[Grade 7 Math] Positive and Negative Numbers — Let's Expand Our Mathematical World!
Hello! Today, we’re going to study the very first topic in middle school math: "Positive and Negative Numbers."
You might be feeling a bit nervous, thinking that math is going to get a lot harder than elementary school arithmetic, but don't worry—you’ve got this!
Once you master this chapter, you’ll be able to perform calculations you once thought were "impossible," and your view of the world will expand significantly. Let's take it slow and move at your own pace.
1. Numbers Smaller Than 0? "Positive Numbers" and "Negative Numbers"
In elementary school, we only dealt with numbers 0 or greater. However, the world around us is full of "numbers smaller than 0."
Take winter temperatures, for example. When the temperature is below 0 degrees, we say "minus 5 degrees," right? That is a negative number.
■ Check the Terminology!
- Positive Number: A number greater than 0. Indicated by a \(+\) (plus) sign. (Example: \(+5\), \(+1.2\))
- Negative Number: A number smaller than 0. Indicated by a \(-\) (minus) sign. (Example: \(-3\), \(-0.5\))
- Natural Number (Positive Integer): Among positive numbers, these are integers like \(1, 2, 3, \dots\). Be careful—"0" is not considered a natural number!
【Pro Tip】 0 is neither a "positive number" nor a "negative number." It is the neutral reference point exactly in the middle.
■ Words with Opposite Properties
By using negative numbers, we can represent opposite meanings.
Example: If "a profit of 500 yen" is represented as \(+500\) yen, then "a loss of 300 yen" can be represented as \(-300\) yen.
★ Key Point: Opposite meanings can be represented with a minus sign!
"Moving \(+3km\) to the East" ⇔ "Moving \(-3km\) to the East" is the same as "Moving \(3km\) to the West."
2. The Number Line and Absolute Value
A line with numbers arranged horizontally is called a number line.
- The further right you go, the larger the number becomes.
- The further left you go, the smaller the number becomes.
- The 0 in the center is called the origin.
■ What is Absolute Value?
The absolute value is the distance from 0 (the origin) on the number line.
Since it represents a distance, it is always a number 0 or greater, regardless of whether the original number was positive or negative.
Example: The absolute value of \(+3\) is \(3\), and the absolute value of \(-3\) is also \(3\).
It’s easy to remember as: "The number after removing the sign (\(+\) or \(-\) )"!
■ Comparing Numbers
It’s easiest to visualize on the number line.
While \(2 < 5\), in the negative world, \(-5 < -2\).
Keep this rule in mind: "For negative numbers, a larger absolute value means a smaller number." If you think of it in terms of debt, having "a debt of 2 yen (\(-2\))" is better (larger) than having "a debt of 5 yen (\(-5\))."
3. Addition and Subtraction
This is the first major challenge! It's easiest to think of this as a "team battle."
■ Rules for Addition
- Numbers with the same sign (\(+\) and \(+\) or \(-\) and \(-\)):
This is an alliance between teammates. Add their absolute values and keep the common sign.
Example: \((-3) + (-5) = -8\) (A team of minus 3 joins a team of minus 5, resulting in a total of minus 8!) - Numbers with different signs (\(+\) and \(-\)):
This is a tug-of-war between the plus team and the minus team! Use the sign of the larger team (the one with the greater absolute value) and subtract the smaller number from the larger one.
Example: \((+5) + (-2) = +3\) (If 5 pluses fight 2 minuses, 3 pluses remain!)
Example: \((+2) + (-6) = -4\) (If 2 pluses fight 6 minuses, 4 minuses remain!)
■ Rules for Subtraction
The golden rule for subtraction is to "Change the sign of the number being subtracted and turn it into addition."
Just remember: "Subtracting a negative is the same as adding a positive."
\( (+5) - (+2) \rightarrow (+5) + (-2) = +3 \)
\( (+3) - (-4) \rightarrow (+3) + (+4) = +7 \) ← This is a common trap!
★ Key Point: Get rid of the parentheses to make it cleaner!
Once you get used to the calculations, you can write them without parentheses:
\( 5 + (-3) = 5 - 3 = 2 \)
\( 2 - (-4) = 2 + 4 = 6 \)
4. Multiplication and Division
The rules for multiplication and division are actually much simpler than addition.
■ How to Determine the Sign
- Multiplication/Division with the same sign: The answer is always \(+\) (plus).
- Multiplication/Division with different signs: The answer is always \(-\) (minus).
\( (+2) \times (+3) = +6 \)
\( (-2) \times (-3) = +6 \) (Multiplying two minuses makes a plus!)
\( (+2) \times (-3) = -6 \)
\( (-6) \div (+2) = -3 \)
■ The Trap of Exponents
Multiplying the same number repeatedly is called an exponent. Be careful with how you write it!
- \( (-3)^2 = (-3) \times (-3) = 9 \) (The entire parenthetical value is squared)
- \( -3^2 = -(3 \times 3) = -9 \) (Only the 3 is squared, and the minus is applied at the end)
【Common Mistake】 These two are very common test traps. Always check whether or not there are parentheses!
5. Calculations with Multiple Operations
When addition, subtraction, multiplication, and division are all mixed together, there is a set order of operations:
- If there are exponents, calculate those first.
- If there are parentheses, calculate what's inside them first.
- Calculate multiplication and division.
- Finally, calculate addition and subtraction.
Example: \( 5 + (-2) \times 3 = 5 + (-6) = -1 \)
Summary & Final Thoughts
Today’s key points:
1. Negative numbers are numbers smaller than 0. Absolute value is the distance from 0!
2. Addition is a "team battle." For subtraction, "change it to addition" first!
3. For multiplication/division, an odd number of minus signs results in a minus, and an even number results in a plus!
4. Never overlook the difference between \( (-3)^2 \) and \( -3^2 \)!
You might feel a bit confused by negative numbers at first, but by solving problems repeatedly, it will become as natural as riding a bike.
Just take it one step at a time and double-check your signs as you go. You've got this!