations and logic operations easier to perform.

4. Can you use standard notation for fractional binary numbers?

Yes, standard notation can also be used for fractional binary numbers. In this case, the exponent represents the negative power of 2. For example, the binary number 0.101 can be written as 2^-1 + 2^-3, which is equivalent to 0.5 + 0.125 = 0.625.

5. Are there any limitations to using standard notation for binary numbers as exponents?

One limitation of using standard notation for binary numbers as exponents is that it can only represent positive powers of 2. This means that it cannot be used to represent negative numbers or fractions with negative exponents, as these would require the use of negative powers of 2.

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