# Don’t Miss These Subtleties

This is a list of subtleties. These may appear to be *trivial. However, they are actually very important–upon which, seemingly more interesting facts/concepts are built.. The successful student not only knows these facts automatically, but can also explain why.

1. 100% is 1.
• This is consistent (well, it should be, all math is consistent) with the idea that to turn a percent into a decimal you move the decimal point 2 places.
• Percents bigger than 100% are numbers bigger than 1.  For example 150% is 1.5 (which is one and a half).
2. $\sqrt 1 = 1$
• This is true for the same reason as $\sqrt9 = 3, 3^2=9$$1^2=1$.
• Note (a.)  n<1 then $\sqrt n > n$ and (b.) n>1 then $\sqrt n < n$
3. $\frac{n}{n}=1$
• This is the whole pizza, cut into pieces, with no pieces eaten!
• This makes $1- \frac{2}{7}$ easy.  It is $\frac{5}{7}$.
4. The decimal point in a whole number is to the right of the number.  5 = 5.  Don’t ever doubt this.
• The number immediately to the left of the decimal point is the units place (the ones place).
5. $1\% = .01= \frac{1}{100}$
• 1% means 1 out of a 100.  Everything is built on that.
6. $x^0=1$ (where $x\neq 0$)
• See the diagram at the right.
• Also, 1 is the starting point for multiplication. (0 is the starting point for addition.) That is, if you’ve multiplied 0 times (you haven’t multiplied yet) then you are at 1.
7. $x=1x=\frac{x}{1}$
• I actually prefer to not use 1x and x over 1. However, it is good to always remember that the coefficient of x is 1 and that x can be written as $\frac{x}{1}$.
• It is also a good reminder that when something is multiplied ‘on the side’ of a fraction, it is actually multiplied on the top.
• For example, $x\cdot\frac{a}{b}=\frac{xa}{b}$

*These facts have a tendency to appear so simple they don’t offer much to think about. The student should ask (as they always should) WHY and find ways to connect the idea to other ideas.