List of Notes - Category: Integers

Negative Numbers

Negative numbers appear when you lose more than you gain.

For example, suppose you have $5, but you owe your friend $11. You can give him back $5, but you still owe him $11-$5=$6. In this case we say that you have -$6.

What is Integer Number

Integer numbers are whole numbers plus negative numbers.

So, -4, -10, 0, 2, 23 are all integer numbers.

Thus, integers can be

  • positive (they are bigger than 0) {1,2,3,4,5,...}
  • zero {0}
  • negative (the are less than 0) {..., -5,-4,-3,-2,-1}

Together we can write integers as {..., -5,-4,-3,-2,-1,0,1,2,3,4,5,...}.

Rounding Integers

Integers are rounded in exactly the same way as whole numbers are rounded, because negative sign (minus sign) doesn't influence process of rounding.

So, we just go through a couple of examples.

Example 1. Round -456 to the nearest ten.

Number Line with Integers

Number line for integers is similar to the number line for whole numbers, but we extend it for negative numbers (to the left of zero).

Although this picture shows only integers from -5 to 5, but arrows indicate that number line contains all integers.

Ordering and Comparing Integers

Integers are compared almost in the same way as whole numbers, but with addition of some rules.

Steps for comparing integers:

  1. If we compare numbers with different signs, then negative number is less than positive.
  2. If numbers are both positive then this is the case when we compare whole numbers.
  3. If numbers are both negative then we compare numbers without signs. The bigger positive number, the smaller negative. For example, if we compare -3 and -5, then we compare 3 and 5 (numbers without signs). Since `3<5` then `-3> -5`.

Example 1. Compare 4567 and -12345.

Adding Integers

Integers are added in the same fashion as whole numbers, except that certain rules should be applied.

If you add integers with different signs, i.e. one is positive and another is negative, then subtract negative number without minus from positive number.

Adding Integers on a Number Line

With the help of number line we can add integers.

It is better to show on examples how to do this.

Example 1. Find -1+4.

We draw -1 on a number line. Since we add positive number (4) we move 4 units to the right. Result is 3. So, -1+4=3.

Subtracting Integers

Integers are subtracted in the same fashion as whole numbers, except that certain rules should be applied.

  • If you subtract negative integer from positive then just add numbers ignoring any minuses, i.e `color(blue)(a-(-b)=a+b)`.
  • If you subtract positive integer from negative, add numbers ignoring any minuses and then place minus in front of result, i.e. `color(green)(-a-b=-(a+b))`.

Example 1. Find `46-(-21)`.

Subtracting Integers on a Number Line

With the help of number line we can subtract integers.

It is better to show on examples how to do this.

Example 1. Find -1-4.

We draw -1 on a number line. Since we subtract positive number (4) we move 4 units to the left. Result is -5. So, -1-4=-5.

Multiplying Integers

Integers are multiplied in the same fashion as whole numbers, except that certain rules should be applied.

That's why it is strongly recommended, that you read first Multiplying Whole Numbers!!!

If you multiply integers with different signs, i.e. one is positive and another is negative, then multiply numbers ignoring minus and place minus in front of the result.

Dividing Integers

Integers are divided in the same fashion as whole numbers, except that certain rules should be applied.

Word of Caution. Remember, that we can't divide by 0.

Another interesting property is that `color(red)(0/a=0)` for any number `a`. For example, `0/5=0`.

Exponents and Integers

Let's learn about positive integer exponents.

We already know how to multiply integers.

Indeed, you've learned, that `2*2=4`, `2*2*2=8`, `2*2*2*2=16`.

But what if you want to multiply same number certain number of times?