Let’s meet the first tool in your mathematical toolbox.

Integers are simply whole numbers.

Some are positive.

Some are negative.

And one of them—zero—sits exactly in the middle.

You’ve been using integers for years.

Today’s goal isn’t just to recognize them.

It’s to become fast, confident, and accurate when working with them.

Because even though integers look simple, they appear inside many of the questions you’ll solve later in this handbook.

🎯 Why Do Integers Matter?

At first glance, integers may seem too easy.

You might even wonder,

“Will the SAT really ask about integers?”

Usually, not by themselves.

But integers quietly appear inside many different topics.

A student may lose points on a function question because of a sign error.

Another may miss a geometry question after adding negative numbers incorrectly.

The mathematics wasn’t difficult.

The arithmetic was.

Strong foundations prevent simple mistakes.

📋 Let’s Meet Some Integers

Remember: Every integer is a real number, but not every real number is an integer.

💡 Did You Notice?

Every integer belongs to exactly one of these three groups:

• Positive integers
• Negative integers
• Zero

There is no fourth category.

Whenever you’re asked to classify an integer, begin by asking:

Is it positive, negative, or zero?

This simple habit prevents many common mistakes.

🧐 Dr. Aytekin Says

Don’t memorize examples.

Instead, ask yourself one simple question:

Does this number have a decimal or fractional part?

If the answer is yes, it is not an integer.

If the answer is no, it is an integer.

Simple questions often lead to the clearest thinking.

⚠️ Watch Out!

Many students believe that 0 is positive.

✘ It isn’t.

Others think 0 is negative.

✘ That isn’t true either.

Zero is an integer, but it is neither positive nor negative.

This small fact appears more often than you might expect.

🕷️ Detective Mode

Which of the following are integers?

• A. -17
• B. 6.3
• C.
• D. -5/2

Before calculating anything, ask yourself:

What does each expression represent?

That’s a habit strong problem-solvers develop early.

🚀 Speed Boost

Some values appear so often that recognizing them instantly saves valuable time.

You don’t need to memorize this table overnight.

As you work through more questions, these values will become familiar naturally.

🎯 SAT Connection

Integers are rarely the main topic of a question.

Instead, they quietly support many other topics, including:

• Linear equations
• Functions
• Exponents
• Coordinate geometry
• Statistics

You’ll use the ideas from this spread many times—even when the question isn’t about integers.

📝 Quick Check

Circle all the integers.

🔴 Challenge

Without using a calculator, which expression has the greatest value?

• A. -8
• B. -3
• C. 0
• D. 2 − 5

Take a moment to think before choosing.

Sometimes the simplest-looking question rewards the most careful reading.

✅ Answers & Solutions

Quick Check

🔴 Challenge

Answer: C

Let’s compare the values.

• A = -8
• B = -3
• C = 0
• D = 2 − 5 = -3

On a number line, since 0 is greater than every negative number, the correct answer is C.

💡 What Can We Learn?

This question isn’t really testing arithmetic.

It’s testing whether you understand how negative numbers are ordered.

Many students immediately calculate, but the more important idea is recognizing that 0 is greater than every negative number.

Whenever you’re comparing numbers, imagine placing them on a number line. The farther to the right a number is, the greater its value.

🌟 Looking Ahead

In the next spread, we’ll meet fractions.

Fractions may look different from whole numbers, but they’re simply another way of representing quantities.

Once you understand a few key ideas, you’ll discover that many fraction questions are much easier than they first appear.

📝 Quick Check

If yes, you already know the basics. Read this spread quickly and focus on the examples and practice questions.

If not, no problem—this spread will help you build confidence.

1. Which number is greater: -4 or -7?
2. Is 0 positive or negative?
3. Which is smaller: 5 or -5?
4. Which number is farther from zero: -8 or 3?
5. Arrange these numbers from smallest to largest: 4, -2, 0, 7, -5

📌 Essential Rules

• ✔ Numbers become greater as you move to the right.
• ✔ Numbers become smaller as you move to the left.
• ✔ Every positive number is greater than every negative number.
• ✔ Zero is neither positive nor negative.
• ✔ On a number line, the farther right a number is, the greater its value.

