Introducing Integers and Negative Numbers

Understanding integers, particularly negative numbers, is a critical leap in your mathematical journey. It moves you beyond the world of counting and whole quantities into a system that can represent opposites, deficits, and directions. Mastering integers and their operations is not just an academic exercise; it is the essential foundation for algebra, coordinate graphing, and modeling real-world situations like debt, temperature changes, and elevation.

What Are Integers?

Integers are the set of whole numbers, their opposites, and zero. This set is represented as {... -3, -2, -1, 0, 1, 2, 3, ...}. The numbers to the right of zero (1, 2, 3,...) are positive integers. The numbers to the left of zero (-1, -2, -3,...) are negative integers. Zero is neutral; it is neither positive nor negative. The negative sign (-) does not mean "bad"; it is simply a symbol indicating direction or opposite. For instance, if +5 represents a gain of $5,then-5representsalossof$5. If +200 feet represents an elevation above sea level, then -200 feet represents a depth below sea level.

The most powerful tool for visualizing integers is the number line. It is a horizontal line with zero at the center, positive numbers extending to the right, and negative numbers extending to the left. The number line makes several key ideas clear:

• Numbers increase in value as you move to the right.
• Every number has an opposite (or additive inverse) on the other side of zero at the same distance. The opposite of 4 is -4. The opposite of -7 is 7.
• Distance from zero is a crucial concept, known as absolute value.

Absolute Value and Comparing Integers

Absolute value is the distance a number is from zero on the number line, regardless of direction. Distance is never negative. The absolute value of a number $n$ is written as $|n|$. For example, $|5|=5$ and $|-5|=5$. Both are 5 units away from zero. This concept is vital for understanding operations later.

Comparing integers (using <, >, or =) is straightforward on a number line: the number further to the right is always greater.

• Any positive number is always greater than any negative number (e.g., $1>-100$).
• For two negative numbers, the number with the smaller absolute value is actually greater. Think of temperature: -2°C is warmer than -10°C. Therefore, $-2>-10$.

Adding and Subtracting Integers

These operations can be modeled on the number line or understood with consistent rules.

For Addition:

1. Same Signs: Add the absolute values and keep the common sign.

• $(-6)+(-2)=-8$ (Think: a debt of $6plusadebtof$2 is a total debt of $8).
• $(+5)+(+3)=+8$

1. Different Signs: Subtract the smaller absolute value from the larger absolute value. The answer takes the sign of the number with the larger absolute value.

• $(-7)+(+3)$. Absolute values: 7 and 3. $7-3=4$. The negative number (-7) has the larger absolute value, so the sum is negative: $-4$.
• $(+4)+(-9)=-5$.

For Subtraction: A crucial rule: Subtracting an integer is the same as adding its opposite. Change the subtraction sign to an addition sign, and change the sign of the number that follows. Then, follow the rules for addition.

• $5-8=5+(-8)=-3$
• $-4-(-6)=-4+(+6)=2$ (This is often the most confusing: subtracting a negative is like adding a positive).
• $-3-5=-3+(-5)=-8$

Multiplying and Dividing Integers

The rules for multiplication and division are identical and very consistent. They depend only on the signs of the numbers involved.

1. Same Signs: The product or quotient is positive.

• $(+6)\times (+2)=+12$

$(-6)\times (-2)=+12$
$(-15)÷(-3)=+5$

1. Different Signs: The product or quotient is negative.

• $(-6)\times (+2)=-12$

$(+6)\times (-2)=-12$
$(-15)÷(+3)=-5$

A helpful way to remember this is with a simple phrase: "Like signs make a positive, unlike signs make a negative." This rule applies exclusively to multiplication and division. Do not apply it to addition or subtraction.

Common Pitfalls

1. Confusing the Rules for Operations: The most frequent error is using the multiplication sign rule ("like signs make positive") for addition or subtraction. Remember: For addition/subtraction, you must consider absolute values and opposites. For multiplication/division, you only need to check if the signs are the same or different.

• Incorrect: $-5+(-2)=+7$ (applied multiplication rule).
• Correct: $-5+(-2)=-7$ (add absolute values, keep common sign).

1. Misinterpreting Double Negatives in Subtraction: The expression $8-(-5)$ causes trouble. Remember that subtraction means "add the opposite." The opposite of $-5$ is $+5$.

• Incorrect: $8-(-5)=3$ or $8-(-5)=-13$.
• Correct: $8-(-5)=8+(+5)=13$.

1. Misunderstanding Absolute Value: Absolute value is about distance, not about "removing the sign" or always making a number positive. It outputs a non-negative number. The error often appears inside larger expressions.

• Incorrect: $|-3|=-3$.
• Correct: $|-3|=3$. Then, $-|-3|=-3$ because you take the absolute value first (getting 3), then apply the negative sign in front.

1. Order of Operations with Negatives: When evaluating something like $-3^2$, the exponent applies before the negative sign (unless parentheses are used). This is different from $(-3{)}^2$.

• Incorrect: $-3^2=9$.
• Correct: $-3^2=-9$ because it means $-(3^2)=-(9)$. For the result to be positive 9, it must be written as $(-3{)}^2$.

Summary

• Integers include all positive and negative whole numbers and zero, represented powerfully on a number line.
• Absolute value ($|n|$) measures a number's distance from zero and is always non-negative.
• Comparing Integers: Numbers to the right on the number line are greater. Negative numbers with smaller absolute values are greater (e.g., $-2>-10$).
• Addition Rules: Same signs? Add and keep the sign. Different signs? Subtract and take the sign of the number with the larger absolute value.
• Subtraction Rule: Subtract an integer by adding its opposite: $a-b=a+(-b)$.
• Multiplication/Division Rule: Like signs give a positive result. Unlike signs give a negative result.