Positive Integers

What Are Positive Integers?

A positive integer is a whole number that is greater than zero. These are the numbers you first learned when you started counting: $1,2,3,4,5$, and so on. They continue forever in the positive direction, which is why we often write them as a set like this: $\{1,2,3,...\}$, where the three dots mean 'and so on infinitely'.

It's important to know what they are not. Positive integers do not include fractions (like $\frac{1}{2}$), decimals (like $2.5$), or negative numbers (like $-3$). The number zero ($0$) is also not a positive integer; it's considered neutral, neither positive nor negative.

Think of it this way: the family of numbers called integers includes three groups:

• Positive Integers: $\{1,2,3,4,...\}$
• Negative Integers: $\{...,-4,-3,-2,-1\}$
• Zero: $\{0\}$

Our focus in this lesson is on that first group: the friendly, everyday counting numbers.

Where Do We See Positive Integers in Real Life?

You use positive integers all the time, probably without even thinking about it! They help us quantify and understand the world around us. Here are just a few examples of positive integers at work:

• Counting: The most basic use! If you have $8$ apples, $25$ students in your class, or $365$ days in a year, you are using positive integers.
• Scores in Games: When you score $10$ points in basketball or earn $500$ points in a video game, those scores are positive integers.
• Measurement: Measuring your height as $52$ inches, the distance to your school as $3$ miles, or the weight of your dog as $15$ pounds all involve positive integers (when we round to the nearest whole number).
• Temperature: On a sunny day, the temperature might be $75$ degrees Fahrenheit or $24$ degrees Celsius. These are positive integers representing degrees above zero.
• Money: Having $\mathrm{\$}20$ in your wallet means you have a positive amount of money.
• Addresses and Floors: Living at $123$ Main Street or taking an elevator to the $10$th floor uses positive integers to identify a location.

Every time you count, measure, or label something with a whole number greater than zero, you are tapping into the power of positive integers.

How Can We Visualize Positive Integers on a Number Line?

A number line is a fantastic tool for understanding integers. It's a straight line with numbers placed at equal intervals along its length. It helps us see the relationship between numbers clearly.

Here’s how it works:

1. The Center Point is Zero: The number line is centered around zero ($0$).
2. Positives to the Right: All the positive integers are placed to the right of $0$. As you move to the right, the numbers get larger. So, $5$ is to the right of $4$, and $10$ is to the right of $9$.
3. Negatives to the Left: All the negative integers are placed to the left of $0$. As you move to the left, the numbers get smaller.

For our lesson, we are focused on the right side of the number line. If you want to find the number $6$, you start at $0$ and move $6$ steps to the right. To find $2$, you move $2$ steps to the right. This visual setup makes it very easy to compare numbers. Any number to the right of another number on the line is the greater number.

How Do You Compare and Order Positive Integers?

Comparing positive integers means determining which one is larger, smaller, or if they are equal. We use special symbols for this:

• $>$ means "greater than". For example, $9>5$ reads as "9 is greater than 5."
• $<$ means "less than". For example, $3<7$ reads as "3 is less than 7."
• $=$ means "equal to". For example, $6=6$.

A helpful trick is to imagine the $>$ and $<$ symbols as an alligator's mouth. The alligator is always hungry and wants to eat the bigger number! So its mouth always opens towards the larger value.

When ordering a group of positive integers, you are simply arranging them from smallest to largest (ascending order) or largest to smallest (descending order). On the number line, ordering from smallest to largest is the same as reading the numbers from left to right.

How Do You Add, Subtract, Multiply, and Divide Positive Integers?

Performing operations with positive integers is straightforward because the rules are simple and consistent. Let's review the four basic operations.

Addition (+)

Adding positive integers means combining them. On a number line, you start at the first number and move to the right by the amount of the second number.

For example, to solve $4+3$, you start at $4$ on the number line and move $3$ units to the right. You land on $7$. So, $4+3=7$.

Subtraction (-)

Subtracting positive integers means taking one away from another. On a number line, you start at the first number and move to the left by the amount of the second number. This is where things can get interesting.

Case 1: Subtracting a smaller number from a larger one.
If you calculate $9-5$, you start at $9$ and move $5$ units to the left. You land on $4$, which is another positive integer.

