Numbers Definitions and Examples
Introduction
In mathematics, a number is a value that represents a quantity, such as 2, 3, or 4. We use numbers all the time in our everyday lives. For example, we use them to count how many people are in a room or to measure how much water we need to fill a container. But numbers can also have other meanings in math. In this blog post, we’ll take a look at some of the different ways numbers can be used and explore some examples to help you better understand their meaning.
Numbers
There are a few different types of numbers, and we’ll start with the most basic: natural numbers. Natural numbers are counting numbers, like 1, 2, 3, 4, 5. They’re also called whole numbers.
Next, we have negative numbers. Negative numbers are less than zero. For example, -1 is less than 0, and -5 is less than -1 and 0.
Zero is a number too! It’s just not a positive or negative number.
Now let’s talk about decimal numbers. Decimal numbers have a decimal point in them (like 10.50). The decimal point separates the ones place from the tenths place. In the number 10.50, the 5 is in the tenths place because it’s to the right of the decimal point. And since there’s nothing to the right of the 5, that means there aren’t any hundredths or thousandths places either! If there was a number after the 5 like 10.504, then we would say that 4 is in the thousandths place because it’s after the decimal point and to the right of all of the other digits.
Fractions are another type of number that you might see often. They’re written with a slash (/), and they show part of a whole number. For example, if I have one slice of pizza and I cut it into four pieces, each piece would be 1/4 pizza (one-quarter).
What are Numbers?
In mathematics, a number is a concept from number theory, and it is a representation of a quantity using a symbol. In physics and engineering, a number is a measurement of something, such as an electric current or the frequency of a wave.
There are many different types of numbers, including natural numbers (1, 2, 3), whole numbers (0, 1, 2), integers (-5, 0, 10), rational numbers (1/2, 2/3), real numbers (-5.5, 0, 10.9), and complex numbers (3+4i). Each type of number has its own properties and uses.
Natural numbers are the counting numbers {1, 2, 3,…}. They are often used to represent the size of something like how many people are in a room or how many apples are in a basket. Natural numbers can also be used to represent position in space like on a map or in line at the store.
Whole numbers are all the natural numbers {0, 1,…} plus zero. They are often used to represent things that can’t be divided up like money $4 or time 4:00pm. Whole numbers can also be used for ordering like when we put books on shelves in order from shortest to tallest or ordered from left to right.
Integers include all the whole numbers {…,-2,-1,0}, but they also include negative whole numbers {-3,-4,…}.
Introduction to Numbers
Numbers are a critical part of mathematics, and they appear in many different contexts. In this section, we’ll give a brief overview of some of the most important types of numbers and their properties.
The first category of numbers is the natural numbers, which are also sometimes called counting numbers. These are the numbers that we use for counting: 1, 2, 3, 4, 5, and so on. The set of natural numbers goes on forever and has no limit.
The next category is the whole numbers, which include all of the natural numbers along with zero. So, the whole numbers are 0, 1, 2, 3, 4, 5, and so on.
The third category is the integers, which include all of the whole numbers along with their negative counterparts (or opposites). So, the integers are …-3, -2, -1, 0, 1 2 3 4 5… and so on. You can think of them as being like points on a number line: they can be positive or negative (or zero), but they always have a definite value.
The fourth category is the rational numbers , which are any number that can be expressed as a fraction . So rational numbers include things like 1/2 or 3/4 or 22/7… really any number that can be written as one integer divided by another . Rational numbers also include all of the integers , since every integer can be expressed as a rational number.
Pre-number Math
Pre-number math includes addition, subtraction, multiplication, and division. These are the operations that you use to solve problems.
Number Names
In mathematics, a number is a concept from number theory, and it is a representation of a quantity using a symbol. Numbers can be written in many ways, including as Arabic numerals, Roman numerals, Chinese characters, and other ways. There are also certain types of numbers, such as natural numbers, whole numbers, complex numbers, and more. In this article, we will take a look at some of the most common number names and their definitions.
Natural numbers are the counting numbers {1, 2, 3,…}. They are often used for counting or enumerating objects (e.g., “there are six apples on the table”). Whole numbers are the natural numbers plus 0 {0, 1, 2,…}. They are often used for measurement (e.g., “this book is 300 pages long”).
