the real number system
Our idea of number changed over time as we discover new situations or problems that require the use of different types of numbers. We started off using numbers for counting. Such as, how many buffaloes a farmer owns or how many children a man has. These are called natural numbers or counting numbers.
Counting numbers naturally start from 1 and go on to infinity.
1, 2, 3, 4, 5, 6, ... (The three dots at the end is a common math notation meaning that it goes on forever!)
At some point, someone realised (Indian and Persian mathematicians) that we needed a symbol for when we have nothing. When a farmer has no buffaloes, he has zero buffaloes. All natural numbers plus zero are known as whole numbers.
Wait...what is the point of zero?
Now we get to the more abstract part. When the farmer has no buffaloes and owes someone else 5 buffaloes, we could say that the farmer owns negative 5 buffaloes. The set of numbers that include negative and positive whole numbers are called integers.
Why do we need negative numbers to make things more complicated?
Of course now you are thinking in the real world not everything is always in its whole form. Some things, like half a pint of milk or a quarter slice of pizza, are in parts. These are fractions and if we add them to integers we get a set of rational numbers. Fractions in the rational number set can be expressed as a ratio of integers. This means that if you convert a decimal to a fraction, it is rational IF the numerator and denominator are integers. Sounds rational right?
Well, if there are rational numbers, then there must be irrational numbers. These numbers cannot be expressed as a ratio of integers. As decimal numbers, they never terminate or repeat any patterns.
There are also numbers that are imaginary. But we will leave that for another time.
Counting numbers naturally start from 1 and go on to infinity.
1, 2, 3, 4, 5, 6, ... (The three dots at the end is a common math notation meaning that it goes on forever!)
At some point, someone realised (Indian and Persian mathematicians) that we needed a symbol for when we have nothing. When a farmer has no buffaloes, he has zero buffaloes. All natural numbers plus zero are known as whole numbers.
Wait...what is the point of zero?
Now we get to the more abstract part. When the farmer has no buffaloes and owes someone else 5 buffaloes, we could say that the farmer owns negative 5 buffaloes. The set of numbers that include negative and positive whole numbers are called integers.
Why do we need negative numbers to make things more complicated?
Of course now you are thinking in the real world not everything is always in its whole form. Some things, like half a pint of milk or a quarter slice of pizza, are in parts. These are fractions and if we add them to integers we get a set of rational numbers. Fractions in the rational number set can be expressed as a ratio of integers. This means that if you convert a decimal to a fraction, it is rational IF the numerator and denominator are integers. Sounds rational right?
Well, if there are rational numbers, then there must be irrational numbers. These numbers cannot be expressed as a ratio of integers. As decimal numbers, they never terminate or repeat any patterns.
There are also numbers that are imaginary. But we will leave that for another time.