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Real Numbers for GMAT, GRE , SAT. - Number Properties - Real Numbers - Number Line - Basic Operations
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CONTENTS
NUMBER PROPERTIES
BASIC NOTATION
A = B
A + B
A – B
A × B
A ÷ Bx, n, ….
A = B
The Sum of first n consecutive
odd integers is n2.
and 4 dollars and 4 pounds
equivalent to 56 crowns …
A = B
A is equal to B
A is the same as B
A is equivalent to B
The result of A is B
A gives B
A was B
A has B
A + B
… what is the sum of the
digits in that integer?
If the scores were derived by
adding 3.5 points for each
correct answer …
A + B
The sum of A and B
The total of A and B
The combined total of A and B
A increased by BA greater than B
A and B
A received B
A farther than B
A B
is to be reduced to 14
pounds by removing some of
the 20-pound boxes …
A ─ B
The difference of A and B
A decreased by B
The difference between A and B
B fewer than AB is subtracted from A
A received B
B younger than A
A B
Martina earns
her annual income …
The product of all the prime
numbers less than 20 …
A × BTriple (3 times)A multiplied by B
Double, Twice (2 times)
The product of A and B
A times B
A B
Each digit in the two-digit number G
is halved to form a new two-digit
number …
If an object travels 100 feet
, …
A ÷ B
A is how many times B
A divided by B
The quotient of A and B
The ratio of A to B
UNKNOWNS
A certain business produced x
rakes each month form
November through February …
which of the following could be
equal to n?
A B
If a and b are integers and
a is not equal to b.
A is greater than B
A > B
A > B
If n is a prime number
greater than 3, what
is the remainder when …
The area of the right triangle
ABC is greater than the area
of the right triangle KLM.
A is greater than
or equal to B
A ≥ B
A ≥ B
the probability is greater
than or equal to 8/20 …
A+C must be not less than 10.
A < B
Set A consists of all positive
integers less than 100;
How many positive integers less
than 10,000 are there in which the
sum of the digits equals 5?
A ≤ B
The participants age must be at
most 20 years old.
x is less than or equal to 6.
SIGNED NUMBERS
If k is an non-negative integer …
If x and y are positive integers …
If a and b are non-zero integers …
SIGNED NUMBERS
LONG DIVISION
1 2 1
6 7 2 9
- 6
1 2
- 1 2
0 9
- 6
3
1 9 3
5 9 6 5
- 5
4 6
- 4 5
15
- 15
0
LAWS OF OPERATIONS
=+ +
CONMUTATIVE LAW
=
CONMUTATIVE LAW
a b b a
a b b a
CONMUTATIVE LAW
5 3 3 5
6 3 3 6
=+ +
ASSOCIATIVE LAW
=
ASSOCIATIVE LAW
a b c a b c
a b c a b c
4 3 2 4 3 2
6 3 2 6 3 2
ASSOCIATIVE LAW
=+ +
DISTRIBUTIVE LAW (×)
a b c a b a c
a b c a b a c
2 4 3 2 4 2 3
2 4 3 2 4 2 3
DISTRIBUTIVE LAW
=+ +
DISTRIBUTIVE LAW (÷)
237 200 + 30 + 6 + 1
2 2
= 100 + 15 + 3 + ½
= 118 ½
=
b c b c
a a a
b c b c
a a a
15 25 15 25
5 5 5
24 18 24 18
6 6 6
DISTRIBUTIVE LAW
36 36 36
2+4 2 4
6 18 + 9
+
a a a
b c b c
a a a
b c b c
NUMBER PROPERTIES
REAL NUMBERS
REAL NUMBERS
RATIONAL NUMBERS
RATIONAL NUMBERS
Can be integers,
or terminating
decimals or
repeating non-
terminating
decimals.
1/1
2/2
3/3
4/4
5/5
1/2
2/1
1/3
3/1
1/4
2/3
3/2
4/1
1/5
2/4
4/2
5/1
2/5
3/4
4/3
5/2
3/5
5/3
4/5
5/4
Irrational numbers can’t be expressed
precisely as a fraction.
IRRATIONAL NUMBERS
IRRATIONAL NUMBERS
Every irrational number is
a non-integer number.
Irrational numbers are
non-terminating decimals.
2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964288109756659334461284756482337867831
6527120190914564856692346034861045432664821339360726024914127372458700660
6315588174881520920962829254091715364367892590360011330530548820466521384
146951941511609...
12
trillion
digits
1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083…
1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248575675626141415406703029969945
094998952478811655512094373648528…
22
33
INTEGERS
Integers are positive and negative whole
numbers and zero.
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8
INTEGERS
Whole Numbersare positive integers
and zero.
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8
EVEN-ODD NUMBERS
EVEN NUMBERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
ODD NUMBERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
Integers
Natural Numbers
Positive IntegersNegative Integers
Whole Numbers
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Counting Numbers
Any of the numerals from 0 to 9. Form part of a number.
DIGITS
REAL N
UM
BERS
DIGITS IN A NUMBER
5 millions digit
3 hundreds of thousands digit
6 tens of thousands digit
4 units of thousands digit
1 hundreds digit
8 tens digit
9 units digit
7 tenths digit
0 hundredths digit
2 thousandths digit
5,364,189.7025 364 189 702
Rounding a number
means reducing the digits in a number
while trying to keep its value similar.
