Introduction to Integers

Signed Numbers

       We know that       10 - 6=4

       and        17 - 9=8.

       But       4 - 5=?

       Let us observe the followig pattern:

4 - 1=3
4 - 2=2
4 - 3=1
4 - 4=0
4 - 5=owe 1
4 - 6=owe 2

       Since there are numbers like the last two of the above subtraction, special symbols were invented for "owe 1," "owe 2," etc. They weite "owe 1" as "-1", and read it as negative one or minus one. In this way, the numbers corresponding to "owe 2" and "owe 3" are written as -2 and -3, respectively. A number with a "-" sign is called as negative number.

       With the introduction of negative numbers, it is necessary to give a sign to a number unlike positive numbers which you can put a "+" sign or just the number.

• A positive number has a "+" sign.
• A negative number has a "-" sign.

       For convinience, we drop the "+" sign for a positive number. For example, +5=5.

       The signed numbers are also called directed numbers. They are frequently used in our daliy life.

Integers

       The numbers that are

• 1 greater than zero,
• 2 greater than zero,
• 3 greater than zero,

       1, 2, 3, ... are called positive integers

       The numbers that are

• 1 less than zero,
• 2 less than zero,
• 3 less than zero,

-1, -2, -3, ... are called negative integers

       Combining zero and the positive integers, we have the whole numbers 0, 1, 2, 3, ...

       The whole numbers together with the negative integers form the set of integers

{... , -3, -2, -1, 0, 1, 2, 3, ...}

MATH FOCUS

A natural number is any whole number freater than zero, Set of Natural numbers (N)={1, 2, 3, 4, 5, ...} A whole number is a number that is neither 0 or a natural number. Set of Whole numbers (W)={0, 1, ,2 ,3 ,4, 5, ...} The set of integers (Z) consists of positive integers, the number zero, and negative integers. Z={... , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}

Number Line and Ordering of Integers

       A thermometer is at room temperature of +26 dgrees celsius. This means that the room temperature is above 0 degrees celsius. When it is dipped into a breaker of water containing ice and salt, the temperature will drop to about -9 degrees celsius. That means the temperature is below 0 degrees celsius.

On any number line, take note of the following.

• The interval between any integer and next has the same length.
• Each of these equal intervals can represent any number of units.

MATH FOCUS

Numbers to the right of zero are positive numbers. We attach a + sign in front of each of these numbers, that is, +1, +2, +3, +4, ... These numbers are referred to as positive integers numbers to the left of zero are negative numbers. We attach a - sign in front of each of these numbers, that is, -1, -2, -3, -4, ... These numbers are referred to as negative integers.        The number associated with a point on the number line is called the coordinate of the point.

Pairing Opposites on a Number Line

       Consider the pairs of nuumbers 2, -2 and 4, -4. On the number line, the numbers in each pair are at equal distance from zero but lie on opposite sides of zero. They are called opposites of each other, that is, -2 is the opposite of 2.

MATH FOCUS

Numbers with the same distance from zero but in opposite directions from zero are called opposites. A negative number is the opposite of its corresponding positive number.

       Inequality symbols will also be used to show the order or to compare the values of integers.

The symbol < means "is less than."
The symbol > means "is greater than."
The symbol < means "is less than or equal to."
The symbol > means "is greater than or equal to."

MATH FOCUS

Ordering of Integers Step 1: Mark the points which represent integers on the number line, that is, positive integers, negative integers, and zero. Step 2: The number on the right side of 0 is greater than the number on the left.The value of the number increases as we move towards the right and decreases as we move towards the left. Every positive integer is greater than any negative integer. Zero is less than every positive integer. Zero is greater than every negative integer. The greater a positive number, the lesser is its corresponding opposite. There is no smallest integer and there is no greatest integer.

Absolute Values

       Consider the numbers 3 and -3. These two numbers are both 3 units away from zero on the number line. The value 3, representing the distance of each number from zero, is called the absolute value of 3 and -3.

       Thus, the absolute value of -3 is 3 and we write |3|=3.

MATH FOCUS

The absolute value of a number is its distance from zero on the number line. If x represents a number, then the absolute value of x written as |x|={x if x>0, 0 if x=0, -x if x<0.