1. Number Systems

2. Number Systems Natural Numbers
The whole numbers from 1 upwards. (Or from 0 upwards in some fields of mathematics )
The set is {1,2,3,...} or {0,1,2,3,...}
E.g. The positive integers and zero.
Integers
The whole numbers, {1,2,3,...} negative whole numbers {..., -3,-2,-1} and zero {0}. So the set is {..., -3, -2, -1, 0, 1, 2, 3, ...}
(Z is for the German "Zahlen", meaning numbers, because I is used for the set of imaginary numbers).

3. Number Systems Rational Numbers
The numbers you can make by dividing one integer by another (p/q) (but not dividing by zero). In other words fractions.
Q is for "quotient" (because R is used for the set of real numbers).
Examples: 3/2 (=1.5), 8/4 (=2), 136/100 (=1.36), -1/1000 (=-0.001), etc.
Irrational Numbers
Any number that is not a Rational Number.
Cannot be written as p/q where p and q are integers (e.g √2)
Real Numbers
All Rational and Irrational numbers. They can also be positive, negative or zero.
A simple way to think about the Real Numbers is: any point anywhere on the number line (not just the whole numbers).
Examples: 1.5, -12.3, 99, √2, π
They are called "Real" numbers because they are not Imaginary Numbers.

4. Number Systems Imaginary Numbers
Numbers that when squared give a negative result.
If you square a real number you always get a positive, or zero, result. For example 2×2=4, and (-2)×(-2)=4 also, so "imaginary" numbers can seem impossible, but they are still useful!
Examples: √(-9) (=3i), 6i, -5.2i
The "unit" imaginary numbers is √(-1) (the square root of minus one), and its symbol is i, or sometimes j.
i2 = -1
Complex Numbers
A combination of a real and an imaginary number in the form a + bi, where a and b are real, and i is imaginary.
The values a and b can be zero, so the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers.
Examples: 1 + i, 2 - 6i, -5.2i, 4

5. Number Systems Illustration Natural numbers are a subset of Integers
Integers are a subset of Rational Numbers
Rational Numbers are a subset of the Real Numbers
Combinations of Real and Imaginary numbers make up the Complex Numbers.

6. Number systems
Exercise 4.1, Page 93
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7. Questions What are the natural numbers ?
Positive intergers (whole numbers) and zero
What is the symbol for integers ? Z
What is a rational number. Give an example.
Numbers you get by dividing one integer by another.
2/3
Symbol ? Q