Complex numbers are any number that can be expressed in the form a + bi where a and b are real numbers and i is the imaginary unit {?-1}. Complex numbers can be used to solve certain equations that have no solutions in the real world (e.g., x^2 + 1 = 0 has no solutions in the real world but it does have solutions in the complex world).
Integers are all whole numbers {…,-3,-2,-1,0,1,2,…}. They are often used for operations such as addition and subtraction (e.g., “-5 + 3 = -2”).
PEMDAS
PEMDAS is an acronym that stands for parentheses, exponents, multiplication and division (left to right), and addition and subtraction (left to right). This order of operations is used to simplify equations with multiple operations.
Number Systems
There are many different types of number systems in existence, but they can all be broadly classified into two categories: decimal and non-decimal. Decimal systems are based on the number ten, and include the most commonly used number system, the Hindu-Arabic numeral system. Non-decimal systems are based on other numbers, such as 2 (binary), 8 (octal), or 16 (hexadecimal).
The decimal numeral system is used by most people in everyday life. It is sometimes referred to as base 10, because it uses 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent numbers. The Hindu-Arabic numeral system is believed to have originated in India in the 4th or 5th century CE. It spread to the Arab world via trade and was adopted by Europeans during the Renaissance.
The binary numeral system is used in computer science and electronics. It is also known as base 2 because it uses just two digits (0 and 1). Binary numbers are often written using a 0b prefix. For example: 0b1001 = 9
The octal numeral system is used in computing; it is also known as base 8 because it uses 8 digits (0 to 7). Octal numbers are often written using an 0o prefix. For example: 0o755 = 493
The hexadecimal numeral system
Types of Numbers
There are two types of numbers: natural numbers and whole numbers.
Natural numbers are the numbers 1, 2, 3, 4, 5, and so on. They are also called counting numbers because you can use them to count things.
Whole numbers are the numbers 0, 1, 2, 3, 4, 5, and so on. They include all of the natural numbers plus the number 0.
Cardinal Numbers and Ordinal Numbers
Cardinal numbers are the basic counting numbers: 1, 2, 3, 4, 5, etc. They are also known as “counting numbers” because they indicate quantity. Ordinal numbers are used to indicate order or position: 1st, 2nd, 3rd, 4th, 5th, etc.
When we count objects using cardinal numbers, we simply state how many there are: “There are three cats.” When we use ordinal numbers to identify objects or positions, we specify not only how many there are but also their order: “Please line up in order of height, with the tallest person first.”
Here are some more examples of cardinal and ordinal numbers in action:
Cardinal Numbers
1. There are ten students in my class.
2. We need five more volunteers for the bake sale.
3. Can you count to twenty?
4. I have two brothers and three sisters.
5. This is my third time taking this exam.
Ordinal Numbers
1. She finished first in the marathon.
2. He came in second place in the spelling bee competition.
3. We were seated at table six during the banquet.
4. Do you live on the third floor or the fourth floor?
5. Please stand in line according to your birth month, with January being first and December being last.”
Consecutive Numbers
In mathematics, consecutive numbers are numbers that follow each other in order. They have a difference of 1 between each number. For example, the numbers 2, 3, 4, 5, 6 are consecutive because they follow each other in order and have a difference of 1 between each number.
Consecutive numbers can be positive or negative. For example, the numbers -3, -2, -1, 0, 1 are consecutive because they follow each other in order and have a difference of 1 between each number.
Consecutive numbers are often used in mathematical formulas. For example, the sum of three consecutive numbers is always three times the middle number. So, if we take the numbers 2, 3, 4 as an example: 2 + 3 + 4 = 9 This is because: (2 + 3 + 4) / 3 = (9 / 3) = 3 So the sum of three consecutive numbers is always three times the middle number.
Integers
Integers are numbers that can be positive or negative, but cannot be fractional or decimal. Examples of integers include: -5, 0, 3, and 9.
Natural Numbers and Whole Numbers
The natural numbers are the set of all positive integers: 1, 2, 3, 4, and so on. The whole numbers are the set of all integers: that is, the natural numbers plus the number 0. So the whole numbers are 0, 1, 2, 3, and so forth.
Even Numbers and Odd Numbers
In mathematics, even numbers are integers that are divisible by two. That is, if the integer is evenly divided by 2 with no remainder, it is an even number. Even numbers can be positive or negative, but they cannot be fractional or decimal.
