Chapter: Number Systems - Detailed Summary
1. Introduction to Number Systems:
THESE ARE MAIN TOPIC
We use different sets of numbers for various purposes in mathematics.
Understanding these sets and their properties is fundamental.
2. Natural Numbers (N):
These are the counting numbers: 1, 2, 3, 4, ... They extend infinitely.
Example: Counting the number of students in a class (e.g., 30 students).
Key Property: Closed under addition and multiplication (adding or multiplying
two natural numbers always results in a natural number).
3. Whole Numbers (W):
This set includes all natural numbers plus zero: 0, 1, 2, 3, ...
Example: The score in a game can be zero.
Key Property: Similar closure properties as natural numbers for addition and
multiplication.
4. Integers (Z):
Integers encompass all whole numbers and their negative counterparts: ..., -3,
-2, -1, 0, 1, 2, 3, ...
Example: Representing temperature below zero (e.g., -5°C).
Key Property: Closed under addition, subtraction, and multiplication.
5. Rational Numbers (Q):
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, These are numbers that can be expressed as a fraction qp, where p and q are
integers and q is not zero.
Examples:
Fractions: 21, 4−3
Terminating decimals: 0.5 (which is 21), -0.75 (which is 4−3)
Repeating decimals: 0.333... (which is 31)
Key Property: Their decimal representation either terminates or repeats.
6. Irrational Numbers:
These are numbers that cannot be expressed as a simple fraction qp. Their
decimal representations are non-terminating and non-repeating.
Examples:
2≈1.41421356...
π≈3.14159265...
Key Property: Their decimal form goes on forever without a repeating pattern.
7. Real Numbers (R):
The set of real numbers is the union of all rational and irrational numbers. It
includes every number that can be plotted on a number line.
Example: Any number you can think of (-5, 0, 3.14, 7, etc.).
Key Property: Forms a continuous number line.
8. Operations on Real Numbers:
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1. Introduction to Number Systems:
THESE ARE MAIN TOPIC
We use different sets of numbers for various purposes in mathematics.
Understanding these sets and their properties is fundamental.
2. Natural Numbers (N):
These are the counting numbers: 1, 2, 3, 4, ... They extend infinitely.
Example: Counting the number of students in a class (e.g., 30 students).
Key Property: Closed under addition and multiplication (adding or multiplying
two natural numbers always results in a natural number).
3. Whole Numbers (W):
This set includes all natural numbers plus zero: 0, 1, 2, 3, ...
Example: The score in a game can be zero.
Key Property: Similar closure properties as natural numbers for addition and
multiplication.
4. Integers (Z):
Integers encompass all whole numbers and their negative counterparts: ..., -3,
-2, -1, 0, 1, 2, 3, ...
Example: Representing temperature below zero (e.g., -5°C).
Key Property: Closed under addition, subtraction, and multiplication.
5. Rational Numbers (Q):
Untitled 1
, These are numbers that can be expressed as a fraction qp, where p and q are
integers and q is not zero.
Examples:
Fractions: 21, 4−3
Terminating decimals: 0.5 (which is 21), -0.75 (which is 4−3)
Repeating decimals: 0.333... (which is 31)
Key Property: Their decimal representation either terminates or repeats.
6. Irrational Numbers:
These are numbers that cannot be expressed as a simple fraction qp. Their
decimal representations are non-terminating and non-repeating.
Examples:
2≈1.41421356...
π≈3.14159265...
Key Property: Their decimal form goes on forever without a repeating pattern.
7. Real Numbers (R):
The set of real numbers is the union of all rational and irrational numbers. It
includes every number that can be plotted on a number line.
Example: Any number you can think of (-5, 0, 3.14, 7, etc.).
Key Property: Forms a continuous number line.
8. Operations on Real Numbers:
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