🌈 NUMBER SYSTEM 🌈
✨ REAL, IMAGINARY AND COMPLEX NUMBERS featuring PRIME & COPRIME numbers! ✨
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Hello math explorers! Today we meet the number system family — including the special Prime Numbers and Coprime Numbers! Let’s dive in! 🏊♂️
🔵 REAL NUMBERS \( \mathbb{R} \)
Every number on the number line!
\( \frac{p}{q} \), q≠0, p and q are integers, decimals end/repeat
INTEGERS \( \mathbb{Z} \)
🔹 Negative: -3, -2, -1
🔸 Whole \( \mathbb{W} \): 0,1,2,3…
🔹 Natural \( \mathbb{N} \): 1,2,3,4…
PRIME NUMBERS
\( \mathbb{P} \) = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …}
🔴 Only divisible by 1 and itself (and greater than 1)
⭐ 2 is the only even prime! Others are odd.
🍕 FRACTIONS
\( \frac{1}{2}, \frac{3}{4}, \frac{7}{3} \)
Terminating: 0.25, 1.5
Repeating: \( 0.\overline{6} = \frac{2}{3} \)
Any fraction of integers
✅ Integers, fractions, repeating decimals — all RATIONAL!
never-ending, never-repeating
⭐ Cannot be written as a simple fraction!
Rational + Irrational = ALL REAL NUMBERS 🧮
🔮 COMPLEX \( \mathbb{C} \)
\( i = \sqrt{-1} \) is imaginary number
Imaginary unit: \( i^2 = -1 \)
🌀 Purely Imaginary: \( 5i, -2i, i\sqrt{7} \)
\( a + bi \)
Real + Imaginary is called Complex Number
Example: \( 3 + 4i, -2 + 7i \)
COPRIME NUMBERS
Two numbers are coprime (or relatively prime) if their G.C.D (greatest common divisor)/ H.C.F ( highest common factor ) = 1.
📌 They share NO common prime factors! but only Common factor 1
14 & 25
21 & 22
9 & 28
✅ Even if numbers are not prime themselves, they can be coprime!
Example: 15 (3×5) and 28 (2²×7) → no common primes → coprime!
🌟 Fun fact: Consecutive integers are always coprime! (like 14 & 15)
✨ Solvedpapers.net✨
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Natural \( \mathbb{N} \)
1,2,3…
🟢
Whole \( \mathbb{W} \)
0,1,2…
🔢
Integers \( \mathbb{Z} \)
…-2,-1,0,1,2…
👑
PRIME \( \mathbb{P} \)
2,3,5,7,11…
divisors: 1 & itself
🍕
Rational \( \mathbb{Q} \)
fractions, repeating
🌀
Irrational
π, √2, e
🤝
COPRIME
GCD = 1
(8,15), (14,25)
🔮
Complex \( \mathbb{C} \)
a+bi
Where a is real number and bi is imaginary number
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📚 Teacher’s simple summary (with new friends!):
🌱 Natural → counting numbers. 🫚 Whole → add zero. 🔢 Integers → add negatives.
👑 Prime → special natural numbers (only 2 factors: 1 and itself). Example: 2,3,5,7…
🤝 Coprime → two numbers with no common prime factors (GCD=1). They can be composite but still coprime! Like 8 & 15.
🍕 Rational → can be written as fraction. 🌀 Irrational → cannot (π, √2).
🔮 Complex → includes imaginary \( i \).
\( \mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \)
👑 Primes are inside \( \mathbb{N} \), 🤝 Coprime is a relationship between any two integers!
🧮 “Prime numbers are the atoms of mathematics — and coprime numbers are best friends with no common factors!” 🧮