Some examples of even numbers include 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. There are an infinite number of even numbers because there is no highest possible value.
Odd numbers are integers that are not divisible by two. That is, if the integer is not evenly divided by 2 with no remainder, it is an odd number. Odd numbers can be positive or negative as well (for example: -3,-1,1,3), but they cannot be fractional or decimal.
Some examples of odd numbers include 1, 3 , 5 , 7 , 9 , 11 , 13 , 15 , 17 , 19 . As with even numbers there is no highest possible value for odd numbers—there are an infinite number of them.
Prime Numbers and Composite Numbers
Prime Numbers
A prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole number that can be divided evenly into another number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Factors of 23 are 1 and 23. Factors of 29 are 1 and 29.
Composite Numbers
A composite number is a whole number greater than 1 that has more than two factors.
Factors of 4 are 1, 2, 4.
Factors of 6 are 1, 2, 3, 6.
The factors of 8 are 1, 2, 4, 8.
Co-prime Numbers
When two numbers are co-prime, it means that they have no common factors other than 1. In other words, their highest common factor is 1.
For example, the numbers 9 and 16 are co-prime because the highest common factor between them is 1. However, the numbers 15 and 25 are not co-prime because the highest common factor between them is 5.
Co-prime numbers are important in many areas of mathematics, including number theory and cryptography.
Perfect Numbers
A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For example, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
Perfect numbers are rare. The first few perfect numbers are 6, 28, 496 and 8128.
The property of being a perfect number is called perfection.
Fractions and Decimals
When we talk about numbers, we usually think of whole numbers like 1, 2, 3, and 4. But there are other types of numbers too, like fractions and decimals.
Fractions are parts of a whole number. For example, if we have a pizza and we cut it into four equal slices, each slice would be one-fourth (1/4) of the pizza. We can also have fractions that are not equal parts of a whole number. For example, if we have two pieces of candy and one is twice as big as the other, then the bigger piece would be two-thirds (2/3) the size of the smaller piece.
Decimals are numbers that have a point after them. For example, if we say that something is 0.5 (read as “zero point five”), that means it’s half of something. If we say that something is 0.25 (read as “zero point twenty-five”), that means it’s one quarter of something.
Rational Numbers and Irrational Numbers
There are two types of numbers in mathematics: rational numbers and irrational numbers. Rational numbers are any number that can be expressed as a fraction, such as 3/4 or 4/5. Irrational numbers are any number that cannot be expressed as a fraction, such as ?2 or ?.
Both rational and irrational numbers can be positive or negative. All whole numbers are rational numbers, but not all fractions are rational numbers. For example, the fraction 2/3 is a rational number, but the fraction 1/?2 is an irrational number.
All real numbers (numbers that can be represented on a number line) are either rational or irrational. Imaginary numbers, such as i, are not real numbers and therefore are not classified as either rational or irrational.
Real Numbers
In mathematics, real numbers are the ordinary numbers that we use for counting and measuring. They include whole numbers, fractions, and decimals. Real numbers can be positive or negative, and they can be written as either decimal fractions or percentages.
The term “real number” is used in contrast to “imaginary number”. Imaginary numbers are those that cannot be expressed as a real number, such as the square root of -1. Although imaginary numbers have some uses in mathematics, they are not commonly used in everyday life.
Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i2 = ?1. In other words, a complex number is a number that can be written as the sum of a real number and an imaginary number.
The real part of a complex number is the number a, and the imaginary part is the number b. The imaginary unit is denoted by i, and has the property that i2 = ?1. Complex numbers can be added and subtracted like any other numbers, but they must be multiplied and divided using special rules.
The square of a complex number is always a positive real number. This follows from the fact that i2 = ?1. For example, if z = 3 + 4i, then z2 = (3 + 4i)(3 + 4i) = 9 + 12i ? 12i ? 16i2 = 9 ? 16(?1) = 25.
You can also multiply two complex numbers together using the distributive property. For example, if z1 = 2 + 3i and z2 = 4 + 5i, then z1z2 = (2 + 3i)(4 + 5i) = 8 + 10i + 12i + 15i2 = 8 + 22i ? 15(?1) = 23 + 22i.
