how to read binary numbers?

Question

Using decimal representation for numbers,

we read 10 as ten (not as one, zero) we read 1011 as one thousand eleven (not as one, zero, one, one digit by digit).

But using binary representation, how 10 and 1011 should be read? Is digit by digit the only way?

Answer 1

In any discussion of numeral systems, I think it’s important to distinguish between numbers and numerals.

• The number whose name is “ten” is a mathematical object.
• The numeral "10" is a symbol.

When a symbol is used to signify an object or idea, we often read it by saying the name of the signified. For example, I might read a sign with the symbol "🚭" as "no smoking." Similarly, without any additional context, I might read the sentence "10 is an even number" as "ten is an even number."

Following this method, if the context is that we are working in base-2, then it would make sense to read the sentence "10 is an even number" as "two is an even number."

When we want to refer to a symbol itself (often indicated using quotation marks around the symbol), then we often read the name of the symbol, or we describe the symbol. For example, I might read the sentence "'10' has two digits" as "the numeral one-zero has two digits."

There is another method of reading that is specific to numerals used in place value systems. Its purpose is primarily pedagogical. From left to right, we read each digit, followed by its corresponding place value. I've often heard this called "expanded form." For example, in the context of base-2, I might read "10" as "one two and zero ones."

Others can probably suggest even more methods of reading numerals. To summarize, here are the three I've described, applied to the numeral "1011" in the context of base-2.

• Read the name of the number: "eleven."
• Describe the symbol: "one-zero-one-one."
• Read the number in expanded form: "one eight, zero fours, one two, and one one."

Answer 2

There is not solid standardization for this terminology. But one convention is this

\begin{array}{|c|c|c|} \hline \text{1} & \text{one} & \text{bit} \\ \hline \text{10} & \text{two} & \text{crumb} \\ \hline \text{100} & \text{four} & \text{nibble} \\ \hline \text{1000}& \text{eight} & \text{byte} \\ \hline \text{10000}& \text{sixteen} & \text {word}\\ \hline \text{100000}& \text{thirty-two} & \text {dword}\\ \hline \end{array}

and this continues as qword, dqword.

So 101 could be read as "nibble-bit" and 1010 would be read as "byte-crumb." This all feels a little bit like Jabberwocky.

One source for this convention is this Wolfram page

Also see these Microsoft references: word dword qword

This answer has received some down votes based on the objection that this terminology is used typically when describing memory allocations. For example, a byte is any eight bit expression: 00000000 through 11111111. This is a fair objection. However, mathematicians often think of integers as the union of their predecessors, so $$8=\{7,6,5,4,3,2,1,0\}$$

Also I have see undergraduate computer science majors work as mentors in CS Scratch Clubs to help elementary and middle school children learn. I have seen them use base-two blocks and refer to them as bits, crumbs, nibbles, bytes. Kids giggled when they made up things like taking nibble out of bite and being left with two crumbs, etc. They had fun with the language and were learning place value. So in spite of the technical objection described above, the approach does have efficacy in teaching mathematics.