Numbers & Operations
Numbers
Understanding the number system — integers, rational numbers, and how they relate to each other on the number line.
Visualize It
The Number System — How Every Set Fits Inside the Next
1, 2, 3, 4, 5, 6 …
+ 0
+ … −3, −2, −1
+ ½, −¾, 0.6̄, 2.5 …
Cannot be written as p/q
Non-terminating,
non-repeating decimals
Counting (Natural) ℕ
The numbers you count with
1, 2, 3, 4, 5 …
Whole 𝕎
Counting numbers plus zero
0, 1, 2, 3, 4 …
Integers ℤ
Whole numbers plus negatives
… −2, −1, 0, 1, 2 …
Rational ℚ
Any number expressible as p/q
½, −¾, 0.5, 3 …
Irrational ℚ̄
Non-terminating, non-repeating
√2, π, √3, e …
Real ℝ
Every point on the number line
All of the above
Watch & Learn
Watch a clear, friendly video explanation of Numbers:
Watch on YouTube →Opens a YouTube search for the best tutorial videos on this topic.
Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
Which is greater: -7 or -3?
Step-by-Step Solution
Draw a number line in your mind. Numbers increase as you move right.
-3 is to the right of -7 on the number line.
So -3 is greater than -7.
Answer
-3 > -7
Problem
Is √2 rational or irrational?
Step-by-Step Solution
A rational number can be written as a fraction p/q where p and q are integers.
√2 = 1.41421356... — the decimal never repeats or terminates.
Since it cannot be written as a fraction, it is irrational.
Answer
√2 is irrational
Problem
Order from least to greatest: 0.5, -1, 3/4, -2
Step-by-Step Solution
Convert all to decimals: 0.5, -1, 0.75, -2
Place on number line: -2, -1, 0.5, 0.75
Order: -2, -1, 0.5, 3/4
Answer
-2, -1, 0.5, 3/4
Problem
Is -5 an integer, rational number, or both?
Step-by-Step Solution
-5 is a whole negative number, so it is an integer.
Every integer can be written as a fraction: -5 = -5/1.
So -5 is both an integer AND a rational number (and a real number).
Answer
Both an integer and a rational number
Problem
Place these on a number line in order: -3/2, 0, 1.5, -1
Step-by-Step Solution
Convert -3/2 to a decimal: -1.5
Order from left to right: -1.5, -1, 0, 1.5
-3/2 and 1.5 are the same distance from zero but on opposite sides.
Answer
-3/2, -1, 0, 1.5
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
Which is greater: -10 or -4?
Is 0.333... (repeating) rational or irrational?
Order from least to greatest: -5, 2, -1, 0
What type of number is -6?
Is every integer also a rational number?
Is √5 rational or irrational?
Which is greater: -1/2 or -3/4?
Order from least to greatest: 1/2, -2, 0.75, -0.5
Is 0 a natural number, whole number, or integer?
Name a number that is irrational
Is -9 a real number?
Which set is larger: integers or whole numbers?
Finished all 12? Give yourself a pat on the back — then check your work!
Keep going — you're on a roll!
Every topic you master is another step on your journey.
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