Questions tagged [number-systems]

Representations of numeric values in decimal, binary, octal, hexadecimal, and other bases; one's-complement and two's-complement signed numbers; scientific notation; floating-point numbers in digital computers; history of number systems; nonstandard number systems; algorithms for arithmetic within specific number systems or for conversions between number systems.

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Converting a number from base3 to base2 without going through base 10.

Could anyone guide me on how to directly convert a number from base 3 to base 2 without using base 10? For example, converting "2101" (base 3) directly to base 2. Any suggestions, method or ...
• 103
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write the following in sigma notation $a^k+a^{2k} +a^{4k} +a^{8k} +a^{{16}k} ...+a^{{256}k}$

I see the pattern being $2k$ on the power, and I thought the formula would just be $a^{(2k)}$, but i feel like im missing something else here ? Why is it not The summation from $a=1$ to $128 = a^{2k}$...
120 views

Binary with digit 2 allowed

How many ordered 11-tuples $(a_0,a_1,a_2,...,a_{10})$ of integers satisfy the equation $$a_0+2a_1+2^2a_2+...+2^{10}a_{10}=2020$$ where $0\le a_i\le 2$ for all $0\le i\le10$ AMC Mock If $f(n)$ is the ...
• 1,680
1 vote
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What if the number ten was a single digit in the base 10 number system? How would that work? [closed]

I've recently been studying about math, and when learning about decimals. I found myself with this question. An explanation I got from Quora was: Arithmetic operations such as addition, subtraction, ...
• 11
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Reading the book Analysis I (Fourth Edition), by Terence Tao, I found, on page 78, Lemma 4.1.3 (Addition and multiplication are well-defined.) The Lemma says: Let a, b, a', b', c, d natural numbers. ...
• 4,220
48 views

the logic of numeral systems [closed]

i have a question about numeral systems that use the place value concept , in the decimal system the position is a power of ten and it has set of ten unique digits, the ...
• 21
178 views

Digit Sum for Base 2 Alternate Definition

I was looking at the digit sum definition, but I also saw a simpler version that can be used for binary numbers. I'm trying to figure out and understand how the generic formula for any base can be ...
• 131
119 views

Ancient Greek Mathematical Notation

While reading the CMU FLAC overview, I stumbled upon the above image. While I have seen Greek numerals before, I had no idea that there was an entire algebraic symbology developed in Greek (though ...
55 views

Number systems and ring Theory

Are different base number systems defined as separate rings or single ring e.g Z? asking question in other words decimal numeral system different fundamentally from unary or binary or ternary or any ...
• 9
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Does $n = n.\overline 01?$ Similar to $(n-1).\overline 9 = n$ which is just coming form the other direction of $n$.

Just like what the title says. Does: $$n = n.\overline 01?$$ For example, $1.\overline 01 = 1$? Similar to $(n-1).\overline 9 = n$, for example, $0.\overline 9 = 1$. The last statement is true but ...
• 119
437 views

Similarities in the digits of the powers of 2 and 5

Many may have noticed that the negative powers of 5 contain the same digits as the positive powers of 2: This pattern intrigued me. I started to wonder if it exists in different number bases. I soon ...
• 87
61 views

Using modular arithmetic to explain end around carry

I understand that end-around carry occurs because the diminished-radix complement coding has two representations of zero, but I'm having trouble understanding how it follows from the rules of modular ...
• 673
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Are there number systems with fractional or irrational bases?

I'm wondering if there are number systems with bases other than integers? For example, with a fractional, imaginary, irrational, transcendental basis, or with the basis "infinity"? If there ...
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• 958
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Determining if a positional notation can express every natural number

Recently I've researched some positional notations. However, some of them I've invented can't express every natural number. I want to determine which of them can express every natural number, and ...
99 views

Equation with numbers and subscripts

I have no clue how to find the answer in the underlined spot? What does the subscript $9$, $6$ and $3$ mean here? $$888_9+555_6+222_3= \_\_\_\_\_\_\_\_ {}_{12}$$
89 views

Number system where negative numbers are represented as letters

Forgive me for a possibly dumb question, I'm very shallow at math, but is there any number system (or I don't know, it looks more like a number mask system) in which negative numbers are represented ...
72 views

In the unary numeral system, how are the numbers between 1 and -1 labelled?

In the unary numeral system, does 1 - 1 = -1 ? Because if this is true, then how would the numbers between 1 and -1 be represented? Where on a number line would .1 and -.1 be? I am not necessarily ...
325 views

Correct way to find 8's complement?

I'm trying to figure out how to find 8's complement, but am unable to do so. Online it shows that we need to subtract each digit from 7 when the number is in it's octal bit. It gives the 7's ...
97 views

Is there a classification system where the sum of the digits are a factor?

This may be confusing. An example: 12 1+2=3 3 is a factor of 12. 135 1+3+5=9 9 is a factor of 135. 555 5+5+5=15 15 is a factor of 555. What is the name of this classification system or did I create a ...
Negative golden ratio $(–φ)$ as a number system base?
As negative numbers can be used as bases for numeral systems (e.g. negadecimal), and non-integers such as the golden ratio $φ$ can also be used as bases, I have tried to find information on whether ...