Sets of Numbers - WorksheetWorks.com - Free Printable

Let’s go step by step to solve each problem. We’re given a number and asked to say which sets it belongs to:
- R = Real Numbers (all numbers on the number line — including decimals, fractions, negatives, etc.)
- I = Irrational Numbers (real numbers that can’t be written as fractions — like √2 or π)
- Q = Rational Numbers (can be written as fractions — includes integers, terminating/repeating decimals)
- Z = Integers (whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...)
- W = Whole Numbers (non-negative integers: 0, 1, 2, 3, ...)
- N = Natural Numbers (positive integers: 1, 2, 3, ... — sometimes includes 0, but usually not; we’ll follow common school definition: starts at 1)

Note: All natural numbers are whole numbers, all whole numbers are integers, all integers are rational, and all rational numbers are real. Irrational numbers are also real, but NOT rational.

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We’ll go one by one:

(i) 5
→ Positive integer → N, W, Z, Q, R
→ Not irrational → not I
✔ Answer: N, W, Z, Q, R

(ii) -7/8
→ Fraction → rational → Q, R
→ Negative → not W, N
→ Not integer → not Z
→ Not irrational → not I
✔ Answer: Q, R

(iii) 9
√9 = 3 → positive integer → N, W, Z, Q, R
✔ Answer: N, W, Z, Q, R

(iv) 0.142857... (repeating? The bar is over “142857” — so repeating decimal)
Repeating decimals are rational → Q, R
Not integer → not Z, W, N
Not irrational → not I
✔ Answer: Q, R

(v) 0
→ Integer → Z, Q, R
→ Whole number → W
→ Usually NOT considered natural (in most school curricula) → not N
→ Not irrational → not I
✔ Answer: W, Z, Q, R

(vi) (-7)
Square root of negative number → imaginary → NOT real → not in any of these sets except possibly none? But note: the sets listed are subsets of real numbers. So this is not in R, hence not in any of the others.
Wait — let’s check the key:
Key says:
U = Universal set (probably all complex?)
I = Imaginary
B = Real
R = Rational
Z = Integer
But in the instruction: “Write the number with what letter belongs to: N, W, Z, Q, R, I, U” — wait, looking back at image description: actually the key shown is:

Actually, rechecking user input:
The key says:
U = Universal
I = Imaginary
B = Real ← Wait! This is confusing. In standard math:
Usually:
= Real
ℚ = Rational
ℤ = Integer
etc.

But in the worksheet key provided in the image (as described):
It says:
U = Universal
I = Imaginary
B = Real ← Hmm, unusual. Normally B isn't used for Real. Maybe typo? Or perhaps in this context:

Wait — looking again at user's text:
“Key:
U = Universal
I = Imaginary
B = Real
R = Rational
Z = Integer”

This is non-standard. Typically:
Real = ℝ
Rational = ℚ
Integer = ℤ

But here they use:
B = Real
R = Rational
So we must follow THEIR notation.

Also, they ask to write letters from: N, W, Z, Q, R, I, U — but in key they define U, I, B, R, Z — no N or W defined? Contradiction?

Wait — look at the instruction:
“Write the number with what letter belongs to: N, W, Z, Q, R, I, U where x ∈ ___”

And key defines:
U = Universal
I = Imaginary
B = Real
R = Rational
Z = Integer

But N and W are not defined in key? That’s odd.

Perhaps there’s a mistake in transcription. Let me assume based on standard usage AND the fact that the problems include numbers like 5, 0, etc., which are typically classified under N, W.

Looking at problem (i) 5 — if we follow standard:

In many textbooks:
Natural numbers N = {1,2,3,...}
Whole numbers W = {0,1,2,3,...}
Integers Z = {...,-2,-1,0,1,2,...}
Rational Q = fractions
Real R = all rationals + irrationals
Imaginary I = multiples of i
Universal U = everything

But in the key provided in the image (per user), they have:

U = Universal
I = Imaginary
B = Real ← this is strange — why B? Maybe it's a typo and should be R? But then they also have R = Rational.

Wait — perhaps "B" stands for "Base" or something else? Unlikely.

Another possibility: maybe "B" is meant to be "Real", and "R" is "Rational", and they just used different letters.

Also, note: in the list of letters to choose from: N, W, Z, Q, R, I, U — so even though key doesn’t define N and W, we must use them as per standard meaning.

Given that, and since the worksheet likely follows standard definitions despite the key labeling "B" for Real (which might be an error), I think we should proceed with standard meanings:

Assume:
- N = Natural numbers (1,2,3,...)
- W = Whole numbers (0,1,2,3,...)
- Z = Integers
- Q = Rational
- R = Real (even though key says B=Real, but since R is in the answer choices and commonly means Real, and key has R=Rational — conflict!)

This is messy.

Let me re-read user input carefully:

User wrote:
"Key:
U = Universal
I = Imaginary
B = Real
R = Rational
Z = Integer"

And instruction: "Write the number with what letter belongs to: N, W, Z, Q, R, I, U"

So the available letters to write are: N, W, Z, Q, R, I, U

But key defines: U, I, B, R, Z — no N, W, Q defined? Q is probably Rational, same as R? No — key says R = Rational.

