Integers are whole numbers that can be positive, negative, or zero, and they are used whenever quantities can go above or below a reference point. Temperatures, bank balances, elevations, and game scores all use integers to show direction as well as size. A number line makes integer operations easier to see because moving right means the value increases and moving left means the value decreases.
Zero is the neutral anchor that separates positive numbers from negative numbers.
Understanding Math: Integer Operations
A useful way to understand signed numbers is to treat the sign as a direction, not as a decoration. A positive change raises a quantity from its starting value. A negative change lowers it.
For example, a bank account can begin with a balance of negative 12 dollars. A deposit of 20 dollars changes the balance to positive 8 dollars. The deposit does not erase the negative sign by magic.
Its size is greater than the debt, so the account crosses zero. This idea helps with temperature changes, elevator floors, and points gained or lost in games.
Addition becomes easier when you separate two jobs. First, identify whether the numbers point in the same direction or in opposite directions. Numbers with the same direction combine their sizes.
If both numbers are negative, the total remains negative because both changes lower the value. Numbers in opposite directions compete. Compare their distances from zero, which are called absolute values.
The larger distance determines the final direction. A result of negative 15 plus 9 is negative 6 because the downward amount is larger by 6.
This is not a rule to memorize without meaning. It describes two changes that partly cancel each other.
Subtraction needs extra care because the minus sign can have two different roles. It may show that a number is negative, or it may tell you to subtract. Parentheses make this clearer.
In 7 minus negative 3, the first minus sign is an operation. The second belongs to the number negative 3. Subtracting means removing a change.
Removing a positive change lowers the total. Removing a negative change raises the total.
This is why 7 minus negative 3 has the same value as 7 plus 3. Rewriting subtraction as addition of the opposite is a reliable method, especially when several signs appear together.
Real situations can reveal whether an answer makes sense. If the temperature is negative 4 degrees and falls by 6 degrees, the result must be colder than negative 4, so a positive answer cannot be correct. If a submarine is 30 meters below sea level and rises 18 meters, it is still below sea level by 12 meters.
Pay attention to the wording in problems. Words such as loss, decrease, below, owe, and withdraw often represent negative values.
Words such as gain, increase, above, earn, and deposit often represent positive values. Draw a quick number line or write the starting amount and each change separately when the wording feels confusing.
Common mistakes come from ignoring parentheses, combining signs too quickly, or treating every pair of minus signs as positive. Two minus signs only become a plus when one is subtraction and the other is the sign of the number being subtracted. Check each step before moving on.
In longer expressions, simplify inside parentheses first, then follow the usual order of operations. Keep the sign attached to its number while working. Estimation is a strong final check.
If two large negative amounts are combined, the answer should be a large negative amount. If equal positive and negative amounts are combined, the answer should be zero.
Key Facts
- Adding a positive integer moves right on the number line: -3 + 5 = 2.
- Adding a negative integer moves left on the number line: 4 + (-6) = -2.
- Subtracting an integer means adding its opposite: a - b = a + (-b).
- Same signs add and keep the sign: -7 + (-4) = -11 and 6 + 3 = 9.
- Different signs subtract absolute values and keep the sign of the number with greater absolute value: -9 + 5 = -4.
- For multiplication and division, same signs give a positive result and different signs give a negative result: (-4)(-3) = 12 and 20 ÷ (-5) = -4.
Vocabulary
- Integer
- An integer is a whole number that is positive, negative, or zero.
- Number line
- A number line is a straight line that shows numbers in order from least to greatest.
- Opposite
- The opposite of a number is the number the same distance from zero on the other side of the number line.
- Absolute value
- Absolute value is a number's distance from zero, so it is never negative.
- Sign
- A sign tells whether a number is positive or negative.
Common Mistakes to Avoid
- Treating subtraction as always making a number smaller is wrong because subtracting a negative can increase the value, such as 3 - (-5) = 8.
- Adding signs before comparing absolute values is wrong for mixed signs because -8 + 3 requires subtracting 3 from 8 and keeping the negative sign.
- Forgetting that zero is neither positive nor negative is wrong because sign rules for positive and negative numbers do not make zero positive or negative.
- Using addition sign rules for multiplication is wrong because (-2) + (-3) = -5 but (-2)(-3) = 6.
Practice Questions
- 1 Use a number line or integer rules to calculate -6 + 9.
- 2 Evaluate 12 - (-5) + (-8).
- 3 A student says that -4 + 7 must be negative because one number is negative. Explain whether the student is correct and justify your reasoning.