Numbers
Complex Numbers
*ℂ = 𝑥 + 𝑖𝑦 𝑥, 𝑦 ∈ ℝ, 𝑖 2 = −1
*The set of nonzero complex numbers: ℂ∗ =
𝕫∈ℂ𝑧≠0
*if 𝑥 + 𝑖𝑦 = 𝑧 ∈ ℂ, then modulus of 𝑧 is r =
𝑧 = 𝑥 2 + 𝑦2
*The polar form of z ∈ ℂ is 𝑧 = 𝑟ሺ𝑐𝑜𝑠𝜃 +
Real Numbers
Imaginary Numbers
*ℝ = {𝑥|𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑛𝑒}
*ℂ = {𝑥 + 𝑖𝑦|𝑥, 𝑦 ∈ ℝ, 𝑖 2 = −1}
*The set of positive real numbers: ℝ+ = 𝑥 ∈ ℝ 𝑥 > 0
*The set of negative real numbers: ℝ− = 𝑥 ∈ ℝ 𝑥 < 0
*The set of nonnenegative real numbers: ℝ+ =
𝑥∈ℝ𝑥≥0
The set of nonpositive real numbers: ℝ+ =
𝑥∈ℝ𝑥≤0
*The set of nonzero real numbers: ℝ∗ = 𝑥 ∈ ℝ 𝑥 ≠ 0
Nonalgebraic real
numbers
Algebraic real numbers
*ℝ = {𝑥|𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑛𝑒}
Irrational
*The set of positive real numbers: ℝ+ = 𝑥 ∈ ℝ 𝑥 > 0
*𝑄𝑐 = {𝑥|𝑥 𝑖𝑠 𝑛𝑜𝑛𝑟𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔 𝑎𝑛𝑑
−
*The set of negative real numbers: ℝ = 𝑥 ∈ ℝ 𝑥 < 0
𝑛𝑜𝑛𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔 𝑑𝑒𝑐𝑖𝑚𝑎𝑙}
*The set of nonnenegative real numbers: ℝ+ = 𝑥 ∈ ℝ 𝑥 ≥ 0
The set of nonpositive real numbers: ℝ+ = 𝑥 ∈ ℝ 𝑥 ≤ 0
Transcendental numbers
*The set of nonzero real numbers: ℝ∗ = 𝑥 ∈ ℝ 𝑥 ≠ 0
Rational
**The polar form of z ∈ ℂ is 𝑧 =
𝑟 𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃 = 𝑧 𝑒10 where
𝑒10 = 𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃
Irrational
𝑎
𝑏
*ℚ = { |𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0}
*The set of positive rational numbers: ℚ+ = {𝑥 ∈
ℚ|𝑥 > 0}
*The set of nonnegative rational numbers:
ℚ𝑛𝑜𝑛𝑒𝑛𝑒𝑔 = {𝑥 ∈ ℚ|𝑥 ≥ 0}
*𝑄𝑐 =
{𝑥|𝑥 𝑖𝑠 𝑛𝑜𝑛𝑟𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔
𝑎𝑛𝑑 𝑛𝑜𝑛𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔
𝑑𝑒𝑐𝑖𝑚𝑎𝑙}
−
*The set of negative rational numbers: ℚ = {𝑥 ∈
ℚ|𝑥 < 0}
*The set of nonpositive rational numbers: ℚ𝑛𝑜𝑛𝑝𝑜𝑠 =
{𝑥 ∈ ℚ|𝑥 ≤ 0}
*The set of nonezero rational numbers: ℚ∗ = {𝑥 ∈
ℚ|𝑥 ≠ 0}
Integer rational numbers
*Integers: ℤ = {… , −3, −2, −1,0,1,2,3, … }
*The set of positive rational numbers:
Non-integer rational numbers
*The set of positive rational numbers: ℚ+ = {𝑥 ∈
ℚ|𝑥 > 0}
*The set of nonnegative rational numbers: ℚ𝑛𝑜𝑛𝑒𝑛𝑒𝑔 =
{𝑥 ∈ ℚ|𝑥 ≥ 0}
{ℚ+ = 𝒙 ∈ ℚ 𝒙> 0}
*The set of nonnegative integers: ℤ𝑛𝑜𝑛𝑒𝑔 = 𝑥 ∈ ℤ 𝑥 ≥ 0
−
*The set of nonzero integers: ℤ = 𝑥 ∈ ℤ 𝑥 ≠ 0
*The set of negative rational numbers: ℚ = {𝑥 ∈
ℚ|𝑥 < 0}
*The prime numbers: P =
𝑝 ∈ ℕ 𝑝 > 1, 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑝 𝑎𝑟𝑒 1 𝑎𝑛𝑑 𝑖𝑡𝑠𝑒𝑙𝑓
*The set of nonpositive rational numbers: ℚ𝑛𝑜𝑛𝑝𝑜𝑠 =
{𝑥 ∈ ℚ|𝑥 ≤ 0}
*The set of nonezero rational numbers: ℚ∗ = {𝑥 ∈ ℚ|𝑥 ≠
0}
*The set of nonezero rational numbers: ℚ∗ = {𝑥 ∈
ℚ|𝑥 ≠ 0}
+
Natural numbers
Zero
*ℕ = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓, …}
*The empty set: ∅ ={}
Negative integers
*The set of negative integers: ℤ- =
{𝒙 ∈ ℤ 𝒙 < 𝟎} *The set of
nonpositive integers: ℤ𝒏𝒐𝒏𝒑𝒐𝒔 = {𝒙
∈ ℤ 𝒙 ≤ 𝟎}
*The set of negative
rational
−
numbers: ℚ = {𝑥 ∈ ℚ|𝑥 < 0}
*The set of nonpositive rational
numbers: ℚ𝑛𝑜𝑛𝑝𝑜𝑠 = {𝑥 ∈ ℚ|𝑥 ≤
0}