Arithmetic
It
is the fundamental building block of math. The other three subject areas tested
in GAT /GRE Math are all pretty much unthinkable without arithmetic. You’ll
certainly need to know your arithmetic to power through algebra, geometry, and
data analysis problems, but the Math section also includes some pure arithmetic
problems as well. So it makes sense to start Math 101 with a discussion of
numbers and the typical things we do with them.
Common math symbols once can remember for test in case you need a quick refresher, here’s a list of some of the most
commonly used math symbols you should know for the GRE math. We’ll discuss some of
them in this arithmetic section and others later in the chapter.
Symbol
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Name
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Meaning
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<
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Less than
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The quantity to
the left of the symbol is less than the quantity to the right.
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>
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Greater than
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The quantity to
the left of the symbol is greater than the quantity to the right.
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≤
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Less than or
equal to
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The quantity to
the left of the symbol is less than or equal to the quantity to the right.
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≥
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Greater than or
equal to
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The quantity to
the left of the symbol is greater than or equal to the quantity to the right.
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=Squart(3) or ,/''''''
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Square root
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A number which
when multiplied by itself equals the value under the square root symbol.
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| x |
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Absolute value
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The positive
distance a number enclosed between two vertical bars is from 0.
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!
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Factorial
|
|
||
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Parallel
|
|
__!__
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Perpendicular
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In geometry,
two lines separated by this symbol meet at right angles.
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°
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Degrees
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A measure of
the size of an angle. There are 360 degrees in a circle.
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π
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Pi
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The ratio of
the circumference of any circle to its diameter; approximately equal to 3.14.
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Number Terms
The
test makers assume that you know your numbers. Make sure you do by comparing
your knowledge to our definitions below.
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Number
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Definition
|
Example
|
|
Whole numbers
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The set of counting numbers,
including zero
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0, 1, 2, 3
|
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Natural numbers
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The set of whole positive
numbers except zero
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1, 2, 3, 4
|
|
Integers
|
The set of all positive and
negative whole numbers, including zero, not including fractions and decimals.
Integers in a sequence, such as those in the example to the right, are called
consecutive integers.
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–3, –2, –1, 0, 1, 2, 3
|
|
Rational numbers
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The set of all numbers that can
be expressed as integers in fractions—that is, any number that can be
expressed in the form m/n, , where m
and n are integers
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9/10, 7/8, 1/2
|
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Irrational numbers
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The set of all numbers that
cannot be expressed as integers in a fraction
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π,
Square root of 3
1.01010000
|
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Real numbers
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Every number on the number line,
including all rational & irrational numbers
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Every number you can think of
|
Even and Odd Numbers
An
even number is an integer that is divisible by 2 with no remainder, including
zero.
Even
numbers: –10, –4, 0, 4, 10
An
odd number is an integer that leaves a remainder of 1 when divided by 2.
Odd
numbers: –9, –3, –1, 1, 3, 9
Even
and odd numbers act differently when they are added, subtracted, multiplied,
and divided. The following chart shows the rules for addition, subtraction, and
multiplication (multiplication and division are the same in terms of even and
odd).
Addition
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Subtraction
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Multiplication/Division
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even + even = even
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even – even = even
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even × even = even
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even + odd = odd
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even – odd = odd
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even × odd = even
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odd + odd = even
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odd – odd = even
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odd × odd = odd
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Zero,
as we’ve mentioned, is even, but it has its own special properties when used in
calculations. Anything multiplied by 0 is 0, and 0 divided by anything is 0.
However, anything divided by 0 is undefined, so you won’t see that on the GRE math. 1/0 = undefined or infinity
Positive
and Negative Numbers
A
positive number is greater than 0. Examples include 1/2, 15, and 83.4. A negative number is less
than 0. Examples include –0.2, –1, and –100. One tip-off is the negative sign
(–) that precedes negative numbers. Zero is neither positive nor negative. On a
number line, positive numbers appear to the right of zero, and negative numbers
appear to the left:
–5,
–4, –3, –2, –1, 0, 1, 2, 3, 4, 5
Positive
and negative numbers act differently when you add, subtract, multiply, or
divide them. Adding a negative number is the same as subtracting a positive
number:
5 +
(–3) = 2, just as 5 – 3 = 2
Subtracting
a negative number is the same as adding a positive number:
7 –
(–2) = 9, just as 7 + 2 = 9
To
determine the sign of a number that results from multiplication or division of
positive and negative numbers, memorize the following rules.
Multiplication
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Division
|
positive × positive = positive
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positive ÷ positive = positive
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positive × negative = negative & vice versa
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positive ÷ negative = negative & vice versa
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negative × negative = positive
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negative ÷ negative = positive
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Here’s
a helpful trick when dealing with a series of multiplied or divided positive
and negative numbers: If there’s an even number of negative numbers in the
series, the outcome will be positive. If there’s an odd number, the outcome
will be negative.