Factors and Multiples
Factors and multiples are two concepts that are often confused. A factor is a number that divides evenly into another number. A multiple is a number that is the product of two or more factors.
To better understand the difference between these two concepts, let’s look at some examples:
Examples:
1) 2 is a factor of 10 because 10 ÷ 2 = 5.
2) 3 is a factor of 15 because 15 ÷ 3 = 5.
3) 4 is not a factor of 15 because 15 ÷ 4 = 3 with a remainder of 3. However, 4 is a multiple of 15 because 15 x 4 = 60.
4) 6 is a factor of 24 because 24 ÷ 6 = 4.
5) 8 is not a factor of 24 because 24 ÷ 8 = 3 with a remainder of 0. However, 8 is a multiple of 24 because 24 x 8 = 192.
Highest Common Factor (HCF)
The highest common factor (HCF) of two or more numbers is the largest number that divides evenly into all of the numbers.
For example, the HCF of 24 and 36 is 12.
To find the HCF of two numbers:
1. List the factors of each number.
2. Find the largest number that appears in both lists. This is the HCF.
In some cases, it’s easy to spot the HCF straight away. For example, the HCF of 12 and 18 is 6 because 6 is the only number that goes into both 12 and 18 evenly (i.e., without leaving a remainder).
Least Common Multiple (LCM)
The least common multiple (LCM) of two or more positive integers is the smallest positive integer that is evenly divisible by each of the numbers. So, the LCM of 3 and 4 is 12 because 12 is the smallest number that is evenly divisible by both 3 and 4.
You can find the LCM of two or more numbers using either repeated division or prime factorization. To use repeated division, simply keep dividing the larger number by the smaller number until you get a remainder of 0. The last number you divide by before getting a remainder of 0 is your LCM. For example, to find the LCM of 15 and 18:
15 ÷ 3 = 5 with a remainder of 0
18 ÷ 3 = 6 with a remainder of 0
15 ÷ 6 = 2 with a remainder of 3
18 ÷ 6 = 3 with a remainder of 0
15 ÷ 3 = 5 with a remainder of 0
The LCM is 30.
To find the LCM using prime factorization, list out the prime factors for each number and then multiply all of the factors together. The resulting product is your LCM. For example, to find the LCM of 15 and 18:
The prime factors for 15 are 3 and 5
The prime factors for 18 are 2, 3, and 9
The LCM is 90
Prime Factorization
In mathematics, prime factorization or prime decomposition is the factorization of a number into a product of prime numbers. For example, the number 120 can be decomposed into a product of primes as follows:
120 = 2 × 2 × 2 × 3 × 5
The process of factoring a number into itsprime factors is called prime factorization. The unique factorization theorem states that every positive integer has a unique prime factorization.
Properties of Numbers
There are various properties of numbers that help us understand them better. These properties include:
Commutative Property: The commutative property states that when two numbers are added (or multiplied), the order in which they are added (or multiplied) does not affect the result. For example, 3 + 5 = 5 + 3 = 8.
Associative Property: The associative property states that when three or more numbers are added (or multiplied), the order in which they are added (or multiplied) does not affect the result. For example, (5 + 3) + 4 = 5 + (3 + 4) = 12.
Distributive Property: The distributive property states that when one number is multiplied by a sum of two or more numbers, the result is the same as if the number were multiplied by each number individually and then those products were added together. For example, 6 x (5 + 2) = 6 x 5 + 6 x 2 = 30.
Examples on Numbers
When it comes to numbers, there are an endless amount of examples that can be used to demonstrate their definition and function. Here are just a few examples to help better understand numbers:
1. One is the loneliest number – This example demonstrates the definition of a number as a single digit.
2. Two’s company, three’s a crowd – This example demonstrates how numbers can be used in groups or crowds.
3. Four is the magic number – This example demonstrates how numbers can have special meaning or power.
4. Five minutes to live – This example demonstrates how numbers can be used to represent time.
5. Six feet under – This example demonstrates how numbers can represent depth or height.
Conclusion
In mathematics, a number is a concept inspired by the natural numbers and real numbers. It helps to quantify things and to understand their order or magnitude. There are different types of numbers, including natural numbers, whole numbers, integers, rational numbers, irrational numbers, and complex numbers. Each type of number has its own definition and examples.
