Which Choices Are Real Numbers? Check All That Apply
A real number is any number that can be represented on the real number line. Real numbers include integers, whole numbers, fractions, decimals, rational numbers, irrational numbers, and many other numerical values.
Examples of real numbers include: −5, 0, 4, 1/2, −3.75, √2, √3, π.
Important Rule Before Solving
If the square root contains a negative number, the result is not a real number.
√(−25) → Not Real ❌
However, an irrational number such as √2 is still a real number.
Question 1
Which of the following are real numbers? Check all that apply.
Solution
A. 7 → Real number ✅
7 is a positive integer, and every integer is a real number.
B. −12 → Real number ✅
Negative integers are also real numbers.
C. √25 → Real number ✅
5 is a real number.
D. √(−9) → Not a real number ❌
The square root of a negative number is not a real number.
E. 3/5 → Real number ✅
A fraction with real numerator and non-zero denominator is a real number.
F. π → Real number ✅
π is an irrational number, but irrational numbers are still real numbers.
Question 2
Which choices are real numbers? Check all that apply.
Solution
A. √16
4 is real. Therefore, A is correct. ✅
B. √(−16)
It contains the square root of a negative number, so it is not real. ❌
C. 0
0 is a real number. ✅
D. −√49
−7 is real. ✅
E. 2.75
2.75 is a decimal and is therefore a real number. ✅
F. i
The symbol i represents the imaginary unit. It is not a real number. ❌
Question 3
Which of the following are real numbers? Check all that apply.
Solution
A. √81
9 is real. ✅
B. √(−81)
The radicand is negative, so this is not a real number. ❌
C. −7/9
A fraction is a real number. ✅
D. 3.14159
This is a decimal number and is real. ✅
E. √2
√2 is irrational, but irrational numbers are real numbers. ✅
F. 5i
It contains the imaginary unit i, so it is not a real number. ❌
Question 4
Select all the real numbers.
Solution
A. −8 → Real ✅
−8 is an integer.
B. √36
6 is real. ✅
C. √(−36) → Not real ❌
The square root has a negative radicand.
D. −11/4 → Real ✅
It is a fraction of two integers.
E. π → Real ✅
π is an irrational real number.
F. i√5 → Not real ❌
The expression contains i.
Question 5
Which choices represent real numbers? Check all that apply.
Solution
A. √100
10 is real. ✅
B. −√121
−11 is real. ✅
C. √(−100)
This involves the square root of a negative number, so it is not real. ❌
D. 22/7 → Real ✅
It is a rational number.
E. −3.6 → Real ✅
It is a decimal number.
F. 4i → Not real ❌
It is an imaginary number.
Question 6
Which of the following are real numbers? Check all that apply.
Solution
A. √144
12 is real. ✅
B. √(−144) → Not real ❌
The radicand is negative.
C. −15 → Real ✅
−15 is an integer.
D. 8/3 → Real ✅
It is a rational number.
E. √7 → Real ✅
√7 is irrational but real.
F. 2i → Not real ❌
Question 7
Which choices are real numbers? Check all that apply.
Solution
A. 0.125 → Real ✅
It is a terminating decimal.
B. −√64
−8 is real. ✅
C. √(−25) → Not real ❌
D. −5/2 → Real ✅
It is a rational number.
E. √3 → Real ✅
√3 is irrational but real.
F. 7i → Not real ❌
Question 8
Select all the real numbers.
Solution
A. √225
15 is real. ✅
B. −20 → Real ✅
C. √(−4) → Not real ❌
D. 6.25 → Real ✅
E. 13/2 → Real ✅
It is a rational number.
F. i√2 → Not real ❌
Question 9
Which choices are real numbers? Check all that apply.
Solution
A. √1
1 is real. ✅
B. √(−1)
√(−1) is represented by i, so it is not real. ❌
C. −π → Real ✅
π is real, so −π is also real.
D. 4.75 → Real ✅
E. −9/10 → Real ✅
F. 3i → Not real ❌
Question 10
Which of the following are real numbers? Check all that apply.
Solution
A. √400
20 is real. ✅
B. −√169
−13 is real. ✅
C. √(−49) → Not real ❌
The radicand is negative.
D. 12.5 → Real ✅
E. 17/3 → Real ✅
17/3 is a rational number.
F. 9i → Not real ❌
Real Numbers — Quick Revision
| Number Type | Example | Real? |
|---|---|---|
| Positive integer | 8 | ✅ Yes |
| Negative integer | −8 | ✅ Yes |
| Zero | 0 | ✅ Yes |
| Fraction | 3/5 | ✅ Yes |
| Decimal | 2.75 | ✅ Yes |
| Irrational number | √2 | ✅ Yes |
| Pi | π | ✅ Yes |
| Square root of negative number | √(−9) | ❌ No |
| Imaginary number | 3i | ❌ No |
Most Important Rule
Remember This:
√(Positive Number) → Usually Real
√(Negative Number) → Not Real
i → Not a Real Number
√2 and π → Irrational but Real
Final Answer Pattern
When you see a question saying:
“Which choices are real numbers? Check all that apply.”
Check for integers, fractions, decimals, positive square roots, and irrational numbers such as √2 and π.
Do not select expressions containing the square root of a negative number or the imaginary unit i.