House Address

My twin lives at the reverse of my house number. The difference between our house numbers ends in two. What are the lowest possible numbers of our house numbers?

That number (difference) must be divisible by 9 and end in two. \(\displaystyle \cancel{9, 18, 27, 36, 45, 54, 63,}\) 72 - got it
 
khansaheb said:
My twin lives at the reverse of my house number. The difference between our house numbers ends in two. What are the lowest possible numbers of our house numbers?

That number (difference) must be divisible by 9 and end in two. \(\displaystyle \cancel{9, 18, 27, 36, 45, 54, 63,}\) 72 - got it
Click to expand...
You found the difference is 72, but how to get to the numbers themselves? :)
 
khansaheb said:
Since we are looking for is lowest possible 72 < = N <= 99. A quick spreadsheet gave the answer to be 91
Click to expand...
I found the solution without a spreadsheet. I’ll share my solution in a few days if no one posted by then.
 
This is how I did it.
Firstly, I assumed they were two digit numbers 10a+b and 10b+a where a and b are single digit numbers.
The difference 10a+b - (10b+a) = 9a-9b = 9(a-b) is a multiple of 9 and we are told it ends in a 2.
The smallest number that is a multiple of 9 and ends in a 2 is 72.
So, if 9(a-b)=72, then a-b=8.
Given a and b are both single digit numbers with a difference of 8, they must be 9 and 1 OR 8 and 0.
Hence the solution 91 and 19 OR 80 and 08 but I assume the second option is not allowed because a house number would be written as 8 not 08.
 
Harry_the_cat said:
This is how I did it.
Firstly, I assumed they were two digit numbers 10a+b and 10b+a where a and b are single digit numbers.
The difference 10a+b - (10b+a) = 9a-9b = 9(a-b) is a multiple of 9 and we are told it ends in a 2.
The smallest number that is a multiple of 9 and ends in a 2 is 72.
So, if 9(a-b)=72, then a-b=8.
Given a and b are both single digit numbers with a difference of 8, they must be 9 and 1 OR 8 and 0.
Hence the solution 91 and 19 OR 80 and 08 but I assume the second option is not allowed because a house number would be written as 8 not 08.
Click to expand...
Just for the record, you are being redundant by saying a single digit number. Just saying a digit will suffice. (It just bothers me every time I see that)
 
Steven G said:
Just for the record, you are being redundant by saying a single digit number. Just saying a digit will suffice. (It just bothers me every time I see that)
Click to expand...
Hmmm. In my mind a digit (and a numeral for that matter) are the symbols used to represent numbers, but not numbers themselves. I'd like to hear what others think.
 
Harry_the_cat said:
Hmmm. In my mind a digit (and a numeral for that matter) are the symbols used to represent numbers, but not numbers themselves. I'd like to hear what others think.
Click to expand...
I agree. I would call 12 the decimal representation of a number.
14 (base 8), 1100 (binary), C (hexidecimal), 102.010221... (base [imath]\pi[/imath]), {x | 2x - 15 = 9}, [imath]2 \binom{4}{2}[/imath]
are examples of other representations of the same number.

However, a digit (singular) would be part of a representation.
1 is the 8's digit (in base 8) of 12
4 is the 1's digit (in base 8) of 12

etc.

That's the way I learned it, and this agrees with the Merriam-Webster definition, anyway:
digit: any of the Arabic numerals 1 to 9 and usually the symbol 0.

Wikipedia defines the Mathematical use here. It's essentially the same definition as Merriam-Webster.

Specific sources might disagree, but I would think this might be called a standard definition.

-Dan
 
Last edited:
Steven G said:
Just for the record, you are being redundant by saying a single digit number. Just saying a digit will suffice. (It just bothers me every time I see that)
Click to expand...
It's not a bit redundant.

"Digit" is about the symbols, "number" is about the quantity denoted, and "single-digit number" means "a number that is written with one symbol, so that the number is between 0 and 9 inclusive" (or maybe you wouldn't include 0). This assumes that all numbers are conventionally written in base ten, of course, as they are.

If she had said "a and b are digits", that would have been wrong, because a and b are variables representing numbers, and are not themselves digits.

Another valid way to say, "two digit numbers 10a+b and 10b+a where a and b are single digit numbers." would have been "two-digit numbers 10a+b and 10b+a, where a and b are the values of their digits."
 
I think that we need to recognize that the vocabulary around numbers and numerals is a product of history and so is sometimes confusing.

To prevent confusion, I think what we should say when we NEED to be exact is:

A numeral system is a set of symbols, called digits, and rules for combining digits to represent numbers exactly and concisely. There are several numeral systems in use. If the type of numeral system is not specified explicitly or implicitly by context, what is intended is the decimal numeral system, which is the most commonly used such system. The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

A numeral is a single digit or an ordered sequence of digits that represents a unique number in a numeral system. If the type of numeral system is not specified explicitly or implicitly by context, what is intended is the decimal numeral system.

I do not claim those as descriptive definitions in normal speech but suggest them as presecriptive definitions for technical speech.

Thus digits and numerals are technically symbols rather than numbers, and a digit may be used as a numeral or as a component of a numeral.

But in normal speech, there is usually no need to be very careful. There will seldom be confusion if we write "17 is the number named 'seventeen'" even though what is technically correct is to say "17 is a concantenation of digits that, in the decimal numeral system, form the numeral representing the number named 'seventeen'." Nor do I think there is likely to be confusion if we say in most contexts "one-digit number" instead of saying "non-negative integer less than ten."
 
JeffM said:
what is technically correct is to say "17 is a concatenation of digits that, in the decimal numeral system, form the numeral representing the number named 'seventeen'."
Click to expand...
On the other hand, it might be more correct to say "'17' is a concatenation of digits ...; 17 is the number represented by that numeral".

I say that in the spirit of Lewis Carroll:

en.wikipedia.org

Haddocks' Eyes - Wikipedia

en.wikipedia.org en.wikipedia.org

Use–mention distinction - Wikipedia

en.wikipedia.org en.wikipedia.org
 
You must log in or register to reply here.
Top