Natural Numbers

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A natural number is a positive whole number from 1 onwards. Negative numbers are not considered natural numbers. Some examples are 1, 67, 450, 23005 and 2000000. Natural numbers are often represented on a number line;

Natural Numbers Number line example StudySmarter Natural Number Line, Thomas-Gay, StudySmarter Originals

Number classes

Natural numbers can also be part of other number classes, and the diagram below shows how they are all related;

Natural Numbers Number Classes StudySmarter Number Classes, Thomas-Gay - StudySmarter Originals

Properties of natural numbers

Natural numbers have four different properties;

Closure property – this means that when two or more natural numbers are multiplied or added together, it will result in a natural number. For example, $2 + 2 = 4$ or $3 \times 2 = 6$ .
Associative property – suggests that when three natural numbers are added or multiplied together, it will result in the same answer no matter how they are grouped. For example, 3 + (2 + 5) = 10 and (3 + 2) + 5 = 10. This also works when they are multiplied, $3 \times (2 \times 5) = 30$ and $(3 \times 2) \times 5 = 30$ .
Commutative property – this property says that when two natural numbers are multiplied or added together, they will result in the same answer no matter their order. For example, 4 + 8 = 12 and 8 + 4 = 12. This also works when they are multiplied, $4 \times 8 = 32$ and $8 \times 4 = 32$ .
Distributive property – when three natural numbers are being multiplied using brackets, you can also do this by multiplying the numbers separately. For example, 5 (2 + 3) = 25 and $5 \times 2 + 5 \times 3 = 25$ .

What is the natural numbers symbol?

A set of natural numbers is often represented by the symbol $ℕ$ .

$ℕ = {1, 2, 3, 4, 5 . . .}$

How to find the sum of natural numbers

The list of natural numbers creates an arithmetic sequence. There is a formula that you can use to help you find the sum of a sequence of natural numbers:

$\sum_{1}^{n} = \frac{(n (n + 1))}{2}$

In the above formula, n represents the number of terms. The sequence will start with 1. It is also important to note that the sum of all natural numbers is infinity.

Sigma $\sum_{}$ , is a notation used to represent the sum of terms

Find the sum of the first 50 natural numbers.

To do this, you would first start by looking at your formula, identifying the n from the question and substituting it into the formula;

$\sum_{1}^{n} = \frac{(n (n + 1))}{2}$

Since you are finding the sum of the first 50 terms, n = 50, therefore;

$\sum_{1}^{50} = \frac{(50 (50 + 1))}{2}$

Now you can simply solve the formula to find your answer;

$\sum_{1}^{50} = \frac{2550}{2}$

$\sum_{1}^{50} = 1275$

Find the sum of the first 100 natural numbers.

As before, you need to identify the n for the formula from the question. In this case, n=100, and now you can substitute it into the formula and solve the question:

$\sum_{1}^{n} = \frac{(n (n + 1))}{2}$

$\sum_{1}^{100} = \frac{(100 (100 + 1))}{2}$

$\sum_{1}^{100} = \frac{10100}{2}$

$\sum_{1}^{100} = 5050$

Learn more about arithmetic Sequences and Series.

Natural Numbers - Key takeaways

A natural number is a positive whole number higher than 1.
Natural numbers are often represented on a number line.
The notation $ℕ$ is used to represent a set of natural numbers.
The formula $\sum_{1}^{n} = \frac{(n (n + 1))}{2}$ can be used to find the sum of the first n terms of natural numbers.

Frequently Asked Questions about Natural Numbers

Natural numbers are positive whole numbers from 1 onwards.

The difference between natural and whole numbers is that whole numbers include 0, whereas natural numbers start from 1.

Consecutive natural numbers are numbers that follow one another, for example, 1, 2, 3, 4, 5, 6.

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What is a natural number?

A positive whole number.

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Natural Numbers

Number classes

Properties of natural numbers

What is the natural numbers symbol?

How to find the sum of natural numbers

Natural Numbers - Key takeaways

Frequently Asked Questions about Natural Numbers

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Natural Numbers

Number classes

Properties of natural numbers

What is the natural numbers symbol?

How to find the sum of natural numbers

Natural Numbers - Key takeaways

Frequently Asked Questions about Natural Numbers

What are natural numbers?

What is the difference between natural and whole numbers?

What does consecutive natural numbers mean?

Learn with 1 Natural Numbers flashcards in the free StudySmarter app

Join over 22 million students in learning with our StudySmarter App

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