Real Number System Review

Every number you have ever used in math class belongs to a family called the real numbers. Understanding how these families are organized will help you think more clearly in Grade 11 math, whether you choose science, management, or humanities.

The Number Families

Think of the real number system as a set of nested boxes -- each bigger box contains all the smaller ones inside it.

Natural Numbers (N): The counting numbers you learned first: 1, 2, 3, 4, 5, ...

Whole Numbers (W): Natural numbers plus zero: 0, 1, 2, 3, 4, ...

Integers (Z): Whole numbers plus negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...

Rational Numbers (Q): Any number that can be written as a fraction p/q where p and q are integers and q is not zero. This includes decimals that terminate (like 0.75) or repeat (like 0.333...).

Irrational Numbers: Numbers that cannot be expressed as a fraction. Their decimal forms go on forever without repeating. Examples include the square root of 2, the square root of 3, and pi.

Key Relationship: N is inside W, W is inside Z, Z is inside Q, and Q together with irrational numbers makes up the real numbers (R).

The Number Line

Every real number corresponds to exactly one point on the number line, and every point on the number line corresponds to exactly one real number.

Worked Example: Classify the number -7/3.

• It is a fraction of two integers, so it is rational.
• It equals approximately -2.333..., which is negative, so it is not a whole number or natural number.
• It is not an integer because it falls between -3 and -2.

Did You Know? Ancient mathematicians in the Indian subcontinent developed the concept of zero, which separates natural numbers from whole numbers -- a contribution that changed mathematics forever.

Key Takeaways

• Real numbers include both rational and irrational numbers
• Every rational number can be written as a fraction; irrational numbers cannot
• The number line is a visual tool to represent all real numbers
• Classification depends on whether a number can be expressed as p/q