Wait — perhaps "Q" is not defined in key, but in the answer choices it's included. Likely, Q = Rational, same as R? That would be redundant.

I think there's a mislabeling in the key. Probably:

Standard interpretation expected:

Despite key saying "B = Real", since "R" is in the answer choices and commonly denotes Real, and "Rational" is often denoted by Q, I suspect the key has a typo.

Moreover, in problem (iv) 0.142857... repeating — that's 1/7, rational.

To resolve, let's look at typical such worksheets.

Commonly:

- N: natural
- W: whole
- Z: integer
- Q: rational
- R: real
- I: irrational (but here key says I=imaginary — big difference!)

Ah! Here's the issue: in the key, I = Imaginary, but in many contexts I = Irrational. But here it's specified as Imaginary.

Also, B = Real — very unusual.

Perhaps in this curriculum:

They use:
- U: universal set (all numbers)
- I: imaginary numbers
- B: real numbers (maybe "B" for "base" or "basic"?)
- R: rational numbers
- Z: integers

And additionally, they expect us to know:
- N: natural numbers
- W: whole numbers
- Q: ? — not defined, but probably rational, same as R? Confusing.

Wait — in the answer choices, they list Q, and in key R = Rational, so perhaps Q is not used? But it's in the list.

Another idea: perhaps "Q" is a typo and should be "B"? But unlikely.

Let's count the letters in the instruction: "N, W, Z, Q, R, I, U" — seven letters.

Key defines five: U, I, B, R, Z.

Missing definitions for N, W, Q.

Probably, we are to assume:
- N = natural numbers
- W = whole numbers
- Q = rational numbers (same as R? But R is already defined as rational)

This is inconsistent.

Perhaps "R" in the answer choices stands for Real, and "Q" for Rational, ignoring the key's "B=Real, R=Rational".

That makes more sense because otherwise "B" is not in the answer choices.

Look: the instruction says to write letters from: N, W, Z, Q, R, I, U

"B" is NOT in that list! So probably the key's "B = Real" is irrelevant or a distractor, and we should use standard:

- R = Real numbers
- Q = Rational numbers
- Z = Integers
- I = Irrational numbers? But key says I = Imaginary — conflict.

For √(-7), it's imaginary, so if I = Imaginary, then it belongs to I.

For other numbers, if they are real, they belong to R.

But then what about rational vs irrational? If R = Real, then both rational and irrational are in R, but we need to distinguish.

The key has R = Rational, so perhaps in this system:

- R = Rational
- And Real is not directly represented, but since all rational are real, and irrational are also real, but irrational is not defined separately.

This is too confusing.

Let me try to infer from the problems.

Take (i) 5: clearly natural, whole, integer, rational, real.

If we must choose from N,W,Z,Q,R,I,U, and assuming:

- N: natural
- W: whole
- Z: integer
- Q: rational
- R: real
- I: irrational or imaginary? Key says imaginary.
- U: universal

For 5, it is not imaginary, not necessarily universal (universal includes everything, so technically yes, but usually we don't put U unless specified).

Typically in such exercises, U is not selected unless the number is complex or something.

For √(-7), it is imaginary, so I.

For 0.142857... repeating, it is rational, so Q and R (if R=real).

But if R=real, then all real numbers are in R, and rational are subset.

The key says R = Rational, so perhaps in this context, "R" means rational, and "real" is implied or not labeled.

But then for irrational numbers, what letter? Not defined.

For example, √2 is irrational — would it be only in R (real) but not in R (rational)? Confusing.

Perhaps the "R" in the answer choices is for Real, and "Q" for Rational, and we ignore the key's "B=Real, R=Rational" as a mistake.

I think that's the best approach, because otherwise it's impossible.

Moreover, in many online sources, similar worksheets use:

- N: natural
- W: whole
- Z: integer
- Q: rational
- R: real
- I: irrational
- C: complex, etc.

Here, I is defined as Imaginary in the key, so for √(-7), it should be I.

For real numbers, they are in R (real).

Rational numbers are in Q.

So let's assume:

- N: natural numbers (1,2,3,...)
- W: whole numbers (0,1,2,3,...)
- Z: integers
- Q: rational numbers
- R: real numbers
- I: imaginary numbers (as per key)
- U: universal set (all numbers, so every number is in U, but usually not selected unless specified)

In practice, for real numbers, we don't select U, as it's too broad.

For √(-7), it is imaginary, so I, and also in U.

But typically, we select the most specific sets.

Let's proceed with this assumption.

Also, for irrational numbers, they would be in R (real) but not in Q, and not in I (since I is imaginary, not irrational).

But in this worksheet, is there any irrational number? Let's see.

(vii) √13 — 13 is not perfect square, so irrational — so it should be in R (real) and not in Q, and not in I (imaginary).

But if I is only for imaginary, then irrational numbers have no special letter except R.

