The Number System

Natural numbers:

       These number are 1,2,3,...
Note: We use the notation "..." to represent that the pattern continues infinitely. For example, the pattern here is that we add the value 1 to the previous number to get the next number. We get the number 2 by adding the value 1 to the number 1. We get to 3 by adding the value 1 to the number 2.

Whole Numbers:

      These numbers are 0,1,2,3,...

Integers:

      These numbers are ...,-3,-2,-1,0,1,2,3,...
Note: A natural number is a whole number, and a whole number is an integer.

Rational Numbers:

      These numbers are found when an integer is divided by another nonzero integer (you can't divide by zero). In other words, fractions make up the rational numbers.

Irrational Numbers:

      These numbers are numbers that cannot be expressed as a fraction. For example, , , , and are all irrational numbers. Note: is not an irrational number because this number is really just 2 which can be written as a fraction, namely 2/1.

Even and Odd Numbers:

An even number is a number that can be divided by 2 evenly. For example, the numbers -6,-4,-2,-0,2,4,8, and 64 are all even numbers. There are an infinite number of even numbers. An odd number is a number that cannot be divided evenly by 2. For example, the numbers -5,-3,-1,1,3,19, and 57 are all odd numbers. There are an infinite number of odd numbers. A quick trick in determining whether a number is even or odd is to just look at the last digit.

Example:

The number 1,456,234,154 is an even number, because the last digit (which is 4) is even. The number 1,345 is odd because the last digit (which is 5) is odd.

Digits:

There are ten digits: 0,1,2,3,4,5,6,7,8,9. Every integer is made up of digits.

Example:

Consider the number 1,245. The digits that make up that number are 1,2,4, and 5. Since four digits make up the number 1,245 it is called a four digit number. The 1 is called the thousands digit, the 2 is called the hundreds digit, 4 is called the tens digit, and 5 is called the ones (or units) digit because 1,245 can be written as follows:

1 1000 = 1000

2 100   =  200

4 10     =   40

5 1       =    5
_____
1245

Consecutive Integers:

Consecutive integers are integers listed in increasing size without any integers missing in between.

Example:

-2,-1,0,1,2,3 are consecutive integers. Note: 0,-1,-2,-3 are not consecutive integers because they are decreasing in size.

Prime and Composite Numbers:

A prime number is a positive integer that is divisible only by itself and by 1. For example, here is a list of the first 10 prime numbers: 2,3,5,7,11,13,17,19,23, and 29. Note: 0 and 1 are not considered prime numbers, 2 is the smallest prime, 2 is the only even prime (all other prime numbers are odd), there are an infinite number of prime numbers, not all odd numbers are primes (i.e. consider 9), and a number that is not prime is said to be composite.

Ordering Numbers:

There are six standard symbols used to put some type of ordering on numbers:

• < read as "less than"
• > read as "greater than"
• < read as "less than or equal to"
• > read as "greater than or equal to"
• = read as "equal to"
• read as "not equal to"

Example:

The following statements below are all true statements:

-3 < 0

4 > -1

34 < 100

3 = 3

5 4

Note: -5 > -3 is not a true statement because -5 is not greater than -3 (it is less than) since -5 appears to the left of -3 on the number line.

Absolute Value:

The absolute value of a number denoted by the symbol | |, is the number of units the number is from 0 on the number line.

Example:

The absolute value of 3 is 3 since 3 is three units from 0 on the number line. We write this as |3| = 3. Also, Since -3 is also 3 units from 0 on the number line, we also have |-3| = 3. A easy trick to remember is that the absolute value "strips off negative signs".