# Concepts of Modern Mathematics (Ian Stewart) - 751=7.107+2?

In Chapter 3 Ian Stewart talks about Short Cuts in the Higher Arithmetics, one section is on modular arithmetics.

When talking about the days of the week;
$0= \text{Sunday}\\ 1= \text{Monday}\\ 2= \text{Tuesday}\\ 3= \text{Wednesday}\\ 4= \text{Thursday}\\ 5= \text{Friday}\\ 6= \text{Saturday}\\ 7= \text{Sunday}\\ \vdots\\ 7n= \text{Sunday}\\ 7n+1= \text{Monday}\\ 7n+2= \text{Tuesday}\\ 7n+3= \text{Wednesday}\\ 7n+4= \text{Thursday}\\ 7n+5= \text{Friday}\\ 7n+6= \text{Saturday}\\ 7n+7= 7(n+1) = 7n = \text{Sunday}$

In a simple statement $4+5=2 \dfrac bc$ the cycle (day $9$ is the same as day $2$).

In a later statement,

"What is $751$ days after Thursday" we rephrase it as $4+751 = ?$ We can observe that $751=7.107+2$

That is where I am getting lost. What is Ian Stewart doing to make this a true statement? I've attached the three pages on this topic, the statement in question is towards to bottom of page 3.

Thank you!!!

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That is a typo in the book. It should say:

Now 751 isn't in our table, but we observe that $$751 = 7\cdot107+2$$ etc.

Rather than a decimal point, that should be a multiplication symbol.

EDIT: After some Google-ing, I've found that some cultures use "$.$" for multiplication and "$\cdot$" for the decimal point. Thus, the expression could also be written: $$751 = 7\times 107+2 \quad \text{or}\quad 751 = (7)(107)+2$$

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Is your "it should say" not the same as it says originally? Certainly I can't see a difference at a glance so it might be worth clarifying the difference... Though having typed this is it that the . is slightly higher? Either way +1 for the last line which says all that needs saying. :) – Chris Sep 1 '13 at 16:10
@Chris Perhaps this is a USA convention compared to other countries, but my understanding is that a vertically centered dot is standard for multiplication, and a dot in line with the bottom of the line represents a decimal point (and not multiplication). – apnorton Sep 1 '13 at 16:47
my comment was mainly because it took me a while to notice the difference between what you'd written and the original so may be worth spelling it more explicitly. As for canonicity, I'm not sure. When writing stuff down by hand I'd tend to put mulitplication dots at the bottom and decimal points in the middle but not sure if there is a different convention for print (there often is) and I'm in thailand at the moment without any maths books to hand to check what they do. :) I suspect the problem though is that the OP had not seen a dot as a multiplication symbol before in any position. :) – Chris Sep 1 '13 at 16:51
@Chris Good point. I've added some other forms. :) – apnorton Sep 1 '13 at 17:00

He is saying that since $751=7\times 107+2$, the day is two days after Thursday which is Saturday. The dot means multiplication here.

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Note that $7\times 107=749$ - so $749$ days represents exactly $107$ weeks, and the $750^{th}$ day falls on the same day of the week as the $1^{st}$ day.

$751$ days after Thursday is $107$ weeks and two days. Thursday is the fourth day of the week, so the answer is the sixth day of the week (two days on).

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