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Number

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
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- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
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- Candidate Test
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- Continuity
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- Derivative Functions
- Derivative of Exponential Function
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- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
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- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
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- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
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- Implicit Differentiation Tangent Line
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- Linear Approximations and Differentials
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- Maclaurin Series
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- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
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- Probability and Statistics
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- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
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- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
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- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
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- Angle Measure
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- Approximation and Estimation
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- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
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- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
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- Fractions
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- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
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- Graphs
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- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
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- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
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- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
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- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
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- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
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- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
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- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
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- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
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- Probability Generating Function
- Quantitative Variables
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- Residual Sum of Squares
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- Sample Mean
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- Single Variable Data
- Skewness
- Spearman's Rank Correlation Coefficient
- Standard Deviation
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- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

If we think about it for a moment, **numbers** are everywhere in our daily lives. They help us to think logically and to keep track of the things we do. For example, numbers help us with **simple tasks** like calculating the time that it takes you to get from home to your place of work, the amount of money that you need to pay for your shopping, and the amount of bags that you need to carry your shopping home. In addition to this, they are also especially useful to solve **more complex problems** in the world of sciences and engineering, like calculating the amount of fuel that will be needed to get a rocket into space, or the number of lorries that a warehouse needs to transport their customer's orders safely and on time.

In this article, we will define what numbers are, and we will explore the main types of numbers that you can find so that you can recognise them more easily. We will also explain what is studied by number theory, the different number systems and the concept of number sequence.

Numbers are considered the heart of Mathematics, and rightly so because without numbers Maths would simply not exist.

A **number** is a mathematical concept that represents quantity, which has many applications such as counting, measuring, labelling, and performing calculations that help us to solve problems, among others.

Numbers come in many varieties. Let's see some examples below.

Examples of numbers of different types:

As you can see, there are many types of numbers, let's identify each one in the following section so that you can recognise them more easily.

Numbers can be classified into different groups according to the types of numbers that they include. Let's take a look.

**Natural numbers** are also known as counting numbers, because these are the numbers that you first learn how to count with. They include all positive numbers greater than zero. That is 1, 2, 3, 4, 5, 6, and so on.

They are represented with the letter N. The set notation for natural numbers is as follows:

.

Whole numbers are closely related to natural numbers, with one main difference, as you can see from the definition below.

**Whole numbers** are basically the** natural numbers plus zero**. They do not include negative numbers, fractions or decimals.

They are represented with the letter W, and their set notation is shown below:

.

All natural numbers are whole numbers, but not all whole numbers are natural numbers, this is the case for zero. Let's see this in a diagram.

Natural and whole numbers can be represented on the number line as follows:

The **integer numbers** include all positive numbers, zero and negative numbers. Again, integers do not include fractions or decimals.

They are represented with the letter Z, and their set notation is as follows:

.

If we expand the previous diagram to include integers, it will look like this:

Looking at the diagram above, we can say that **not all integers are natural and whole numbers**, but **all natural and whole numbers are integers**. On the number line, integers can be represented like this:

Some examples of integer numbers are:

**Rational numbers** include all numbers that **can be expressed as a fraction** in the form , where p and q are integers and . This group of numbers **includes fractions and decimals**. Rational numbers are represented with the letter Q.

All integers, natural and whole numbers are rational numbers, as they can be expressed as a fraction with a denominator of 1. For example, 3 can be expressed as a fraction like this .

Some examples of rational numbers are:

Let's now define what we mean by **irrational numbers**.

**Irrational numbers** are numbers that **can't be expressed as a fraction of two integers**. Irrational numbers have **non-repeating decimals** that never end, and have **no pattern** whatsoever. They are represented with the letter Q'.

Some examples of irrational numbers are:

There are numbers with non-terminating decimals that are actually rational. This is the case of numbers with **non-terminating decimals that repeat in a pattern**, as they can be expressed as a fraction of two integers. For example, , the bar above the decimal 1 means that it repeats forever. So, is a rational number.

Read Convert between Fractions and Decimals to learn more about the different types of decimal numbers.

**Real numbers** include all the numbers that you can think of, which you can find in the real world, apart from imaginary numbers. Real numbers are represented with the letter R, and they include all rational and irrational numbers, therefore the set of real numbers can be represented as .

Some examples of real numbers are:

Apart from real numbers, mathematicians came up with a special type of number to be able to solve the square root of negative numbers, included in simple quadratic equations such as . If we try to solve this equation using Algebra, we end up with the following:

But using only real numbers, we can't go any further than this. This is when imaginary numbers come in to play.

