Question 49

Brahmagupta's rules help us understand how to add and multiply positive and negative numbers.

Step 1 — Explaining Addition Rules

Let us use Bela's Building of Fun. Going up means positive (+). Going down means negative (-).

Rule 1: (+) + (+) = (+) Bela goes up 3 floors. Then she goes up 2 more floors. She is now 5 floors higher. On a number line: Start at 0. Move right 3 steps. Then move right 2 more steps. You land on +5.

$(+3) + (+2)$

$= +5$

$\boxed{\text{Bigger gain}}$

Rule 2: (-) + (-) = (-) Bela goes down 2 floors. Then she goes down 4 more floors. She is now 6 floors lower. On a number line: Start at 0. Move left 2 steps. Then move left 4 more steps. You land on -6.

$(-2) + (-4)$

$= -6$

$\boxed{\text{Bigger loss}}$

Rule 3: (+) + (-) = ? Bela goes up 7 floors. Then she goes down 4 floors. She ends up 3 floors above ground. On a number line: Start at 0. Move right 7 steps. Then move left 4 steps. You land on +3.

$(+7) + (-4)$

$= +3$

$\boxed{\text{Result depends on which is bigger}}$

Rule 4: (-) + (+) = ? Bela goes down 6 floors. Then she goes up 2 floors. She ends up 4 floors below ground. On a number line: Start at 0. Move left 6 steps. Then move right 2 steps. You land on -4.

$(-6) + (+2)$

$= -4$

$\boxed{\text{Result depends on which is bigger}}$

Step 2 — Explaining Multiplication Rules

For multiplication, "times a positive number" means doing something repeatedly. "Times a negative number" means doing the opposite action repeatedly.

Rule 5: (+) x (+) = (+) Bela goes up 3 floors. She does this 2 times. She ends up 6 floors higher. On a number line: Start at 0. Move right 3 steps. Do this 2 times. You land on +6.

$(+3) \times (+2)$

$= +6$

$\boxed{\text{Positive result}}$

Rule 6: (-) x (-) = (+) Imagine "going down 3 floors" is a negative action. "Multiplying by negative 2" means doing the opposite of this action, 2 times. The opposite of going down 3 floors is going up 3 floors. Doing this 2 times means going up 6 floors. On a number line: Start at 0. Moving left 3 steps is negative. "Times negative 2" means doing the opposite of moving left 3 steps, 2 times. The opposite is moving right 3 steps. Do this 2 times. You land on +6.

$(-3) \times (-2)$

$= +6$

$\boxed{\text{Positive result}}$

Rule 7: (+) x (-) = (-) Bela goes up 3 floors. "Multiplying by negative 2" means doing the opposite of going up 3 floors, 2 times. The opposite of going up 3 floors is going down 3 floors. Doing this 2 times means going down 6 floors. On a number line: Start at 0. Moving right 3 steps is positive. "Times negative 2" means doing the opposite of moving right 3 steps, 2 times. The opposite is moving left 3 steps. Do this 2 times. You land on -6.

$(+3) \times (-2)$

$= -6$

$\boxed{\text{Negative result}}$

Rule 8: (-) x (+) = (-) Bela goes down 2 floors. She does this 3 times. She ends up 6 floors lower. On a number line: Start at 0. Move left 2 steps. Do this 3 times. You land on -6.

$(-2) \times (+3)$

$= -6$

$\boxed{\text{Negative result}}$

Step 3 — Providing Own Examples

Let us think of money. Earning money is positive (+). Losing money or owing money is negative (-).

Rule 1: (+) + (+) = (+) Example: You found ₹5 on the street. Then you found ₹4 more. You have ₹9 total.

$(+5) + (+4)$

$= +9$

Rule 2: (-) + (-) = (-) Example: You lost ₹3 playing a game. Then you lost ₹6 more. You lost ₹9 total.

$(-3) + (-6)$

$= -9$

Rule 3: (+) + (-) = ? Example: You earned ₹10 for helping. You spent ₹7 on a snack. You have ₹3 left.

$(+10) + (-7)$

$= +3$

Rule 4: (-) + (+) = ? Example: You owed your friend ₹12. You earned ₹5 doing chores. You still owe ₹7.

