Learn about the different types of real numbers and their properties.
Example 1: Rational vs. Irrational Numbers
Understand the difference between rational and irrational numbers.
Real Numbers: The set of all rational and irrational numbers. They can be represented on the number line.
Rational Numbers: Numbers that can be expressed in the form p/q, where p and q are integers and q is not equal to 0.
Examples: 1/2, -3/4, 5 (which is 5/1), 0.75 (which is 3/4).
Irrational Numbers: Numbers that cannot be expressed in the form p/q. Their decimal expansions are non-terminating and non-recurring.
Examples: Square root of 2, pi (approximately 3.14159...).
Key Difference: Rational numbers have decimal expansions that either terminate (like 0.5) or are non-terminating but repeat (like 1/3 = 0.333...). Irrational numbers have decimal expansions that never terminate and never repeat.
Example 2: Decimal Form of Rational Numbers
Convert rational numbers into their decimal form and classify them.
Terminating Decimals: If the division of the numerator by the denominator ends after a finite number of steps, the decimal form is terminating.
Example: 1/4 = 0.25 (The division terminates).
Non-terminating Recurring Decimals: If the division does not end, but a digit or a block of digits repeats periodically, the decimal form is non-terminating recurring (or repeating).
Example: 1/3 = 0.333... (The digit '3' repeats). 2/7 = 0.285714285714... (The block '285714' repeats).
Conversion: To convert a fraction p/q to decimal form, perform long division of p by q.
Example 3: Representing Irrational Numbers on Number Line
Learn how to locate irrational numbers like square root of 2 on the number line.
Step 1: To represent square root of 2 on the number line, we use the Pythagorean theorem.
Step 2: On the number line, mark point O at 0 and point A at 1. The distance OA is 1 unit.
Step 3: Draw a line segment AB of length 1 unit perpendicular to the number line at A.
Step 4: Join OB. Triangle OAB is a right-angled triangle with sides OA = 1 and AB = 1.
Step 5: By the Pythagorean theorem, OB squared = OA squared + AB squared = 1 squared + 1 squared = 1 + 1 = 2. So, OB = square root of 2.
Step 6: With O as the center and OB as the radius, draw an arc that intersects the number line at point P. The point P represents square root of 2 on the number line.
Example 4: Properties of Real Numbers
Understand the fundamental properties of real numbers under addition and multiplication.
Closure Property: The sum and product of any two real numbers is a real number. (a + b is real, a * b is real for any real numbers a, b).
Commutative Property: The order of addition or multiplication does not affect the result. (a + b = b + a, a * b = b * a).
Associative Property: The grouping of numbers in addition or multiplication does not affect the result. ((a + b) + c = a + (b + c), (a * b) * c = a * (b * c)).
Identity Property: There exists an additive identity (0) and a multiplicative identity (1) such that a + 0 = a and a * 1 = a.
Inverse Property: For every real number a, there exists an additive inverse (-a) such that a + (-a) = 0. For every non-zero real number a, there exists a multiplicative inverse (1/a) such that a * (1/a) = 1.
Distributive Property: Multiplication distributes over addition: a * (b + c) = a * b + a * c.
Practice Mode - Real Number Solver
Enter a rational number as a fraction (p/q) to see its decimal form.
Note: This basic solver converts fractions to decimal form. Enter the fraction using '/' (e.g., 1/4, -3/8, 5/3).
Related Concepts
Explore these related mathematical concepts to deepen your understanding of Real Numbers.
Natural Numbers (N)
Counting numbers: {1, 2, 3, ...}
Whole Numbers (W)
Natural numbers plus zero: {0, 1, 2, 3, ...}
Integers (Z)
Whole numbers and their negatives: {..., -2, -1, 0, 1, 2, ...}
Rational Numbers (Q)
Numbers expressible as p/q (p, q integers, q != 0).
Irrational Numbers
Numbers not expressible as p/q (non-terminating, non-recurring decimals).
Terminating Decimal
A decimal that ends after a finite number of digits.
Non-terminating Recurring Decimal
A decimal that repeats a digit or block of digits infinitely.
Number Line
A line where every point corresponds to a unique real number.
Pythagorean Theorem
Relates the sides of a right-angled triangle (a^2 + b^2 = c^2).