Numbers and Arithmetic (Mathematics Curriculum)
What Are Numbers?
Numbers are used to count. They are the first thing we learn in mathematics.
Numbers are represented by symbols called numerals: 1, 2, 3, 4, 5...
The Oxford English Dictionary defines a number as "an abstract entity representing a quantity, used to express how many things are being referred to, or how much there is of some thing."
Here are some examples of numbers.
A History of Numbers
The earliest record of numbers are marks carved into bone and other objects, like this Ishango stick shown on the left.
These would have been very useful as an accounting tool. The trouble is, you need to make as many marks as there are objects to count.
The next breakthrough would have been using symbols (called numerals) to represent numbers.
Roman numerals were used in Europe.
The number 0 was introduced by the Indian mathematician Brahmagupta in 628AD.
Another innovation made in India was the use of place value. The same numeral in a different place can mean a new number: 10 and 100 are different numbers although they make use of the same numerals.
The Hindu numerals were brought, via the Arabs, into the West, where they were adopted. Our modern day numerals are descended from Hindu-Arabic numerals.
Types of Numbers
There are different types of numbers: whole, positive and negative, odd and even, and many more.
In this mini-curriculum, you will learn about the types of numbers.
Types of Whole Numbers
The Number curriculum deals with whole numbers.
There are different types of whole numbers, depending on whether you include 0 and the negative numbers.
A natural number is a whole number from 1 upwards. It is a counting number.
The whole numbers are the numbers 0, 1, 2, 3...
An integer is a whole number that may be positive, negative, or zero.
Positive and Negative Numbers
Numbers can be positive and negative.
In this mini-curriculum, you will learn about positive and negative numbers and how to do arithmetic with negative numbers.
A positive number is a number greater than zero.
A negative number is a number less than zero.
Arithmetic with Negative Numbers
Take extra care when doing arithmetic with negative numbers. Two negatives make a positive.
1 − −1 = 1 + 1 = 2
1 − × −1 = 1
The absolute value of a number is the non-negative value of a number without regard to its sign. It is denoted by || either side of the number.
Even and Odd Numbers
Numbers can be even and odd, depending on whether they can be divided by 2.
In this mini-curriculum, you will learn about even and odd numbers.
An even number is a whole number that can be divided exactly by 2.
An odd number is a whole number that cannot be divided exactly by 2.
Arithmetic with Even and Odd Numbers
Arithmetic with even and odd numbers follow predictable patterns.
even + even = even
even + odd = odd
Numbers are prime numbers if they can only be divided by 1 and themselves.
Otherwise they are composite numbers.
In this mini-curriculum, you will learn about prime and composite numbers.
A prime number is a number that can be divided by only itself and 1.
A composite number is a number that can be divided exactly by at least one number that is not itself or 1.
This means a composite number is any number (other that 1) that is not a prime number.
Fundamental Theorem of Arithmetic
The fundamental theorem of arithmetic states that any number is either a prime number or can be found by multiplying prime numbers together.
This means that any composite number can be written as a product of primes.
Square Numbers and the Square Root
Square numbers are the the result of multiplying a number by itself.
A square root is the number, which when multiplied itself, creates a square number.
In this mini-curriculum, you will learn about square numbers and square roots.
A square number is a number that results from a number being multiplied by itself.
The square root of a number, when multiplied by itself, gives that number. It is denoted by writing a √ symbol in front of a number.
The square root is the opposite of finding the square of a number.
The Number Line
A number line is a line where numbers are written in order, evenly spaced.
Number lines help us add and subtract numbers.
In this mini-curriculum, you will learn about number lines and how to use them.