What is the Number System?

The number system is a method for representing numbers in a consistent way. Whether you’re counting sheep, calculating your bills, or measuring how much pizza is left, you rely on a number system to express all those quantities. Different types of numbers behave differently, and the number system helps us make sense of these differences. Number System is like the backbone of mathematics. It’s a structured way to classify and represent numbers in different forms, making it easier to understand and solve mathematical problems. The number system is not just about counting; it’s about how we use different kinds of numbers for various situations. Let’s break it down even further with examples.

Types of Number Systems and Their Examples:

1. Natural Numbers (N):

These are the simplest numbers, also called counting numbers. They start from 1 and go on infinitely.

Examples:

• Counting the number of apples: 1, 2, 3, 4…
• Number of people in a room: 5, 10, 15…

Note: Natural numbers do not include zero or negative numbers.

2. Whole Numbers (W):

Whole numbers are basically natural numbers but they also include zero.

Examples:

• The number of cookies left in a jar: 0 (when none are left).
• Floors in a building: 0 (ground floor), 1, 2, 3…

Why It’s Useful: Whole numbers are used in situations where the concept of ‘nothing’ or zero needs to be represented.

3. Integers (Z):

Integers are an extension of whole numbers. They include all positive whole numbers, zero, and negative whole numbers. Integers can be visualized on a number line, stretching from negative infinity to positive infinity.

Examples:

• A temperature drop: -5°C.
• Balancing a bank account: -$50 (debt) or +$100 (credit).

Why It’s Useful: Integers help describe situations involving both positive and negative outcomes, such as gains and losses.

4. Rational Numbers (Q):

Rational numbers are numbers that can be written as a fraction of two integers. These numbers include all integers because any integer can be expressed as a fraction (like 5 = 5/1).

Examples:

• ½ (half a pizza).
• 0.75 (three-fourths of a cup of water).
• -⅓ (a debt of one-third of a dollar).

Why It’s Useful: Rational numbers allow us to deal with divisions and fractions, making them ideal for measurements, financial calculations, and more.

5. Irrational Numbers:

These are numbers that cannot be expressed as a fraction of two integers. Their decimal expansions go on infinitely without repeating patterns.

Examples:

• π (Pi): 3.14159… (The ratio of a circle’s circumference to its diameter).
• √2 (Square Root of 2): 1.414213… (The length of the diagonal in a square with side length 1).

Why It’s Useful: Irrational numbers are critical when dealing with geometry, especially with circles and right triangles. These numbers can’t be neatly represented as fractions but are still important in real-world measurements.

6. Real Numbers (R):

Real numbers include both rational and irrational numbers. In simpler terms, real numbers are any number that you can find on the number line. Whether it’s positive, negative, whole, fractional, or irrational, they are all real numbers.

Examples:

• Any rational number like ⅔, 7.25, or -4.
• Any irrational number like π or √2.

Why It’s Useful: Real numbers are used everywhere in everyday life—from measuring distances to understanding time and temperatures.

7. Complex Numbers (C):

Complex numbers extend beyond real numbers. They include a real part and an imaginary part. Complex numbers are written as a+bia + bia+bi, where aaa is the real part and bibibi is the imaginary part (with iii representing the square root of -1).

Examples:

• 3+4i3 + 4i3+4i: A complex number with a real part of 3 and an imaginary part of 4i.
• 0+2i0 + 2i0+2i: A purely imaginary number, since the real part is 0.

Why It’s Useful: Complex numbers are extremely useful in advanced fields like electrical engineering, signal processing, and quantum physics. They allow for the solution of equations that don’t have real number solutions, like the square root of negative numbers.

IMPORTANT TYPE’s

1. Divisibility Rules – Quick Tricks to Check Divisibility

Divisibility rules are simple shortcuts that help you check whether one number can be divided evenly by another without actually doing long division. Here are some key ones:

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
  • Example: 48 is divisible by 2 because its last digit is 8.
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Example: 123 is divisible by 3 because 1 + 2 + 3 = 6, and 6 is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if its last digit is either 0 or 5.
  • Example: 25 is divisible by 5 because it ends in 5.
• Divisibility by 10: A number is divisible by 10 if it ends in 0.
  • Example: 150 is divisible by 10 because it ends in 0.

These rules save time when you need to figure out if a number can be divided by another, especially with large numbers!

