Natural Numbers, Whole Numbers, and Integers

Natural numbers are the counting numbers: 1, 2, 3, 4, … They start at 1 and go up in ones.

Natural numbers do not include negative numbers, zero, decimals or fractions. So −3 is out because it is negative, 0 is out because counting starts at 1 (0 is a whole number, but not a natural number), and 2.5 is out because it is a decimal.

That leaves 7, which is a natural number.

Whole numbers are 0, 1, 2, 3, …: the counting numbers together with zero.

Whole numbers do not include negative numbers, decimals or fractions. So −2 is out because it is negative, 4.7 is out because it is a decimal, and $\frac{1}{2}$ is out because it is a fraction. It sits between 0 and 1, not on a whole-number step.

That leaves 15, which is a whole number.

Natural numbers are the counting numbers $1,2,3,\dots$: positive, with no zero, no negatives and no decimals.

Every value in 1, 5, 20 is a counting number, so this is the correct set. The set 0, 4, 7 fails because 0 is a whole number but not a natural number. The set −3, 1, 10 fails because −3 is negative. The set 2.5, 3, 4 fails because 2.5 is a decimal, not a counting number.

Integers are the whole numbers together with their negatives: $\dots ,-2,-1,0,1,2,\dots$. No fractions and no decimals.

Every value in −4, 0, 3 is an integer, so this list is correct. The list −2, 3/2, 6 fails because $\frac{3}{2}$ is a fraction, not a whole amount. The list 0.2, 1, 7 fails because 0.2 is a decimal. The list −3, −2.8, 5 fails because −2.8 is a decimal, even though it is negative. Being negative is allowed, but the value must still be whole.

Integers are $\dots ,-2,-1,0,1,2,\dots$, so they do include negative numbers. That statement is correct.

Whole numbers are $0,1,2,3,\dots$ and are never negative. Natural numbers are the counting numbers $1,2,3,\dots$, so zero is not one of them. A fraction such as $\frac{3}{2}$ is not an integer, because an integer must be a whole amount.

Natural numbers are $1,2,3,\dots$ and whole numbers are $0,1,2,3,\dots$, so every natural number is also a whole number. That statement is true.

Not all integers are whole numbers, because integers include negatives such as $-4$. For the same reason not all integers are natural numbers. Zero is a whole number, but it is not a natural number, so the last statement is false.