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# Number Systems

There are a variety of number systems, a handful of which are used on a regular basis for basic mathematics in intermediate and high school. These include natural numbers, integers, rational numbers, irrational numbers, real numbers, and more. Continue reading to learn more about the properties of each of these types of numbers.

## Natural numbers

The natural (or counting) numbers are the part of the number system that includes all the positive integers from 1 through infinity. They are used for the purpose of counting. Natural numbers do not include 0, fractions, decimals, or negative numbers.

The set of natural numbers is usually represented by the letter "N".

$N=\left\{1,2,3,4,5,6,7,8,9,10\dots \right\}$

Natural numbers include all whole numbers except for the number 0. In other words, all natural numbers are whole numbers, but not all whole numbers are natural numbers.

There are four properties that natural numbers fit into:

• Closure property – the sum or product of any two natural numbers is a natural number. This does not hold true for subtraction or division.
• Commutative property – the order in which you add or multiply natural numbers does not affect the result. This does not hold true for subtraction or division.
• Associative property – the way natural numbers are grouped in addition or multiplication does not affect the result. This does not hold true for subtraction or division.
• Distributive property – multiplication of natural numbers is always distributive over addition.

## Integers

The integers are a set of numbers consisting of the natural numbers, their additive inverses, and zero. In other words, they include all positive numbers, negative numbers, and 0. They can never be a fraction or decimal. All natural numbers are integers that start from 1 and end at infinity. All whole numbers are integers that start from 0 and end at infinity.

The set of integers is usually represented by the letter "Z". It can also be represented by the letter "J".

$Z=\left\{\dots -3,-2,-1,0,1,2,3,4\dots \right\}$

The sum, product, and difference of any two integers is always an integer. The same is not true for division.

There are four types of numbers that fall under the integer category.

There are rules for integers based on the four basic operations:

Addition rule – If the sign of both integers is the same, the result will have the same sign. Two positives equal a positive and two negatives equal a negative.

$5+5=10$

$-14+\left(-12\right)=-26$

If the two numbers being added have a different sign, it will lead to a subtraction and the result will have the sign of the larger (in absolute value) integer.

$-2+10=8$

$-10+2=8$

Subtraction rule – Keep the sign of the first number the same, change the operator from subtraction to addition, and change the sign of the second number. Once you have applied this rule, follow the rules for adding integers.

Multiplication division rule – If the signs are the same, multiply or divide, and the answer is always positive.

$5\ast 5=25$

$-5\ast \left(-5\right)=25$

If the signs are different, multiply or divide, and the answer is always negative.

$-5\ast 5=-25$

$\frac{25}{-5}=-5$

There are five properties that integers fit into. The first four are the same as natural numbers. The last one is the identity property.

The additive identity property states that any integer added to 0 will give the same number. So 0 is called the additive identity. For any integer x,

$x+0=x=0+x$

The multiplicative identity states that when an integer is multiplied by 1, it will give the integer itself as the product. So 1 is called the multiplicative identity. For any integer x,

$x\ast 1=x=1\ast x$

If any integer is multiplied by 0, the product will be 0.

$x\ast 0=0=0\ast x$

If any integer is multiplied by -1, the product will be the opposite of the number.

$x\ast -1=-x=-1\ast x$

## Rational numbers

Rational numbers can be expressed as a ratio between two integers. For example, the fractions $\frac{1}{3}$ and $\frac{-1211}{19}$ are both rational numbers.

The rational numbers include all the integers because any integer z can be expressed as the ratio $\frac{z}{1}$ .

All decimals that terminate are also rational numbers because, for example, 8.27 can be expressed as $\frac{827}{100}$ . Decimals that have a repeating pattern at some point are also rational. For example, $0.0833333\dots =\frac{1}{12}$ .

Some of the important properties of the rational numbers are as follows:

• The results are always a rational number if you add, subtract, or multiply any two rational numbers.
• A rational number remains the same if you divide or multiply both the numerator and denominator with the same factor.
• If you add 0 to a rational number, the result will be the number itself.
• Rational numbers are closed under addition, subtraction, and multiplication.

## Irrational numbers

An irrational number is one that cannot be written as a ratio, or fraction of integers. In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational. There are equations that cannot be solved by using ratios of integers.

One of the first such equations to be studied was $2={x}^{2}$ . What number times itself equals 2?

$\sqrt{2}$ is about 1.414 because ${1.414}^{2}=1.999396$ , which is close to 2. But you'll never reach 2 exactly by squaring a fraction (or terminating decimal). The square root of 2 is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern.

$\sqrt{2}=1.41421356237309\dots$

Another famous irrational number is the golden ratio, a number that has great importance in biology.

$1+\frac{\sqrt{5}}{2}=1.61803398874989\dots$

Another famous irrational number is pi, the ratio of the circumference of a circle to its diameter.

$\pi =3.14159265358979\dots$

And another famous irrational number is e, the most important number in calculus.

$e=2.718281828455904\dots$

Irrational numbers can be further divided into algebraic numbers, which are the solutions of some polynomial equations (such as $\sqrt{2}$ and the golden ratio), and transcendental numbers, which are not the solutions of any polynomial equation. $\pi$ and $e$ are both transcendental.

## Real numbers

The real numbers are the set of numbers that contain all of the rational numbers and all of the irrational numbers. The real numbers are "all the numbers" on the number line. There are infinitely many real numbers, just as there are infinitely many numbers in each of the other sets of numbers. But it can be proved that the infinity of the real numbers is a bigger infinity!

The smaller, or countable, infinity of the integers and rationals is sometimes called $\mathrm{\aleph }0$ (aleph-naught), and the uncountable infinity of the reals is called $\mathrm{\aleph }1$ or aleph-one.

There are actually even bigger infinities, but you would want to take a set theory class to learn about those.

## Complex numbers

The complex numbers are the set { $a+{b}_{i}$ | a and b are real numbers}, where i is the imaginary unit $\sqrt{-1}$ .

The complex numbers include the set of real numbers. The real numbers, in the complex system, are written as $a+{0}_{i}=a$ where a is a real number.

This set is sometimes written as C for short. The set of complex numbers is important because for any polynomial $p\left(x\right)$ with real number coefficients, all the solutions of $p\left(x\right)=0$ will be in C.

## ..and beyond

There are even bigger sets of numbers that are used by mathematicians. For example, the hyper-real numbers or the quaternions, discovered by William H. Hamilton in 1845, form a number system with three different imaginary units!

Cylinder

## Get help learning about number systems

From natural numbers to real numbers to the imaginary numbers of quaternions, the concepts learned in number systems can be challenging for students to remember and use practically. Getting help from a private tutor is an effective and efficient way to help clear up any misunderstandings your student may have when it comes to math concepts including number systems.

Working with a math tutor helps your student in many different ways. A private tutor will move at your student's pace, taking the time your student needs to make sure they understand each concept thoroughly before moving on to the next concept. They are right there to answer questions as your student thinks of them so they don't waste time completing equations the wrong way. Contact the Educational Directors at Varsity Tutors to see how tutoring can help your student today.

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