Understanding Numbers: A Comprehensive Guide
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Understanding Numbers

A Comprehensive Guide to Number Systems & Operations

🔢 Types of Numbers: Natural, Whole, Integers 🔢

Natural Numbers (N)

  • Used for counting: (1, 2, 3, …).
  • Smallest natural number: 1.
  • There are infinite natural numbers.

Whole Numbers (W)

  • Natural numbers including zero (0).
  • Smallest whole number: 0.
  • There is no largest whole number.

Integers (Z or I)

  • Natural numbers, their negatives, and zero.
  • Smallest and largest integers cannot be determined.

짝 Even & Odd Numbers 짝

Even Numbers (E)

  • Integers divisible by 2.
  • Example: E = …, –4, –2, 0, 2, 4, 6, 8,…
  • Smallest even natural number: 2.
  • There is no largest even number.

Odd Numbers (O)

  • Integers which are not divisible by 2.
  • Example: O = …, –3, –1, 1, 3, 5, 7,…
  • Smallest odd natural number: 1.
  • There is no largest odd number.

🌟 Prime & Composite Numbers 🌟

Prime Numbers

  • Natural numbers with exactly two factors: 1 and itself.
  • Lowest prime number: 2.
  • 2 is the only even prime number.
  • All primes (except 2 and 3) are of the form $6N \pm 1$.
  • Primes less than 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Composite Numbers

  • Natural number with at least one divisor other than 1 and itself.
  • Can be factorized into prime factors.
  • Example: $24 = 2 \times 2 \times 2 \times 3$.
  • Smallest composite number: 4.

🤝 Co-prime Numbers & Key Facts 🤝

Co-prime Numbers

  • Numbers with no common factors other than 1.
  • Example: 14 and 15 (common factor is only 1).

💡 Important Numerical Facts:

1 is neither prime nor composite.
1 is an odd integer.
0 is neither positive nor negative.
0 is an even integer.
2 is prime & even both.
All prime numbers (except 2) are odd.

📏 Real Numbers: Rational & Irrational 📏

Real Numbers (R)

  • All numbers representable on a number line.
  • $R^+$: Positive real numbers.
  • $R^-$: Negative real numbers.
  • Real Numbers = Rational Numbers + Irrational Numbers.

Rational Numbers (Q)

  • Any number in the form $p/q$, where p and q are integers and $q \neq 0$.

Irrational Numbers (Q’ or Qc)

  • Numbers which are not rational or which cannot be put in the form $p/q$, where p and q are integers and $q \neq 0$.

➗ Fractions & Decimal Rounding ➗

Fraction Formula

Fraction = Numerator / Denominator

Decimal fraction: Denominator is a power of 10.

Example: $2/10 = 0.2$, $9/100 = 0.09$

Rounding Off Decimal Numbers:

  • Rounding Down: If the next digit after the required place is less than 5, keep the digit as is.
  • Rounding Up: If the next digit after the required place is 5 or greater than 5, increase the required place digit by 1.
  • Example: 3.4578, 21.358940789

✅ Divisibility Rules: 2, 3, 4, 5, 6 ✅

Divisible by 2: Unit digit is either even or 0.
Divisible by 3: Sum of digits is divisible by 3.
Divisible by 4: Number formed by the last 2 digits is divisible by 4, or if the last two digits are 0’s.
Divisible by 5: Unit digit is either 5 or 0.
Divisible by 6: Simultaneously divisible by 2 and 3.

🔍 Divisibility Rules: 7, 11, 13 🔍

Divisible by 7:

  • Multiply unit digit by 2, subtract from remaining digits. If the result is divisible by 7, the original number is. Repeat if needed.
  • Example: 16807 → 1680 – ($7 \times 2$) = 1666. 1666 → 166 – ($6 \times 2$) = 154. 154 → 15 – ($4 \times 2$) = 7. (Divisible by 7).

Divisible by 11:

  • If the difference of the sum of digits at even places and odd places is 0 or a multiple of 11.
  • Example: 12342. Sum of even place digits (2+4=6). Sum of odd place digits (1+3+2=6). Difference = 6 – 6 = 0. (Divisible by 11).

Divisible by 13:

  • Multiply unit digit by 4, add to remaining digits. If the result is divisible by 13, the original number is. Repeat if needed.
  • Example: 2197 → 219 + ($7 \times 4$) = 247. 247 → 24 + ($7 \times 4$) = 52. (Divisible by 13).

➕ Division Algorithm ✖️

Core Formula

Dividend = (Divisor $\times$ Quotient) + Remainder

Where:

  • Dividend: The number which is being divided.
  • Divisor: The number which performs the division process.
  • Quotient: Greatest possible integer as a result of division.
  • Remainder: Rest part of dividend which cannot be further divided by the divisor.

Special Case: Remainder Sum

If two different numbers $x$ and $y$ when divided by a certain divisor $D$ leave remainder $r_1$ and $r_2$ respectively. When the sum of them is divided by the same divisor, the remainder is $r_3$. Then,

Divisor $(D) = r_1 + r_2 – r_3$

⚛️ Law of Indices ⚛️

$a^m \times a^n = a^{m+n}$ Example: $5^3 \times 5^4 = 5^{3+4} = 5^7$
$a^m / a^n = a^{m-n}$ if $m > n$ Example: $6^5 / 6^2 = 6^{5-2} = 6^3$
$a^m / a^n = 1 / a^{n-m}$ if $m < n$ Example: $4^3 / 4^8 = 1 / 4^{8-3} = 1 / 4^5$
$a^m / a^n = a^0 = 1$ if $m = n$ Example: $3^4 / 3^4 = 3^{4-4} = 3^0 = 1$
$(a^m)^n = a^{mn} = (a^n)^m$ Example: $(6^2)^4 = 6^{2 \times 4} = 6^8 = (6^4)^2$
$(ab)^n = a^n \cdot b^n$ Example: $(6 \times 4)^3 = 6^3 \times 4^3$
$(a/b)^n = a^n / b^n$ (where $b \neq 0$) Example: $(5/3)^4 = 5^4 / 3^4$
$a^{-n} = 1 / a^n$ Example: $5^{-3} = 1 / 5^3$
$a^0 = 1$ (for any real number a) Example: $8^0 = 1$

√ Law of Surds √

$(a^{1/n})^n = a$ Example: $(8^{1/3})^3 = 8$
$a^{1/n} \cdot b^{1/n} = (ab)^{1/n}$ Example: $5^{1/3} \cdot 8^{1/3} = (5 \times 8)^{1/3} = (40)^{1/3}$
$(a^{1/n})^{1/m} = a^{1/(mn)}$ Example: $(5^{1/3})^{1/5} = 5^{1/(3 \times 5)} = 5^{1/15}$
$\sqrt[n]{a} = a^{1/n}$ Example: $\sqrt[3]{5} = 5^{1/3}$
$\sqrt[n]{a^m} = a^{m/n}$ Example: $\sqrt[7]{5^3} = 5^{3/7}$
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