📖 Every Number Has an Address

Imagine a long road where every number has its own address.

Moving to the right means the numbers become larger.

Moving to the left means they become smaller.

Figure 3.1 — Number Line (Illustration)

One simple picture explains almost every comparison you’ll make with integers.

💡 Think About Temperature

Many students think,

“8 is bigger than 3, so surely -8 is bigger than -3.”

Not quite.

Think about the weather.

Which day is colder?

• 🥶 -8°C
• 🙂 -3°C

Everyone knows -8°C is colder.

That means -8 < -3.

Whenever you’re comparing negative numbers, imagine a thermometer or a number line.

📊 Quick Examples

⚠️ Watch Out!

Which number is greater?

-12 or -5

Many students answer -12 because 12 is larger than 5.

But that’s not how negative numbers work.

On the number line,

-5 is farther to the right.

Therefore, -5 > -12.

Always compare positions, not just digits.

🧐 Dr. Aytekin Says

Whenever you compare negative numbers,

don’t compare the digits first.

Instead, picture the number line and ask yourself:

Which number is farther to the right?

That simple habit will help you avoid many common mistakes.

🎯 Guided Example

Arrange these numbers from least to greatest:

6, -9, 0, 2, -1

🧠 First Thought

Negative numbers come first.

Among negative numbers, the one farther to the left is smaller.

Then comes zero.

Finally, list the positive numbers from smallest to largest.

✅ Solution

-9, -1, 0, 2, 6

📝 Quick Check

1. 7, -2, 5, 0
2. -6, -1, 3, 8
3. -10, 4, 1, -3

🟢 SAT Practice

Question 1

Which number is closest to zero?

• A. -9
• B. -2
• C. 4
• D. 7

Question 2

Which statement is true?

• A. -8 > -3
• B. 0 < -5
• C. -11 < -4
• D. 5 < -2

Question 3

A submarine is 12 meters below sea level.
A diver is 7 meters below sea level.
Which statement is correct?

• A. The submarine is higher.
• B. The diver is higher.
• C. They are at the same level.
• D. Not enough information.

This is the kind of real-world context the SAT often uses. The mathematics is still just comparing integers.

🔴 Challenge

Without using a calculator, which expression has the smallest value?

• A. -2 + 5
• B. 3 – 7
• C. -8 + 6
• D. -1 – 2

✅ Answers & Solutions

Quick Check Solutions

Arrange from Smallest to Largest Solutions

SAT Practice Solutions

🔴 Challenge Solution

Answer: B

Evaluate each expression:

• A. -2 + 5 = 3
• B. 3 – 7 = -4
• C. -8 + 6 = -2
• D. -1 – 2 = -3

Among these values, -4 is the smallest. Therefore, the correct answer is B (3 − 7).

💡 What Can We Learn?

Whenever you compare several numbers, simplify each expression first.

Then imagine placing the results on a number line.

The smallest number is always the one farthest to the left.

🌟 Looking Ahead

You’ve learned what integers are and how to compare them on a number line.

Next, you’ll learn how to add and subtract integers quickly and accurately.

These operations appear throughout SAT Math, often hidden inside much larger problems. Once you recognize the patterns, they become much easier.

📝 Quick Check

If these feel comfortable, skim the explanations and spend more time on the SAT Practice and Challenge questions.

📌 Essential Rules

Rule 1 — Same Signs

If both numbers have the same sign:

• ✔ Add their absolute values.
• ✔ Keep the common sign.

Rule 2 — Different Signs

If the numbers have different signs:

• ✔ Subtract the smaller absolute value from the larger.
• ✔ Keep the sign of the number with the larger absolute value.

