Number Systems

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).

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.

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

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.

Number systems Exercise 4.1, Page 93 Read page 90 - 94

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