That might be acceptable.

Some systems use "I" for irrational, but here key explicitly says I = Imaginary.

So we'll go with that.

Also, for (xii) -√49 = -7, which is integer.

Now, let's solve each:

(i) 5
- Natural: yes → N
- Whole: yes → W
- Integer: yes → Z
- Rational: yes → Q
- Real: yes → R
- Imaginary: no → not I
- Universal: yes, but usually not selected → skip U
✔ N, W, Z, Q, R

(ii) -7/8
- Rational: yes → Q
- Real: yes → R
- Not integer, not whole, not natural, not imaginary
✔ Q, R

(iii) √9 = 3
Same as 5 → N, W, Z, Q, R
✔ N, W, Z, Q, R

(iv) 0.142857... (repeating)
This is 1/7, rational → Q, R
✔ Q, R

(v) 0
- Whole: yes → W
- Integer: yes → Z
- Rational: yes → Q
- Real: yes → R
- Natural: usually no → not N
- Imaginary: no
✔ W, Z, Q, R

(vi) √(-7)
Imaginary number → I
Also in universal set → U
Is it real? No. Rational? No. Etc.
✔ I, U

(vii) √13
13 is prime, not perfect square, so irrational → real but not rational
So: R (real)
Not rational, not integer, etc.
Not imaginary.
✔ R

(viii) -26
Integer → Z
Rational → Q
Real → R
Not whole, not natural
✔ Z, Q, R

(ix) 1/3
Rational → Q, R
✔ Q, R

(x) √25 = 5
Same as (i) → N, W, Z, Q, R
✔ N, W, Z, Q, R

(xi) 25
Positive integer → N, W, Z, Q, R
✔ N, W, Z, Q, R

(xii) -√49 = -7
Integer → Z, Q, R
✔ Z, Q, R

(xiii) -48
Integer → Z, Q, R
✔ Z, Q, R

(xiv) 48
Positive integer → N, W, Z, Q, R
✔ N, W, Z, Q, R

(xv) 0.3
Terminating decimal → rational → Q, R
✔ Q, R

(xvi) -8.2
Decimal, can be written as -82/10 = -41/5 → rational → Q, R
✔ Q, R

(xvii) -√(100)/10 = -10/10 = -1
Integer → Z, Q, R
✔ Z, Q, R

(xviii) -π
Pi is irrational, so -pi is also irrational → real but not rational
So: R
Not imaginary.
✔ R

(xix) 22/7
Fraction → rational → Q, R
(Note: 22/7 is approximation of pi, but itself is rational)
✔ Q, R

(xx) -65,000.011
Decimal, can be written as fraction → rational → Q, R
✔ Q, R

Now, double-check (vi) √(-7): definitely imaginary, so I and U.

For (vii) √13: irrational, so only R (real), assuming no separate letter for irrational.

Similarly for (xviii) -π: only R.

Is there any number that is only in U? Probably not, since U is universal.

Usually, we don't select U unless the number is not in any other set, but here all numbers are in some set.

For √(-7), it is in I and U.

Now, let's compile final answers.

But wait, in the key, they have "B = Real", but we're using R for real. Since "B" is not in the answer choices, we ignore it.

Also, for irrational numbers, they are in R (real), and that's fine.

One more thing: in (iv) 0.142857..., if it's repeating, it's rational, correct.

All seem good.

Final Answers:

(i) N, W, Z, Q, R
(ii) Q, R
(iii) N, W, Z, Q, R
(iv) Q, R
(v) W, Z, Q, R
(vi) I, U
(vii) R
(viii) Z, Q, R
(ix) Q, R
(x) N, W, Z, Q, R
(xi) N, W, Z, Q, R
(xii) Z, Q, R
(xiii) Z, Q, R
(xiv) N, W, Z, Q, R
(xv) Q, R
(xvi) Q, R
(xvii) Z, Q, R
(xviii) R
(xix) Q, R
(xx) Q, R

Now, to present neatly.

Note: For (vi), some might argue that imaginary numbers are also in universal set, so U is correct.

For others, no U needed.

Also, ensure order: probably alphabetical or as listed, but not specified, so any order is fine, but typically we list from smallest set to largest or vice versa. But for consistency, I'll list as N,W,Z,Q,R for naturals, etc.

In the answer, we can list in the order of the letters given: N,W,Z,Q,R,I,U — but only those applicable.

So for (i): N,W,Z,Q,R

For (vi): I,U

Etc.

I think this is correct.

Final Answer:
(i) N, W, Z, Q, R
(ii) Q, R
(iii) N, W, Z, Q, R
(iv) Q, R
(v) W, Z, Q, R
(vi) I, U
(vii) R
(viii) Z, Q, R
(ix) Q, R
(x) N, W, Z, Q, R
(xi) N, W, Z, Q, R
(xii) Z, Q, R
(xiii) Z, Q, R
(xiv) N, W, Z, Q, R
(xv) Q, R
(xvi) Q, R
(xvii) Z, Q, R
(xviii) R
(xix) Q, R
(xx) Q, R