Rule 3 — Subtracting a Negative

Subtracting a negative is the same as adding.

This simple rule appears frequently throughout SAT Math.

🧐 Dr. Aytekin Says

Don’t try to memorize lots of different cases.

Instead, ask yourself two questions:

1. Are the signs the same or different?
2. Which number has the larger absolute value?

Those two questions will guide almost every integer addition problem.

🎯 Guided Example 1

Evaluate: -14 + 9

🧠 First Thought

The signs are different.

Subtract the absolute values: 14 − 9 = 5.

The larger absolute value is 14, which is negative.

✅ Solution: -5

🎯 Guided Example 2

Evaluate: 12 − (-7)

🧠 First Thought

Subtracting a negative becomes addition.

Rewrite the expression: 12 + 7.

✅ Solution: 19

📊 Quick Examples

⚠️ Watch Out!

Students sometimes write 5 − (-3) = 2 because they simply subtract.

Instead, slow down for one second.

A minus sign followed by a negative changes the operation.

A one-second pause can prevent an easy mistake.

😊 A Memory Trick

Two negative signs standing next to each other sometimes feel like they’re saying, “Let’s stop being so negative!” 😄

So whenever you see −(−), think (+).

It’s a playful way to remember the rule—but always understand why it works.

👀 Read Carefully

The SAT rarely asks -8 + 5 by itself.

Instead, it often hides the same arithmetic inside a story.

For example:

A hiker descends 8 meters and then climbs 5 meters.

The context changes. The mathematics doesn’t.

Learning to recognize the underlying operation is an important SAT skill.

📝 Quick Check

1. 12 + (-8)
2. -15 + 12
3. 16 − (-7)
4. -11 − 8
5. -18 + 20

🟢 SAT Practice

Question 1

A submarine is 18 meters below sea level. It rises 11 meters. What is its new position?

• A. 7
• B. -7
• C. 29
• D. -29

Question 2

During a game, Sophia loses 12 points in the first round and gains 17 points in the second round. What is her overall score change?

• A. -29
• B. 5
• C. 29
• D. -5

Question 3

The temperature is -6°C. During the afternoon it increases by 9°C. What is the new temperature?

• A. 15
• B. 3
• C. -15
• D. -3

🔴 Challenge

Without using a calculator, find the value of:

-7 + 12 − (-4) + (-9) − 5

🚀 Speed Boost

When an expression contains several positive and negative numbers, try grouping them mentally.

Positive numbers: group together.

Negative numbers: group together.

Now combine the totals.

This strategy often makes longer calculations faster and reduces mistakes.

🎯 SAT Connection

Integer arithmetic is rarely the main topic on the SAT.

Instead, it quietly appears inside:

• Linear equations
• Coordinate geometry
• Functions
• Statistics
• Exponents
• Word problems

The more automatic these calculations become, the more attention you can give to the problem’s main idea.

✅ Answers & Solutions

Quick Check Solutions

SAT Practice Solutions

🔴 Challenge Solution

Answer: 1

One approach is to evaluate from left to right.

A faster approach is to group positive and negative numbers.

Positive numbers: 12 + 4 = 16

Negative numbers: -7 + (-9) + (-5) = -21

Then combine the totals: 16 − 21 = -5 (Note: re-evaluating expression totals: -7 + 12 – (-4) + (-9) – 5 => -7 + 12 + 4 – 9 – 5 = 1.

🌟 Looking Ahead

You’ve learned how to identify, compare, order, add, and subtract integers.

These skills quietly support many SAT questions, even when integers aren’t the main topic.

But numbers aren’t always whole numbers.

What if you eat half a pizza? 🍕

Or a recipe calls for 3/4 cup of sugar? 🧁

Or a store advertises 25% off? 🏷️

Those situations lead us to our next topic:

➡️ Fractions

You’ll discover that fractions are simply another way of representing numbers—and once you understand a few key ideas, they’re much friendlier than their reputation suggests.