**Imaginary numbers** are the root of negative numbers.

We know that we can't take the square root of negative numbers, because there is no number that when squared will result in a negative number. In this case, we need to use imaginary numbers. To do this, we say that . Let’s see this more clearly with an example.

**Solve using imaginary numbers.**

can be written as

If we substitute with ,

then we can say that

Now, let's explore a few other concepts related to the topic of numbers.

What do we mean by number theory?

**Number theory** is the branch of Mathematics that studies positive integer numbers, their properties and relationships.

The types of numbers studied by number theory, can be classified into some of the categories shown in the table below.

Name | Definition | Examples |

Even | Integer numbers that can be divided by 2. | |

Odd | Integer numbers that cannot be divided by 2. | |

Square | Integer numbers that result from multiplying a number by itself. | |

Cube | Integer numbers that result from multiplying a number by itself 3 times. | |

Prime | This type of numbers have only 2 factors, because they are only divisible by itself and 1. | |

Composite | This type of numbers have more than 2 factors. | |

Perfect | Integer numbers that result from the sum of their proper divisors. | 6 is a perfect number, because if you add the divisors of 6, which are 1, 2, and 3, you get 6 as a result. Other examples include 28, 496, 8128, ... |

Fibonacci | Series of integers, in which each number results from adding the two preceding numbers, starting from 1. |

When you see a list of numbers in order, and you can identify a pattern between them, then you have found a **number sequence**.

A **number sequence **is a list of numbers in either ascending or descending order that follows a pattern or rule to obtain the following number (term) in the sequence.

Number sequences can be **finite**, if they have an end, and** infinite, **if they have no end.

Here are some examples to help you recognise a number sequence.

**Identify if the following lists of numbers represent a sequence:**

a) 1, 3, 5, 7, 9, ...

Yes, this is a number sequence, because it is a list of numbers in ascending order, and there is a consistent pattern to find the following term in the sequence (add 2).

b) 4, 0, 3, 1, 7, ...

No, this is not a number sequence, because it is a list of numbers in **no specific order** whatsoever.

c) 2, 4, 6, 8, 0, ...

No, this is not a number sequence, because **the pattern is not consistent**. Even though the 2^{nd}, 3^{rd} and 4^{th} terms are obtained by adding 2 to the previous term, the zero (0) in the 5^{th} term breaks the pattern.

Read Sequences to learn more about the different types of number sequence.

**Number systems** are systems that represent numbers using a specific set of digits and letters.

The most common number system, that we use on a regular basis, is the **decimal system**, which uses the digits 0 to 9. In the table below, you can also see other types of number systems that are mostly used by computers.

Name | Base | Digits/Letters | Example |

Decimal | 10 | (0 - 9) | 15 |

Binary | 2 | (0, 1) | 15 in binary:(1111)_{2} |

Octal | 8 | (0 - 7) | 15 in octal:(17)_{8} |

Hexadecimal | 16 | (0 - 9, A - F) | 15 in hexadecimal:(F)_{16} |

The table below shows you the equivalence between the different number systems for the numbers from 0 to 15:

Decimal | Binary | Octal | Hexadecimal |

0 | 0000 | 0 | 0 |

1 | 0001 | 1 | 1 |

2 | 0010 | 2 | 2 |

3 | 0011 | 3 | 3 |

4 | 0100 | 4 | 4 |

5 | 0101 | 5 | 5 |

6 | 0110 | 6 | 6 |

7 | 0111 | 7 | 7 |

8 | 1000 | 10 | 8 |

9 | 1001 | 11 | 9 |

10 | 1010 | 12 | A |

11 | 1011 | 13 | B |

12 | 1100 | 14 | C |

13 | 1101 | 15 | D |

14 | 1110 | 16 | E |

15 | 1111 | 17 | F |

This is just an introduction to the different types of number systems, you can also convert numbers between the different number systems and perform arithmetic operations with them, but that goes beyond the scope of this article.

- Natural numbers, also known as counting numbers, are all numbers starting from 1.
- Whole numbers are all natural numbers plus zero.
- Integer numbers include all negative and positive whole numbers.
- Rational numbers can be expressed as a fraction of two integers, while irrational numbers can't.
- Real numbers include all rational and irrational numbers.
- Real numbers are all the numbers that you can think of, apart from imaginary numbers.
- Number theory is the branch of Mathematics that studies positive integer numbers, their properties and relationships.
- A number sequence is a list of numbers in either ascending or descending order that follow a pattern or rule to obtain the following number (term) in the sequence.
- The four main types of number systems are: decimal, binary, octal and hexadecimal.

Some examples of numbers of different types include:

-3, 0, 2, 3.8, 3/4, π, and √2.

- Natural
- Whole
- Integers
- Rational
- Irrational
- Real
- Imaginary

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