If there is one thing you can count on, it’s your toes. Really, fingers and toes are naturally some of the first objects humans count. You learned to count fingers, toe, and toys when you were very little. You counted using natural numbers.
Natural numbers are foundations of mathematics.
In algebra, Natural numbers are defined as the counting numbers; positive integers beginning with and increasing by forever. Zero is not a natural number.
Another definition of natural numbers is whole, positive numbers. Natural numbers are never negative numbers or fractions, so not all rational numbers are natural numbers.
In math, the symbol for a set of natural numbers is .
When mathematicians describe a group or set of integers, they use brackets and ellipses like this: .
The ellipsis means the set continues in either one or two directions, getting smaller or getting larger in a predictable way.
A set of natural numbers looks like this:
The first five natural numbers are . Notice the set begins with , not .
A set of natural numbers will always be a set of positive integers.
Look at your fingers. You can mentally count using the natural numbers to find you have (in most cases) eight fingers and two thumbs.
Feet? Two feet; ten toes. Hairs on your head? Well, that may take longer, but on average you will have 100,000 of those, from this part of the set of whole numbers:
Natural numbers are called “natural” because they are a natural way to count objects using one-to-one correspondence. We have one number for every object, no matter what we are counting, real or imagined.
Here are exactly nine countable examples:
In no case does the counting process of these items begin with , which is a problem.
Most mathematicians, teachers, and professors consider a whole number but not a natural number. Some, though, do consider a natural number:
{0, 1, 2, 3, 4, 5 …}
Its use in physics, for example, allows for the zeroth law of thermodynamics.
If you are uncertain how your textbook, teacher, or professor uses (is it a whole number, a natural number, or something else?), ask.
For that class, course or textbook, go with what you are told but understand mathematics is often as much opinion as precision, so another course, textbook, or class could view differently.
Natural numbers can combine using operations:
Here are four examples to demonstrate these qualities:
Here are exactly eight challenges to see if you know your natural numbers:
We know you naturally want to peek, but don't! Work these out first, then look at the answers below.
After working your way through this lesson and video, you should know:
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