Operations with Integers

Numbers Go Both Ways 🔢

An integer is any whole number, including zero and the negatives: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ (no fractions or decimals!).

You already meet negative numbers all the time:

• Temperature: it can be $-8$ degrees on a cold morning
• Money: owing your friend $5 is like having $-5$ dollars
• Elevation: a diver $20$ meters below sea level is at $-20$ m

The key idea is direction. On a number line, positive means moving right and negative means moving left. In this lesson you'll learn to add, subtract, multiply, and divide integers with confidence.

Absolute Value: How Far From Zero 📏

The absolute value of a number is its distance from zero on the number line. Distance is never negative, so absolute value is always positive or zero. We write it with bars: $|\ \ |$.

• $|5| = 5$ (the number $5$ is $5$ steps from zero)
• $|-5| = 5$ (the number $-5$ is also $5$ steps from zero)
• $|0| = 0$

This is super useful for adding integers with different signs, because we'll compare which number is "bigger" in distance from zero.

⭐ The Two Rules for Adding Integers

When you add integers, first look at the signs.

Rule 1 — Same Signs: Add the absolute values and keep the shared sign.

• $5 + 3 = 8$ (both positive → answer positive)
• $-5 + (-3) = -8$ (both negative → answer negative)

Rule 2 — Different Signs: Subtract the smaller absolute value from the larger, and take the sign of the number with the larger absolute value.

• $7 + (-3)$: subtract $7 - 3 = 4$, and $7$ wins (it's bigger), so the answer is $+4$
• $-7 + 3$: subtract $7 - 3 = 4$, and $-7$ wins, so the answer is $-4$

Think of it like a tug-of-war: positives pull right, negatives pull left, and the stronger side wins!

Quick Concept Check ✅

Let's make sure the adding rules stuck before we move on.

Worked Examples: Adding ➕

Example 1 — Same signs: $-6 + (-4)$

• Both signs are negative, so add the absolute values: $6 + 4 = 10$
• Keep the shared sign (negative): answer $= -10$ ✅

Example 2 — Different signs: $9 + (-12)$

• The signs differ, so subtract absolute values: $12 - 9 = 3$
• $-12$ has the larger absolute value, so the answer is negative: answer $= -3$ ✅

Worked Examples: Subtracting ➖

The golden rule for subtraction: Add the opposite!

$a - b = a + (-b)$

Change the subtraction to addition, then flip the sign of the second number. Now use the adding rules you already know.

Example 1: $5 - 8$

• Add the opposite: $5 + (-8)$
• Different signs → subtract: $8 - 5 = 3$, and $-8$ wins → answer $= -3$ ✅

Example 2: $-3 - 7$

• Add the opposite: $-3 + (-7)$
• Same signs → add: $3 + 7 = 10$, keep negative → answer $= -10$ ✅

Example 3: $4 - (-6)$ (watch the double negative!)

• Add the opposite of $-6$, which is $+6$: $4 + 6$
• answer $= 10$ ✅

Tip: Two minus signs next to each other turn into a plus: $4 - (-6) = 4 + 6$.

Your Turn ✍️

Solve each one. Type just the number, including a minus sign if the answer is negative (like $-7$).

• Box 1: $-6 + (-9) = ?$
• Box 2: $8 + (-15) = ?$
• Box 3: $-2 - (-10) = ?$

Guided Practice: Pick the Answer 🎯

Work each one carefully. Decide whether to use the same-sign or different-sign rule!

Finish Each Statement 🧩

Choose the option that correctly completes each sentence about integer operations.

Integers in the Real World 🌎

Negative numbers describe anything that can go below a starting point — money owed, temperatures below zero, or going underground.

Temperature change: At dawn it was $-5$°C. By noon it rose $8$ degrees.

• New temperature: $-5 + 8 = 3$°C 🌡️

Bank account: Jordan has $20, then spends $32 (overdraft!).

• Balance: $20 - 32 = 20 + (-32) = -12$, so Jordan owes $12 (a balance of -$12).

Multiplying for repeated change: A submarine descends $15$ meters each minute for $4$ minutes. Descending is negative:

• $4 \times (-15) = -60$, so it ends at $-60$ meters (different signs → negative).

The math is the same as before — just decide which operation the story describes, then apply the integer rules.

Solve the Story Problems 📖

Type just the number, with a minus sign if the answer is negative.

• Box 1: The temperature is $-3$°C and drops another $6$ degrees. What is the new temperature?
• Box 2: A diver is at $-12$ m and rises $5$ m. What is the diver's new depth?
• Box 3: A game costs $7$ points each time you lose. If you lose $3$ times, what is your total score change? (Use multiplication.)

One More Real-World Problem 💳

🏆 Putting It All Together

You've now worked with all four operations on integers. The trickiest part is keeping track of signs — here's a summary to lock it in:

The big shortcut for $\times$ and $\div$: count the negative signs. An even number of negatives gives a positive answer; an odd number of negatives gives a negative answer. For example, $(-2) \times (-3) \times (-1)$ has three negatives (odd), so the result is negative: $-6$.

Final Challenge: Mixed Operations 🎓

These mix every operation together. Decide on the right rule for each — take your time!