Case 2: Subtracting a larger number from a smaller one.
If you calculate $5-9$, you start at $5$ and move $9$ units to the left. You will pass $0$ and land on $-4$. The result is a negative integer!

Multiplication (×)

Multiplying positive integers is a form of repeated addition. For example, $4\times 3$ is the same as adding $4$ three times ($4+4+4$) or adding $3$ four times ($3+3+3+3$). Both equal $12$.

The result of multiplying two positive integers is always another, larger positive integer (unless one of the numbers is $1$).

Division (÷)

Dividing positive integers means splitting a number into equal groups. For example, $20÷4$ asks, "How many groups of $4$ can you make from $20$?" The answer is $5$.

Sometimes, a number doesn't divide perfectly. For example, if you have $21$ cookies to share among $4$ friends, each friend gets $5$ cookies ($4\times 5=20$), but there is $1$ cookie left over. This leftover part is called the remainder.

When positive integers divide evenly, the result (the quotient) is a positive integer. When they don't, the result can be expressed with a remainder or as a fraction/decimal, which is not an integer.

What Are Some Key Properties of Positive Integers?

There are some special rules, or properties, that positive integers follow for certain operations. Understanding them can make solving problems much easier.

The Commutative Property

This property applies to addition and multiplication. It means you can swap the order of the numbers without changing the result.

• For Addition: $a+b=b+a$. For example, $7+3=10$ is the same as $3+7=10$.
• For Multiplication: $a\times b=b\times a$. For example, $5\times 6=30$ is the same as $6\times 5=30$.

Note: This property does NOT work for subtraction or division! $7-3=4$, but $3-7=-4$.

The Associative Property

This property also applies to addition and multiplication. It means that when you are adding or multiplying three or more numbers, you can group them in any way without changing the result. We use parentheses $()$ to show grouping.

• For Addition: $(a+b)+c=a+(b+c)$. For example, $(2+4)+5=6+5=11$ is the same as $2+(4+5)=2+9=11$.
• For Multiplication: $(a\times b)\times c=a\times (b\times c)$. For example, $(3\times 2)\times 4=6\times 4=24$ is the same as $3\times (2\times 4)=3\times 8=24$.

The Distributive Property

This property links multiplication and addition. It says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

• Formula: $a\times (b+c)=(a\times b)+(a\times c)$.
• Example: To solve $5\times (10+2)$, you can do it two ways:
  1. Add first: $5\times (12)=60$.
  2. Distribute first: $(5\times 10)+(5\times 2)=50+10=60$.
  Both methods give the same answer! This is very useful in algebra.

What Are Some Common Mistakes to Avoid?

When first learning about positive integers, students sometimes make a few common mistakes. Being aware of them is the first step to avoiding them!

• Forgetting that $0$ is not positive. Zero is a special integer that is neither positive nor negative. The positive integers start with $1$.
• Confusing integers with all numbers. Remember, integers are only whole numbers. Numbers with fractions or decimals, like $8\frac{1}{2}$ or $1.25$, are not integers.
• Mixing up the $>$ and $<$ signs. A great way to remember is that the 'alligator mouth' always opens to eat the larger number. So in $12>9$, the open side faces the $12$.
• Thinking subtraction is commutative. The order matters in subtraction. $10-4$ is $6$, but $4-10$ is $-6$. They are not the same! The same is true for division.
• Ignoring order of operations. When a problem has multiple operations, like $3+2\times 5$, remember to follow the correct order (PEMDAS/BODMAS). You must multiply before you add: $3+10=13$.

Quick Summary Reference

Here are the most important points to remember about positive integers:

• Definition: Positive integers are whole numbers greater than $0$. They are the counting numbers: $1,2,3,4,...$
• Smallest Value: The smallest positive integer is $1$. There is no largest positive integer.
• On the Number Line: They are located to the right of $0$. Numbers increase in value as you move to the right.
• Key Operations Rules:
  • Positive + Positive = Positive
  • Positive × Positive = Positive
  • When subtracting, if you subtract a larger number from a smaller one, the result will be negative.
• Exclusions: Positive integers do not include $0$, negative numbers, fractions, or decimals.