Real Numbers for GMAT, GRE , SAT. - Number Properties - Real Numbers - Number Line - Basic Operations

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3. CONTENTS

4. NUMBER
PROPERTIES

5. BASIC NOTATION
A = B
A + B
A – B
A × B
A ÷ B
x, n, ….

6. A = B
The Sum of first n consecutive
odd integers is n2.
and 4 dollars and 4 pounds
equivalent to 56 crowns …

7. 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

8. 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 …

9. 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

10. A  B
is to be reduced to 14
pounds by removing some of
the 20-pound boxes …

11. 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

12. A  B
Martina earns
her annual income …
The product of all the prime
numbers less than 20 …

13. A × B
Triple (3 times)A multiplied by B
Double, Twice (2 times)
The product of A and B
A times B

14. 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
, …

15. A ÷ B
A is how many times B
A divided by B
The quotient of A and B
The ratio of A to B

17. UNKNOWNS
A certain business produced x
rakes each month form
November through February …
which of the following could be
equal to n?

19. A  B
If a and b are integers and
a is not equal to b.

20. A is greater than B
A > B

21. 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.

22. A is greater than
or equal to B
A ≥ B

23. A ≥ B
the probability is greater
than or equal to 8/20 …
A+C must be not less than 10.

26. 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?

28. A ≤ B
The participants age must be at
most 20 years old.
x is less than or equal to 6.

30. SIGNED NUMBERS

31. If k is an non-negative integer …
If x and y are positive integers …
If a and b are non-zero integers …
SIGNED NUMBERS

32. LONG DIVISION

33. 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

34. LAWS OF
OPERATIONS

35. =+ +
CONMUTATIVE LAW

36. =
CONMUTATIVE LAW

37. a b b a
a b b a
  
  
CONMUTATIVE LAW
5 3 3 5
6 3 3 6
  
  

38. =+ +
ASSOCIATIVE LAW

39. =
ASSOCIATIVE LAW

40.    
   
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

41. =+ +
DISTRIBUTIVE LAW (×)

42.  
 
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

43. =+ +
DISTRIBUTIVE LAW (÷)

44. 237 200 + 30 + 6 + 1
2 2
= 100 + 15 + 3 + ½
= 118 ½
=

45. 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

46. 36 36 36
2+4 2 4
6  18 + 9
 +

47. a a a
b c b c
a a a
b c b c
 

 


48. NUMBER PROPERTIES

49. REAL
NUMBERS

51. REAL NUMBERS

52. RATIONAL NUMBERS

54. 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

55. Irrational numbers
can’t be expressed
precisely as a fraction.

56. IRRATIONAL NUMBERS

57. IRRATIONAL NUMBERS
Every irrational number is
a non-integer number.

58. Irrational numbers are
non-terminating decimals.
2.7182818284590452
35360287471352662497757247093699959574
9669676277240766303535475945713...

59. 3.1415926535897932
384626433832795028841971693
993751058209749445923078164062862089986280348253
4211706798214808651328230664709384460955058223172535940812848111745028410
2701938521105559644622948954930381964288109756659334461284756482337867831
6527120190914564856692346034861045432664821339360726024914127372458700660
6315588174881520920962829254091715364367892590360011330530548820466521384
146951941511609...
12
trillion
digits

60. 1.41421356237309504
88016887242096980785696718753769480731766797379
90732478462107038850387534327641572735013846230912297024924836
05585073721264412149709993583141322266592750559275579995050115
27820605714701095599716059702745345968620147285174186408891986
0955232923048430871432145083…
1.73205080756887729
35274463415058723669428052538103806280558069794
51933016908800037081146186757248575675626141415406703029969945
094998952478811655512094373648528…
22
33

61. INTEGERS

62. Integers
are positive and
negative whole
numbers and zero.
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8

63. INTEGERS

64. Whole Numbers
are positive integers
and zero.
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8

65. -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
EVEN-ODD NUMBERS

66. 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

67. 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

68. 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

70. Any of the numerals from 0 to 9.
Form part of a number.
DIGITS

71. REALNUMBERS

72. 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

73. Rounding a
number
means reducing the
digits in a number
while trying to keep
its value similar.

75. 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.

76. Rounded to the nearest ten:
1,256.23
Rounded to the nearest tenth:
1,256.23
ROUNDING NUMBERS
 1,260.00
 1,256.20

77. TRANSITIVE PROPERTY
If a < b and b < c, then a < c
a b c

78. TRANSITIVE PROPERTY
If a > b and b > c, then a > c
c b a

79. a
b
c
a < b and b < c, then a < c

82. REAL NUMBERS

83. NUMBER
LINE

84. 01 1
NUMBER LINE

85. 01 1
x < 1

86. 01 1
1 < x < 0

87. 01 1
0 < x < 1

88. 01 1
1 < x

89. NUMBER LINE

90. INEQUALITIES
> Greater than
 Greater than or equal to
> Less than
 Less than or equal to

91. 4 7
4 < x < 7
INEQUALITIES

92. 4 ≤ x ≤ 7
4 7
INEQUALITIES

93. 4 < x ≤ 7
4 7
INEQUALITIES

94. 4 ≤ x < 7
4 7
INEQUALITIES

95. 4 ≤ x
4
INEQUALITIES

96. 4 < x
4
INEQUALITIES

97. x ≤ 7
7
INEQUALITIES

98. x < 7
7
INEQUALITIES

99. 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

100. a b c 
2 2 2a b c    
5 5 5a b c    
2 2 2a b c 
SOLVING INEQUALITIES

101. INEQUALITIES

102. PROPERTIES
OF ZERO

103. 1 10
NUMBER LINE

104. ZERO PROPERTIES

105. NON-POSIITIVE NUMBERS
1 0

106. NON-NEGATIVE NUMBERS
10

107. Any number multiplied
by zero is zero.

109. Zero added to any
number is the same as
the original number.

114. Zero divided by any
number not equal to
zero is zero.

116. Dividing by zero is
undefined.

118. PROPERTIES OF ZERO

119. PROPERTIES
OF ONE

120. Any number divided
by itself equals 1.

122. A number multiplied
or divided by 1 results
the same number.

125. The reciprocal of a
number a is 1 .
a

130. The product of a
number by its
reciprocal is 1.

132. Dividing by a number
a is the same as
multiplying the
number by 1 .
a

134. PROPERTIES OF ONE

135. ABSOLUTE
VALUE

136. |n|=
n, when n ≥ 0
 n, when n < 0

137. |3| |3|

138. |n| ≥ 0
n2 = |n|

139. ABSOLUTE VALUE

140. OPERATIONS
WITH SIGNED
NUMBERS

141. ADDICTION /
SUBTRACTION

142. ADDICTION /
SUBTRACTION

143. MULTIPLICATION

144. MULTIPLICATION

145. DIVISION

146. DIVISION

147. POWERS

148. POWERS

149. POWERS

150. OPERATIONS WITH SIGNED
NUMBERS

151. POWERS OF 10

153. POWERSOF10

154. POWERSOF10

155. POWERSOF10

157. SCIENTIFIC NOTATION

158. SCIENTIFIC NOTATION

159. SCIENTIFIC NOTATION

160. POWERS OF 10

161. BASIC
OPERATIONS

162. means things like add,
subtract, multiply,
divide, squaring, etc.

163. Order of Operations
which procedures should
be performed first in a
given mathematical
expression.

165. Operate parentheses
from inside out.
PEMDAS

172. 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

173. 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
0
x 1
x 2
x 3
x 4
x 5
x 6
x x

174. 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
0
x 1
x 2
x x

175. BASIC OPERATIONS

176. SUMMARY

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