Understand classification of real, rational, and irrational numbers. Math 10-C Alberta mathematics.
Lesson 2.1 of Number Systems & Operations in Math 10C — Alberta curriculum lessons.
A number system organizes every number into categories based on what kind of number it is. Think of it like a set of nested boxes — each box fits inside a bigger one, and every number belongs somewhere.
Natural Numbers — N. The natural numbers are the counting numbers — the ones you would naturally use to count objects.
1, 2, 3, 4, 5, 6,
They start at 1 and go on forever. There are no negatives, no fractions, no decimals, and no zero.
Whole Numbers — W. The whole numbers are the natural numbers plus zero.
0, 1, 2, 3, 4, 5,
The only difference between whole numbers and natural numbers is that 0 is included.
Why Does Zero Get Its Own Category?. Zero was not always considered a number. It was invented much later than counting numbers to represent "nothing." So mathematicians keep it in its own layer of the number system.
Integers — I. The integers include all whole numbers and their negatives.
, -3, -2, -1, 0, 1, 2, 3,
Still no fractions or decimals — just whole values stretching infinitely in both directions.
Rational Numbers — Q. A rational number is any number that can be written as a fraction (a)/(b), where a and b are integers and b ≠ 0.
This includes:
All integers — because 5 = (5)/(1)
Terminating decimals — like 0.75 because 0.75 = (3)/(4)
Repeating decimals — like 0.333 because 0.333 = (1)/(3)
If a decimal terminates (ends) or repeats a pattern, it is rational.
Quick test: Can you write it as a fraction of two integers? If yes, it is rational.
Irrational Numbers. Irrational numbers cannot be written as a fraction of two integers. Their decimal forms go on forever without repeating a pattern.
Examples:
π = 3.14159265 (never repeats, never ends)
√(2) = 1.41421356 (never repeats, never ends)
√(3) = 1.73205080 (never repeats, never ends)
Common Mistake — Long Decimal Does Not Mean Irrational. Students sometimes think any long decimal is irrational. That is not true!
The decimal 0.333333 goes on forever, but it repeats, so it equals (1)/(3) and is rational.
An irrational number's digits never settle into a repeating cycle.
Real Numbers — R. The real numbers include everything — all rational and irrational numbers together.
Every number you will encounter in this course is a real number.
Example 1 — GCF of 18 and 28. Step 1: Find the prime factorization of each number.
18 = 2 · 3^2
28 = 2^2 · 7
Step 2: Identify the shared primes.
Both have the prime factor 2. The lowest power of 2 between 2^1 (from 18) and 2^2 (from 28) is 2^1.
The factor 3 only appears in 18, and 7 only appears in 28 — those are not shared.
Step 3: Multiply the shared primes at their lowest powers.
Answer: GCF(18, 28) = 2
Example 2 — GCF of 50 and 63. Step 1: Prime factorizations.
50 = 2 · 5^2
63 = 3^2 · 7
Step 2: Identify shared primes.
These two numbers share no common prime factors at all.
When this happens the numbers are called co-prime (or relatively prime), and their GCF is 1.
Answer: GCF(50, 63) = 1
Example 3 — GCF of 45 and 84. Step 1: Prime factorizations.
45 = 3^2 · 5
84 = 2^2 · 3 · 7
Step 2: Identify shared primes.
The only shared prime is 3. The lowest power of 3 is 3^1.
Answer: GCF(45, 84) = 3
Example 4 — GCF of 28, 49, and 63 (Three Numbers). Step 1: Prime factorizations.
28 = 2^2 · 7
49 = 7^2
63 = 3^2 · 7
Step 2: Identify primes shared by all three.
The only prime that appears in all three numbers is 7. The lowest power of 7 across all three is 7^1.
Answer: GCF(28, 49, 63) = 7
Example 5 — GCF of 90 and 30. Step 1: Prime factorizations.
90 = 2 · 3^2 · 5
30 = 2 · 3 · 5
Step 2: Shared primes at lowest powers.
2^1 · 3^1 · 5^1
Answer: GCF(90, 30) = 2 · 3 · 5 = 30
Notice: 30 divides evenly into 90 (because 90 ÷ 30 = 3). Whenever one number is a multiple of the other, the GCF equals the smaller number.
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