270 MCGRAW-HILL’S SAT
Integers and Real Numbers
On the SAT, you only need to deal with two kinds of
numbers: integers(the positive and negative whole
numbers,... , 3, 2, 1, 0, 1, 2, 3,.. .) and real num-
bers(all the numbers on the number line, including
integers, but also including all fractions and deci-
mals). You don’t have to know about wacky numbers
such as irrationalsor imaginaries.
The SAT only uses real numbers. It will never
(1) divide a number by 0 or (2) take the square
root of a negative number because both these
operations fail to produce a real number.
Make sure that you understand why both these
operations are said to be “undefined.”Don’t assume that a number in an SAT prob-
lem is an integerunless you are specifically told
that it is. For instance, if a question mentions
the fact that x> 3, don’t automatically assume
that xis 4 or greater. If the problem doesn’t
say that xmust be an integer, then xmight be
3.01 or 3.6 or the like.The Operations
The only operations you will have to use on the SAT
are the basics: adding, subtracting, multiplying, divid-
ing, raising to powers, andtaking roots. Don’t worry
about “bad boys” such as sines, tangents, or
logarithms—they won’t show up. (Yay!)
Don’t confuse the key words for the basic op-
erations: Summeans the result of addition, dif-
ference means the result of subtraction,
productmeans the result of multiplication,
and quotientmeans the result of division.The Inverse Operations
Every operation has an inverse, that is, another oper-
ation that “undoes” it. For instance, subtracting 5 is
the inverse of adding 5, and dividing by 3.2 is the in-
verse of multiplying by 3.2. If you perform an oper-
ation and then perform its inverse, you are back to
where you started. For instance, 135 4.5 ÷ 4.5 135.
No need to calculate!
Using inverse operations helps you to solve
equations. For example,
3 x 7 38
To “undo” 7, add 7 to both sides: 3 x 45
To “undo” 3, divide both sides by 3: x 15Alternative Ways to Do OperationsEvery operation can be done in two ways, and
one way is almost always easier than the other.
For instance, subtracting a number is the same
thing as adding the opposite number.So sub-
tracting 5 is the same as adding 5. Also, di-
viding by a number is exactly the same thing
as multiplying by its reciprocal.So dividing by
2/3 is the same as multiplying by 3/2. When
doing arithmetic, always think about your op-
tions, and do the operation that is easier! For
instance, if you are asked to do 45 ÷1/2, you
should realize that it is the same as 45 2,
which is easier to do in your head.The Order of OperationsDon’t forget the order of operations: P-E-MD-
AS. When evaluating, first do what’s grouped
in parentheses(or above or below fraction bars
or within radicals), then do exponents (or
roots) from left to right, then multiplication
or divisionfrom left to right, and then do ad-
ditionor subtractionfrom left to right. What is
4 – 6 ÷ 2 3? If you said 3, you mistakenly did
the multiplication before the division. (Instead,
do them left to right). If you said 3 or 1/3,
you mistakenly subtracted before taking care
of the multiplication and division. If you said
5, pat yourself on the back!
When using your calculator, be careful to use
parentheses when raising negatives to powers.
For instance, if you want to raise –2 to the 4th
power, type “(–2)^4,” and not just “–2^4,” be-
cause the calculator will interpret the latter as
–1(2)^4, and give an answer of –16, rather than
the proper answer of 16.Lesson 1: Numbers and Operations