2. Factors and Multiples – Breaking Down Numbers

• Factors are numbers that divide evenly into another number. For example, 1, 2, 4, and 8 are factors of 8 because they divide evenly into 8.
  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12.
• Multiples are the result of multiplying a number by any whole number.
  • Example: Multiples of 5 are 5, 10, 15, 20, 25…

Understanding factors and multiples helps in finding greatest common factors (GCF) and least common multiples (LCM), which are super useful in fractions and division problems.

3. Prime Numbers – Building Blocks of All Numbers

• Prime numbers are numbers greater than 1 that have only two factors: 1 and the number itself. In other words, you can’t divide them evenly by any other numbers.
  • Examples: 2, 3, 5, 7, 11 are prime numbers.
• Composite numbers are numbers that have more than two factors.
  • Example: 8 is a composite number because its factors are 1, 2, 4, and 8.

Prime numbers are like the “atoms” of the number system, used to build all other numbers through prime factorization.

4. Prime Factorization – Breaking Down a Number into Primes

Prime factorization means breaking down a composite number into the prime numbers that multiply together to make that number.

• Example: The prime factorization of 18 is 2 × 3 × 3 (or 2×322 \times 3^22×32).

This concept is useful for simplifying fractions and solving problems involving GCF and LCM.

5. Greatest Common Factor (GCF) – The Biggest Shared Factor

The Greatest Common Factor is the largest number that divides evenly into two or more numbers.

• Example: The GCF of 12 and 16 is 4 because 4 is the largest number that both 12 and 16 can be divided by.

GCF helps when simplifying fractions.

6. Least Common Multiple (LCM) – The Smallest Shared Multiple

The Least Common Multiple is the smallest number that two or more numbers divide into evenly.

• Example: The LCM of 4 and 6 is 12 because 12 is the smallest number that both 4 and 6 divide evenly into.

LCM is useful for adding and subtracting fractions with different denominators.

7. Even and Odd Numbers – Basic Number Properties

• Even numbers are divisible by 2 (like 2, 4, 6, 8).
• Odd numbers are not divisible by 2 (like 1, 3, 5, 7).

Knowing whether a number is even or odd helps with various math problems, especially when working with patterns, division, and problem-solving.

8. Square and Cube Numbers – Perfect Powers

• A square number is a number multiplied by itself.
  • Example: 42=164^2 = 1642=16 (because 4 × 4 = 16).
• A cube number is a number multiplied by itself twice.
  • Example: 33=273^3 = 2733=27 (because 3 × 3 × 3 = 27).

These numbers are important in geometry, algebra, and problem-solving.

9. Remainders and Modulus – What’s Left After Division

When you divide two numbers and they don’t divide evenly, the remainder is what’s left over.

• Example: When 10 is divided by 3, the quotient is 3 and the remainder is 1 (because 10÷3=310 ÷ 3 = 310÷3=3 with a remainder of 1).

The modulus operation (often written as “mod”) gives you the remainder after division.

• Example: 10mod  3=110 \mod 3 = 110mod3=1.

Understanding remainders and modulus is useful in algorithms, programming, and everyday problem-solving.

10. Exponents and Powers – Multiplying a Number by Itself

• An exponent tells you how many times to multiply a number by itself. For example, 232^323 (read as “2 to the power of 3”) means 2×2×2=82 × 2 × 2 = 82×2×2=8.
• Powers of a number can help solve large multiplication problems quickly without doing it step by step.

EXAMPLE QUESTION’s

1. Find the largest 4-digit number divisible by 12, 15, and 18.

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2. What is the smallest number which, when divided by 8, 12, and 15, leaves a remainder of 5 in each case?

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3. How many numbers between 1 and 100 are divisible by both 3 and 5?

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4. Find the greatest number that will divide 140, 196, and 244 leaving the same remainder in each case.

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5. If the product of two numbers is 180 and their GCF is 6, find their LCM.

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6. Find the smallest number divisible by all numbers from 1 to 10.

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7. What is the unit digit of 71257^{125}7125?

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8. Find the sum of all prime numbers less than 30.

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9. If a number when divided by 27 gives a remainder of 7, what will be the remainder when the square of the number is divided by 27?

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10. Find the number of trailing zeros in 100!100!100! (100 factorial).

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