When
negative signs and parentheses collide, it can get pretty ugly. However, the
principle is simple: A negative sign outside parentheses is distributed across
the parentheses. Take this question:
3 +
4 – (3 + 1 – 8) = ?
You’ll
see a little later on when we discuss order of operations that in complex
equations we first work out the parentheses, which gives us:
3 +
4 – (4 – 8)
This
can be simplified to:
3 +
4 – (– 4)
As
discussed earlier, subtracting a negative number is the same as adding a
positive number, so our equation further simplifies to:
3 +
4 + 4 = 11
An
awareness of the properties of positive and negative numbers is particularly
helpful when comparing values in Quantitative Comparison questions, as you’ll
see later in
Remainders
A
remainder is the integer left over after one number has been divided by
another. Take, for example, 92 ÷ 6. Performing the division we see that 6 goes
into 92 a total of 15 times, but 6 × 15 = 90, so there’s 2 left over. In other
words, the remainder is 2.
Divisibility
Integer
x is said to be divisible by integer y when x divided by y yields a remainder
of zero. The GRE sometimes tests whether you can determine if one number is
divisible by another. You could take the time to do the division by hand to see
if the result is a whole number, or you could simply memorize the shortcuts in
the table below. Your choice. We recommend the table.
Divisibility
Rules
1
|
All whole
numbers (0,1,2,3,…..) are divisible by
1. 0/1 =0, 1/1=1…..etc
|
2
|
A number is
divisible by 2 if it’s even. 2/2, 4/2,
6/2, 8/2,……etc 1,2,3,4,
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3
|
A
number is divisible by 3 if the sum of its digits is divisible by 3. This
means you add up all the digits of the original number.
If that total is divisible by 3, then so is
the number. For example, to see whether 83,503 is divisible by 3, we
calculate 8 + 3 + 5 + 0 + 3 = 19.
19 is not divisible by 3, so neither is 83,503. 3x1=3, 3x2=6, 3x3=9, 3x4=12, 3x5= 15,
3x6=18……
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4
|
A
number is divisible by 4 if its last two digits, taken as a single number,
are divisible by 4. For example, 179,316 is divisible by 4 because 16 is
divisible by 4.
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5
|
A number is divisible by 5 if
its last digit is 0 or 5. Examples
include 0, 430, and –20.
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6
|
A number is divisible by 6 if
it’s divisible by both 2 and 3. For example, 663 is not divisible by 6
because it’s not divisible by 2. But 570 is divisible by 6 because it’s
divisible by both 2 and 3 (5 + 7 + 0 = 12, and 12 is divisible by 3).
2x3 =6 3x2= 3 suck it up
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7
|
7
may be a lucky number in general, but it’s unlucky when it comes to
divisibility. Although a divisibility rule for 7 does exist, it’s much harder
than dividing the original number by 7 and seeing if the result is an
integer. So if the GRE happens to throw a “divisible by 7” question at you, you’ll just have to suck it
up and do the math.
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8
|
A number is divisible by 8 if
its last three digits, taken as a single number, are divisible by 8. For
example, 179,128 is divisible by 8 because 128 is divisible by 8.
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9
|
A number is divisible by 9 if
the sum of its digits is divisible by 9. This means you add up all the digits
of the original number. If that total is divisible by 9, then so is the
number. For example, to see whether 531 is divisible by 9, we calculate 5 + 3
+ 1 = 9. Since 9 is divisible by 9, 531 is as well.
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10
|
A number is divisible by 10 if
the units digit is a 0. For example, 0, 490, and –20 are all divisible by 10.
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11
|
This one’s a bit involved but
worth knowing. (Even if it doesn’t come up on the test, you can still impress
your friends at parties.) Here’s how to tell if a number is divisible by 11:
Add every other digit starting with the leftmost digit and write their sum.
Then add all the numbers that you didn’t add in the first step and write
their sum. If the difference between the two sums is divisible by 11, then so
is the original number. For example, to test whether
803,715 is divisible by 11,
we first add 8 + 3 + 1 = 12.
To do this, we just started with
the leftmost digit and added alternating digits. Now we add the numbers that
we didn’t add in the first step: 0 + 7 + 5 = 12. Finally, we take the
difference between these two sums: 12 – 12 = 0. Zero is divisible by all
numbers, including 11, so 803,715 is divisible by 11.
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12
|
A number is divisible by 12 if
it’s divisible by both 3 and 4. For example, 663 is not divisible by 12
because it’s not divisible by 4.
162,480
is divisible by 12 because it’s
divisible by both 4 (the last two digits, 80,
are divisible by 4) and 3
(1 + 6 + 2 + 4 + 8 + 0 = 21, and 21 is
divisible by 3).
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