natural numbers and the operations

Here’s an explanation of natural numbers and their operations in a notes format, designed for ease of understanding with emojis:

Natural Numbers and Operations 🔢

• Natural Numbers (N)

  • Natural numbers are primarily used for counting objects (e.g., 7 balls ⚽️⚽️⚽️⚽️⚽️⚽️⚽️, 7 pencils ✏️✏️✏️✏️✏️✏️✏️).
  • They are denoted by the symbol N.
  • In this context, the set of natural numbers includes 0 (i.e., {0, 1, 2, 3, 4, …}).
    • 💡 Historically, there can be some confusion in different books about whether 0 is included, but here, it always is.

• Arithmetic Operations

  • Addition (+):
    • This operation combines two or more numbers to find their sum.
    • Example: 5 + 2 = 7 ➕.
  • Subtraction (-):
    • This operation finds the difference between two numbers.
    • A key point: If you subtract a larger number from a smaller number (e.g., 5 - 6), you go below zero 📉.
    • This need to represent “less than nothing” led to the expansion of numbers to include negative numbers, forming the set of integers.
  • Multiplication (×):
    • Multiplication is essentially repeated addition.
    • Example: 7 times 4 means making 4 copies of 7, which is 7 + 7 + 7 + 7 = 28 ✖️.
    • Notation: Besides ‘x’, a dot (e.g., 7 ⋅ 3) or simply writing symbols together (e.g., ‘mn’ for m times n) can represent multiplication.
    • Sign Rule:
      • Negative × Positive = Negative (e.g., -7 × 4 = -28).
      • Negative × Negative = Positive (e.g., -7 × -4 = +28).
      • If there’s an even number of minus signs, the result is positive. If there’s an odd number, it’s negative.
  • Exponentiation (Powers):
    • Exponentiation is repeated multiplication.
    • “Squared” (m²): Means ’m’ multiplied by itself twice (m × m). This term comes from the ability to arrange these items into a square (e.g., 6 × 6 = 36 items forming a 6x6 square) 🟩.
    • “Cubed” (m³): Means ’m’ multiplied by itself three times (m × m × m). This comes from arranging items in a 3D cube (e.g., 3 × 3 × 3 forming a cube) 📦.
    • Higher Powers (m^k): For powers beyond 3 (like m^4, m^5, etc.), we simply say “m to the power k” or “the kth power of m” because we can’t visually imagine objects in more than 3 dimensions to name them accordingly.
  • Division (÷):
    • Division is repeated subtraction.
    • Example: 20 mangoes divided among 5 friends (20 ÷ 5 = 4) means you can subtract 5 from 20 exactly 4 times to reach 0 🍎.
    • Quotient and Remainder: If numbers don’t divide evenly (e.g., 19 mangoes for 5 friends), you get a quotient (the number of times you can divide without a fractional part, e.g., 3) and a remainder (what’s left over, e.g., 4).
    • Modulo Operator (mod): This notation (e.g., 19 mod 5 = 4) is used to explicitly state the remainder when one number is divided by another.

• Factors

  • A number ‘a’ is a factor of ‘b’ if ‘b’ can be divided by ‘a’ evenly, leaving no remainder (i.e., b mod a = 0).
  • This also means that ‘b’ is a multiple of ‘a’.
  • Factors usually come in pairs (if ‘a’ divides ‘b’ ‘k’ times, then ‘k’ also divides ‘b’).
  • Perfect Squares: For numbers that are perfect squares (a number multiplied by itself, e.g., 36 = 6 × 6), the square root (e.g., 6) is a factor whose “pair” is itself, leading to an odd number of factors. Numbers that are not perfect squares have an even number of factors.

• Prime Numbers

  • A prime number is a natural number that has exactly two distinct factors: 1 and itself.
  • 1 is not a prime number because it only has one factor (1 itself).
  • The smallest prime number is 2 (factors: 1, 2).
  • After 2, no other even numbers can be prime because they are all multiples of 2.
  • Numbers that are not prime (and not 1) are called composite numbers; they have more than two factors.
  • Any integer can be uniquely factorized into a product of prime numbers (e.g., 12 = 2 × 2 × 3 or 2² × 3) 🧩.