ROUNDING A NUMBER
Decide
which is the
last digit to
keep.
Increase it
by 1 if the
next digit is
5 or more.
Leave it the
same if the
next digit is
less than 5.
Rounded to the nearest ten:1,256.23
Rounded to the nearest tenth:1,256.23
ROUNDING NUMBERS
1,260.00
1,256.20
TRANSITIVE PROPERTY
If a < b and b < c, then a < c
a b c
TRANSITIVE PROPERTY
If a > b and b > c, then a > c
c b a
abc
a < b and b < c, then a < c
REAL NUMBERS
NUMBER LINE
01 1
NUMBER LINE
01 1
x < 1
01 1
1 < x < 0
01 1
0 < x < 1
01 1
1 < x
NUMBER LINE
INEQUALITIES
> Greater than
Greater than or equal to
> Less than
Less than or equal to
4 7
4 < x < 7
INEQUALITIES
4 ≤ x ≤ 7
4 7
INEQUALITIES
4 < x ≤ 7
4 7
INEQUALITIES
4 ≤ x < 7
4 7
INEQUALITIES
4 ≤ x
4
INEQUALITIES
4 < x
4
INEQUALITIES
x ≤ 7
7
INEQUALITIES
x < 7
7
INEQUALITIES
2 3 9
2 9 3
2 6
3
x
x
x
x
2 3 9
2 9 3
2 6
3
x
x
x
x
SOLVING INEQUALITIES
a b c
2 2 2a b c
5 5 5a b c
2 2 2a b c
SOLVING INEQUALITIES
INEQUALITIES
PROPERTIES OF ZERO
1 10
NUMBER LINE
ZERO PROPERTIES
NON-POSIITIVE NUMBERS
1 0
NON-NEGATIVE NUMBERS
10
Any number multiplied
by zero is zero.
Zero added to any number is the same as the original number.
Zero divided by anynumber not equal to
zero is zero.
Dividing by zero isundefined.
PROPERTIES OF ZERO
PROPERTIES OF ONE
Any number divided
by itself equals 1.
A number multipliedor divided by 1 results
the same number.
The reciprocal of a
number a is 1 .a
The product of a number by its reciprocal is 1.
Dividing by a numbera is the same as multiplying the number by 1 .
a
PROPERTIES OF ONE
ABSOLUTE VALUE
|n|=n, when n ≥ 0
n, when n < 0
|3| |3|
|n| ≥ 0
n2 = |n|
ABSOLUTE VALUE
OPERATIONS WITH SIGNED NUMBERS
ADDICTION / SUBTRACTION
ADDICTION / SUBTRACTION
MULTIPLICATION
MULTIPLICATION
DIVISION
DIVISION
POWERS
POWERS
POWERS
OPERATIONS WITH SIGNED
NUMBERS
POWERS OF 10
PO
WER
S O
F 1
0
PO
WERS O
F 1
0
PO
WERS O
F 1
0
SCIENTIFIC NOTATION
SCIENTIFIC NOTATION
SCIENTIFIC NOTATION
POWERS OF 10
BASIC OPERATIONS
means things like add, subtract, multiply,
divide, squaring, etc.
Order of Operationswhich procedures should
be performed first in a given mathematical
expression.
Operate parentheses
from inside out.
PEMDAS
1 2 3 4 5 6 7 8 9 10 11 12
1 1 2 3 4 5 6 7 8 9 10 11 12
2 2 4 6 8 10 12 14 16 18 20 22 24
3 3 6 9 12 15 18 21 24 27 30 33 36
4 4 8 12 16 20 24 28 32 36 40 44 48
5 5 10 15 20 25 30 35 40 45 50 55 60
6 6 12 18 24 30 36 42 48 54 60 66 72
7 7 14 21 28 35 42 49 56 63 70 77 84
8 8 16 24 32 40 48 56 64 72 80 88 96
9 9 18 27 36 45 54 63 72 81 90 99 108
10 10 20 30 40 50 60 70 80 90 100 110 120
11 11 22 33 44 55 66 77 88 99 110 121 132
12 12 24 36 48 60 72 84 96 108 120 132 144
x
1 1.00 1 1 1 1 1 1 1 1.00
2 0.50 1 2 4 8 16 32 64 1.41
3 0.33 1 3 9 27 81 243 729 1.73
4 0.25 1 4 16 64 256 1,024 2.00
5 0.20 1 5 25 125 625 2.24
6 0.17 1 6 36 216 2.45
7 0.14 1 7 49 343 2.65
8 0.13 1 8 64 512 2.83
9 0.11 1 9 81 729 3.00
10 0.10 1 10 100 1,000 3.16
1
x
0x 1x 2x 3x 4x 5x 6x x
x
11 1 11 121 3.32
12 1 12 144 3.46
13 1 13 169 3.61
14 1 14 196 3.74
15 1 15 225 3.87
16 1 16 256 4.00
17 1 17 289 4.12
18 1 18 324 4.24
19 1 19 361 4.36
20 1 20 400 4.47
0x 1x 2x x
BASIC OPERATIONS
SUMMARY