$(-12) + (+5)$

$= -7$

Rule 5: (+) x (+) = (+) Example: You gain 2 points in a game. You do this 4 times. You gained 8 points.

$(+2) \times (+4)$

$= +8$

Rule 6: (-) x (-) = (+) Example: Imagine losing 5 points is bad. If you "undo" losing 5 points, 3 times, you gain 15 points.

$(-5) \times (-3)$

$= +15$

Rule 7: (+) x (-) = (-) Example: You earn 4 points. But you do the "opposite" of earning, 2 times. This means you lose 8 points.

$(+4) \times (-2)$

$= -8$

Rule 8: (-) x (+) = (-) Example: You lose 3 points in a game. You do this 5 times. You lost 15 points.

$(-3) \times (+5)$

$= -15$

Answer

1. Explanation of Brahmagupta's rules:

• Rule 1: (+) + (+) = (+) If Bela goes up 3 floors and then again up 2 floors, she goes up 5 floors total. Example: (+3) + (+2) = +5. On a number line: Start at 0. Move right 3 steps. Then move right 2 more steps. You land on +5. So, gain + gain = bigger gain.
• Rule 2: (-) + (-) = (-) If Bela goes down 2 floors, then again down 4 floors, she goes down 6 floors total. Example: (-2) + (-4) = -6. On a number line: Start at 0. Move left 2 steps. Then move left 4 more steps. You land on -6. So, loss + loss = bigger loss.
• Rule 3: (+) + (-) = ? If Bela first goes up 7 floors, then down 4 floors, she reaches 3 floors above ground. Example: (+7) + (-4) = +3. On number line: Start at 0. Move right 7 steps. Then move left 4 steps. You land on +3. So, gain + loss = depends on which is bigger.
• Rule 4: (-) + (+) = ? If Bela first goes down 6 floors, then up 2 floors, she ends 4 floors below ground. Example: (-6) + (+2) = -4. On number line: Start at 0. Move left 6 steps. Then move right 2 steps. You land on -4. So, loss + gain = depends on which is bigger.
• Rule 5: (+) x (+) = (+) Going up 3 floors 2 times means total up 6 floors. Example: (+3) x (+2) = +6. Same direction x same direction = positive result.
• Rule 6: (-) x (-) = (+) Going down 3 floors 2 times (a downward move twice) means you come back up — it's positive. Example: (-3) x (-2) = +6. Two negatives make a positive.
• Rule 7: (+) x (-) = (-) Going up 3 floors 2 times in the downward direction means overall downward (negative). Example: (+3) x (-2) = -6. Different directions -> negative result.
• Rule 8: (-) x (+) = (-) Going down 2 floors 3 times -> even more down. Example: (-2) x (+3) = -6. Again, different directions -> negative. 2. Own examples of each rule:
• Rule 1: (+) + (+) = (+) Example: (+5) + (+4) = +9. Explanation: You found ₹5 on the street. Then you found ₹4 more. You have ₹9 total.
• Rule 2: (-) + (-) = (-) Example: (-3) + (-6) = -9. Explanation: You lost ₹3 in one game and ₹6 in another. Total loss = ₹9.
• Rule 3: (+) + (-) = ? Example: (+10) + (-7) = +3. Explanation: You earned ₹10 but spent ₹7. You still have ₹3 left.
• Rule 4: (-) + (+) = ? Example: (-12) + (+5) = -7. Explanation: You owed ₹12 but earned ₹5. You still owe ₹7.
• Rule 5: (+) x (+) = (+) Example: (+2) x (+4) = +8. Explanation: You gain 2 points in a game. You do this 4 times. You gained 8 points.
• Rule 6: (-) x (-) = (+) Example: (-5) x (-3) = +15. Explanation: Imagine losing 5 points is bad. If you "undo" losing 5 points, 3 times, you gain 15 points.
• Rule 7: (+) x (-) = (-) Example: (+4) x (-2) = -8. Explanation: You earn 4 points. But you do the "opposite" of earning, 2 times. This means you lose 8 points.
• Rule 8: (-) x (+) = (-) Example: (-3) x (+5) = -15. Explanation: You lose 3 points in a game. You do this 5 times. You lost 15 points.