Pre-Algebra Workbook 2: Number Sets & Classification

Think about which sets $12$ belongs to: it is positive, has no fractional part, and is not zero. Start with the most specific set (like the counting numbers) and move outward to larger sets.

The number $12$ is a positive whole number with no fractional part.

That means it is a natural number. Because natural numbers are also whole numbers, integers, rational numbers, and real numbers, $12$ belongs to all of those sets.

When we are asked for the best classification, we choose the most specific set it belongs to, which is the set of natural numbers.

So the correct choice is A. Natural number.

Ask: Is $0$ a counting number? Is it included when we start counting $1,2,3,\dots$? Then think about which set is the smallest one that still includes $0$

The number $0$ is not one of the counting numbers $1,2,3,\dots$ so it is not usually classified as natural.

However, $0$ is a whole number, and therefore also an integer, a rational number, and a real number.

The smallest set that contains $0$ is the set of whole numbers.

So the correct choice is B. Whole number.

Negative numbers are not natural or whole. Think about which set is the smallest one that includes all negative whole numbers.

The number $-7$ is negative, so it cannot be a natural number or a whole number.

It is a negative whole number, which means it is an integer. All integers are also rational and real, but integer is the most specific set here.

So the correct choice is C. Integer.

Ask: Is $\frac{3}{4}$ a whole number or integer? Remember that rational numbers are fractions of integers with a nonzero denominator.

The number $\frac{3}{4}$ is not a whole number or an integer because it is not a whole value.

It can be written as a ratio of integers:

$$\frac{3}{4}$$

so it is a rational number. All rational numbers are also real numbers.

The most specific set it belongs to is the set of rational numbers.

So the correct choice is D. Rational number.

Check whether $-2.5$ can be written as a fraction of integers. If it can, it is rational. Then ask which set is the smallest one that includes it.

The number $-2.5$ has a decimal part, so it is not natural, whole, or an integer.

We can rewrite it as a fraction of integers:

$$-2.5 = -\frac{5}{2}$$

This is a ratio of integers, so $-2.5$ is a rational number (and also a real number).

The most specific set it belongs to is the set of rational numbers.

So the correct choice is D. Rational number.

Integers are all the whole numbers and their negatives: $\dots,-3,-2,-1,0,1,2,3,\dots$ Look for the number that has a fractional or decimal part.

Integers are all the whole numbers and their negatives:

$$\dots,-3,-2,-1,0,1,2,3,\dots$$

Among the choices, $3$ $-4$ and $0$ are all integers. The number $2.7$ has a decimal part, so it is not an integer.

So the correct choice is D. 2.7.

Natural numbers are the counting numbers $1,2,3,\dots$ Even numbers are divisible by $2$ with no remainder. Look for a positive counting number that is even.

Natural numbers are the counting numbers $1,2,3,\dots$ and an even number is divisible by $2$

Check each option:

• $0$ is even, but it is not usually counted as a natural number.
• $-6$ is even, but it is negative, so it is not natural.
• $7$ is natural, but it is odd.
• $14$ is a positive counting number and divisible by $2$ so it is even and natural.

So the correct choice is D. 14.

Think about how we count people: $1$ student, $2$ students, etc. We never have $2.5$ students. Which set describes positive counting quantities?

The number of students present must be a positive whole number: $1,2,3,\dots$

These are natural numbers. Natural numbers are also whole, integer, rational, and real, but natural number is the most specific classification here.

So the correct choice is A. Natural number.

Bank balances can include decimals (like $45.75) and can also be negative. Think about which set includes fractions and decimals made from integers.

A bank balance can be positive or negative and may include cents (decimals), such as $45.75.

Any exact decimal with a finite number of decimal places can be written as a fraction of integers, so it is a rational number. All rational numbers are also real.

The best classification for a bank balance is therefore rational number.

So the correct choice is C. Rational number.

Rational numbers can be written as a fraction of integers with a nonzero denominator. Look for the number that cannot be written as a fraction of integers.

Check each choice:

• $\frac{4}{9}$ is a fraction of integers, so it is rational.
• $-3$ can be written as $\frac{-3}{1}$ so it is rational.
• $0.75$ can be written as $\frac{75}{100} = \frac{3}{4}$ so it is rational.
• $\sqrt{2}$ cannot be written as a fraction of integers; it is a classic example of an irrational number.

So the only irrational number in the list is $\sqrt{2}$

The correct choice is A. $\sqrt{2}$.