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Whole Numbers and Integers Whole Numbers Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on) No Fractions! Counting Numbers Counting Numbers are Whole Numbers, but without the zero. Because you can't "count" zero . So they are 1, 2, 3, 4, 5, ... (and so on). Natural Numbers "Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject. Integers Integers are like whole numbers, but they also include negative numbers ... but still no fractions allowed! So, integers can be negative {-1, -2,-3, -4, -5, ... }, positive {1, 2, 3, 4, 5, ... }, or zero {0} We can put that all together like this: Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... } Example, these are all integers: -16, -3, 0, 1, 198 (But numbers like ½, 1.1 and 3.5 are not integers) These are all integers (click to mark), and they continue left and right infinitely: © 2015 MathsIsFun.com v0.77 Some People Have Different Definitions! Some people (not me) say that whole numbers can also be negative, which makes them exactly the same as integers. And some people say that zero is NOT a whole number. So there you go, not everyone agrees on a simple thing! My Standard I usually stick to this: Name Numbers Examples Whole Numbers { 0, 1, 2, 3, 4, 5, ... } 0, 27, 398, 2345 Counting Numbers { 1, 2, 3, 4, 5, ... } 1, 18, 27, 2061 Integers { ... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... } -15, 0, 27, 1102 And everyone agrees on the definition of an integer, so when in doubt say "integer". And when you only want positive integers, say "positive integers". It is not only accurate, it makes you sound intelligent. Like this (note: zero isn't positive or negative): Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... } Negative Integers = { ..., -5, -4, -3, -2, -1 } Positive Integers = { 1, 2, 3, 4, 5, ... } Non-Negative Integers = { 0, 1, 2, 3, 4, 5, ... } (includes zero, see?) Integers Unit 5 > Lesson 1 of 11 Problem: The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the distance from the top of Mt. McKinley to the bottom of Death Valley? Solution: The distance from the top of Mt. McKinley to sea level is 20,320 feet and the distance from sea level to the bottom of Death Valley is 282 feet. The total distance is the sum of 20,320 and 282, which is 20,602 feet. The problem above uses the notion of opposites: Above sea level is the opposite of below sea level. Here are some more examples of opposites: top, bottom increase, decrease forward, backward positive, negative We could solve the problem above using integers. Integers are the set of whole numbers and their opposites. The number line is used to represent integers. This is shown below. Definitions The number line goes on forever in both directions. This is indicated by the arrows. Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line. Whole numbers less than zero are called negative integers. These numbers are to the left of zero on the number line. The integer zero is neutral. It is neither positive nor negative. The sign of an integer is either positive (+) or negative (-), except zero, which has no sign. Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above, +3 and -3 are labeled as opposites. Let's revisit the problem from the top of this page using integers to solve it. Problem: The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the distance from the top of Mt. McKinley to the bottom of Death Valley? Solution: We can represent the elevation as an integers: Elevation Integer 20,320 feet above sea level +20,320 sea level 282 feet below sea level 0 -282 The distance from the top of Mt. McKinley to the bottom of Death Valley is the same as the distance from +20,320 to -282 on the number line. We add the distance from +20,320 to 0, and the distance from 0 to -282, for a total of 20,602 feet. Example 1: Write an integer to represent each situation: 10 degrees above zero a loss of 16 dollars a gain of 5 points 8 steps backward Example 2: Example 3: +10 -16 +5 -8 Name the opposite of each integer. -12 +12 +21 -21 -17 +17 +9 -9 Name 4 real life situations in which integers can be used. Spending and earning money. Rising and falling temperatures. Stock market gains and losses. Gaining and losing yards in a football game. Note: A positive integer does not have to have a + sign in it. For example, +3 and 3 are interchangeable. Summary: Integers are the set of whole numbers and their opposites. Whole numbers greater than zero are called positive integers. Whole numbers less than zero are called negative integers. The integer zero is neither positive nor negative, and has no sign. Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. Positive integers can be written with or without a sign. Absolute Value Absolute Value means ... ... only how far a number is from zero: "6" is 6 away from zero, and "−6" is also 6 away from zero. So the absolute value of 6 is 6, and the absolute value of −6 is also 6 More Examples: The absolute value of −9 is 9 The absolute value of 3 is 3 The absolute value of 0 is 0 The absolute value of −156 is 156 No Negatives! So in practice "absolute value" means to remove any negative sign in front of a number, and to think of all numbers as positive (or zero). Absolute Value Symbol To show that we want the absolute value of something, we put "|" marks either side (they are called "bars" and are found on the right side of a keyboard), like these examples: |−5| = 5 |7| = 7 Sometimes absolute value is also written as "abs()", so abs(−1) = 1 is the same as |−1| = 1 Try It Yourself © 2015 MathsIsFun.com v0.77 Subtract Either Way Around And it doesn't matter which way around we do a subtraction, the absolute value will always be the same: |8−3| = 5 |3−8| = 5 (8−3 = 5) (3−8 = −5, and |−5| = 5) More Examples Here are some more examples of how to handle absolute values: |−3×6| = 18 (−3×6 = −18, and |−18| = 18) −|5−2| = −3 (5−2 = 3 and then the first minus gets you −3) −|2−5| = −3 (2−5 = −3 , |−3| = 3, and then the first minus gets you −3) −|−12| = −12 (|−12| = 12 and then the first minus gets you −12) Number Line Writing numbers down on a Number Line makes it easy to tell which numbers are bigger or smaller. Negative Numbers (-) Positive Numbers (+) (The line continues left and right forever.) Numbers on the left are smaller than numbers on the right. Examples: 5 is smaller than 8 −1 is smaller than 1 −8 is smaller than −5 Numbers on the right are larger than numbers on the left. Examples: 8 is larger than 5 1 is larger than −1 −5 is larger than −8 Try this interactive number line (click to mark): © 2015 MathsIsFun.com v0.77 You can also try the zoomable number line . An Example Example: John owes $3, Virginia owes $5 but Alex doesn't owe anything, in fact he has $3 in his pocket. Place these people on the number line to find who is poorest and who is richest. Having money in your pocket is positive. But owing money is negative. So John has "−3", Virginia "−5" and Alex "+3" Now it is easy to see that Virginia is poorer than John (−5 is less than −3) and John is poorer than Alex (−3 is smaller than 3), and Alex is, of course, the richest! Using The Number Line We can use the number line to help us add. We always move to the right to add. We can use the number line to help us subtract. We always move to the left to subtract. Read How to Use the Number Line to Add and Subtract . Footnote: Absolute Value Absolute Value means to think only about how far a number is from zero. For example "6" is 6 away from zero, but "−6" is also 6 away from zero. So the absolute value of 6 is 6, and the absolute value of −6 is also 6