Basic concepts

In real life, we face various problems, and we need to have the right tools for each of them. For example, to count the number of people in a cafeteria or to accurately measure the temperature of a car engine, we need to understand basic number concepts.

In this topic, you will learn about the different types of numbers, which will help you choose the most appropriate ones for each task. You will also learn a couple of tricks to deal with large amounts of numbers in a simple and effective way.

Basic number sets

Natural numbers are, of course, the best known and used set of numbers because we use them to count. Its importance lies in its simplicity and ease of use. The set of natural numbers is made up of the numbers we have known all our lives and is denoted by:

$$\mathbb{N} = \{1,2, 3, 4, \dots\}$$

There is a discussion about whether $0$ is a natural number or not. Historically, it is not and this is common in many applications even though the formal construction of the natural numbers actually starts with $0$.

On the other hand, if you wanted to represent losses, for example Celsius temperatures below freezing or elevations below sea level, then you would need negative numbers. The natural numbers, zero, and negatives are packed into what we know as the integers. We denote them as:

$$\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$$

Advanced number sets

To tackle more realistic problems, you require more precision than whole numbers. For example, to work with dollars, you should be able to represent the cents, or if you are dealing with the hours, it would be ideal to also include the seconds, each of which would correspond to $\frac{1}{3600}$ of an hour.

The rational numbers are the result of adding all these portions to the integers, thus achieving excellent precision. Each rational number can be written as the quotient of two integers, where the divisor cannot be $0$. We denote the set of all of them as:

$$\mathbb{Q} = \left\{ \frac{m}{n} \text{ with } m \text{ and } n \text{ in } \mathbb{Z} \text{, and } n \neq 0 \right \}$$

Rational numbers have many applications in real life, but if you think of them as a continuous line, you would discover that they leave gaps, that is, there would be numbers that are not rational. A classic example is $\sqrt{2} \approx 1.41$. These gaps are called irrational numbers and if you fill them in, then you get the real numbers, denoted by $\mathbb{R}$, which are the ideal model of a continuous line.

With real numbers, you can approximate real-life measurements as accurately as you want, from speed and temperature to stock prices and the distance between galaxies.

Arguably, the most famous irrational number is $\pi$ whose approximate value is $3.14$. It plays an important role in mathematics and is useful for understanding natural phenomena that contain circular shapes, such as the orbit of planets and even light waves.

The irrational number $e$ has a value of about $2.71$ and has become increasingly famous over the years both for its theoretical importance and for its wide range of applications, from radioactive decay to compound interest to the growth of populations.

Compact notation

What is the sum of the first $5$ natural numbers? At first, you may try to write out $1+2+3+4+5 = 15$, but if you wanted to add the first $100$ numbers, then the notation would be too long and repetitive.

To develop a compact notation, you can first notice that there is a clear pattern: each term of the sum is a natural number. We simply represent this pattern as $i$ (if, for example, we were adding the square of the natural numbers we would use $i^2$). With the pattern identified, all you need to know is where the sum starts and ends (in the example, we start at $1$ and end at $5$). Finally, to represent the sum we use the Greek letter sigma $\Sigma$, where we only indicate where the sum begins, where it ends and what the pattern is:

$$\sum_{i=1}^5 i = 1 + 2 + 3 +4 +5$$

A useful and well-known shortcut is that the sum of the first $n$ natural numbers is equal to $\frac{n (n+1)}{2}$ which represented with our new notation would be:

$$\sum_{i=1}^n i = \frac{n (n+1)}{2}$$Also, an immediate property is that if $a$ is any number, then:

$$\sum_{i=1}^n a \, i = a \sum_{i=1}^n i$$

To see why this happens, you just need to factor:

$$\sum_{i=1}^n a \, i = a + 2a + 3a + \dots + na = a\left(1 + 2 + 3 + \dots + n\right) = a \sum_{i=1}^n i$$

Sometimes you also deal with long multiplications where there is a definite pattern. In these cases you can leverage the same notation, but to indicate that you are multiplying, use the Greek letter pi $\Pi$:

$$\prod_{i=1}^5 i = 1 \times 2 \times 3 \times 4 \times 5$$

Conclusion

• The natural numbers are used to count and are denoted as $\mathbb{N}$.
• Integers extend natural numbers by adding $0$ and its negative versions; we denote them with $\mathbb{Z}$.
• With the rational numbers $\mathbb{Q}$ you can represent more precise quantities than with the integers and therefore deal with more complicated situations such as temperatures and monetary units
• The real numbers $\mathbb{R}$ complement the rational numbers, and with them, you can describe quantities with all the precision you need. Two of the most famous numbers are $\pi$ and $e$, which have great theoretical and practical importance.
• The notation $\Sigma$ is used to represent very long additions in a compact way, while the notation $\Pi$ does the same but with multiplications.