[Math Lair] Number Systems

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The first step in exploring the systems of numbers is the natural numbers. The natural numbers are the numbers used for counting, usually written 1, 2, 3, 4, ... . The "..." means that the natural numbers go on forever, to infinity. When you add or multiply two natural numbers together, the result is always a natural number. However, when subtracting one natural number from another, or dividing one by another, the result is not always a natural number. For example, the answer to 3 − 3 is not a natural number.

For modern-day people, the simplest addition to this system of numbers might be the number zero. This wasn't the way it was in antiquity, though. for example, the ancient Greeks tended to see numbers in terms of geometrical lengths, and a length of zero represented something that just wasn't there. It was not until after the year 500 that a symbol for zero was used regularly, and it was not for a millennium after that that zero was generally considered to be a number in its own right, not just the absence of one. The set of natural numbers plus zero is sometimes referred to as the whole numbers, although the term is sometimes used to refer to various other sets.

After that addition to the family of numbers, we might consider the next simplest addition to be that of the negative integers. Again, like zero, the negative numbers were not fully incorporated into mathematics until Cardano published Ars Magna in 1545.

Rational numbers are more ancient than the negative numbers. They can be found in the mathematics of ancient Egypt, including the Rhind papyrus. Rational numbers include all numbers of the form a/b, where a and b are integers.

The next step would be to the algebraic numbers. These are numbers that can be the solution of an algebraic equation with integer coefficients. They would include, for example, all square roots, cube roots, fourth roots, and so on. They also include some other numbers (for example, the solutions to some equations of degree 5 or higher cannot be expressed using ordinary radicals, but they are still algebraic numbers). These numbers also have a long history. In the sixth century B.C., Pythagoras considered the length of a right triangle whose two shorter sides were both 1. The length of the hypotenuse would then be root 2, and it was found that this number is not a rational number; in other words, it could not be expressed in the form a/b where a and b are integers. Therefore, it is an irrational number. However, root 2 is an algebraic number because it is a solution to the equation x² = 2.

It was not until the nineteenth century that it was shown that some numbers cannot be the solution to any algebraic equation with integer coefficients. These numbers are known as transcendental numbers. π, the ratio of the circumference of a circle to its diameter, is a transcendental number. e, the base of natural logarithms, is another one.

Together, the algebraic numbers and the transcendental numbers form the real numbers. The real numbers are a complete ordered field. "Complete" means that there are no "gaps" in the real numbers; between any two numbers, you can always find another number or, for that matter, an infinite number of them. "Ordered" means that the sizes of its elements can be compared. A field is a system of numbers wherein every number has an additive inverse and every number (other than zero) had a multiplicative inverse, and it obeys the laws of arithmetic.

Up until this point, all of the numbers that we have considered could be plotted on a number line. 2, between 1 and 3. 0, to the left of 1. −1, to the left of zero. ½, halfway between 0 and 1. root 2, a bit more than 40% of the way between 1 and 2. π, a bit less than one-seventh of the way between 3 and 4.

Having got to this point, we might ask ourselves where root −1 can be found on the number line. In other words, we are looking for where to put a number whose square is −1. Can such a number be positive? No, because when you square a positive number, you get another positive number. Can it be negative? No, because when you square a negative number, you get a positive number. If we want to represent this number on a number "line", we will have to extend the number line into another dimension. root −1 is called i, which is an example of an imaginary number. The system of complex numbers consists of all numbers of the form a + bi. We can no longer plot these numbers on a number line; we'll have to plot them on a plane. It turns out that, even though this system of complex numbers uses "imaginary" numbers, it has several real-world applications.

Can we go any further? The multiplication of complex numbers has a simple interpretation as the rotation of a two-dimensional plane. In the nineteenth century, William Rowan Hamilton wondered if the idea could be generalized to three-dimensional or higher-dimensional space. In 1843 he discovered the idea of quaternions. The form of a quaternion is

a + bi + cj + dk
where i, j, and k are hypercomplex numbers. The following rule holds for these hypercomplex numbers:
i² = j² = k² = ijk = −1
The key to quaternions is that the commutative law of multiplication does not hold for them. For example, ij = k, but ji = −k.

This marks the end of the journey through number systems for now. Having gone to quaternions, one might ask: Can we go still farther? Well, we could, moving on to further families of hypercomplex numbers, but they aren't used frequently. We would also have to discard further laws of arithmetic.

You may also be interested in Number System Tutorial.

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