Mathematics: Class 7 (Part 2) — Chapter 2: Operations with Integers

A method to find two numbers given their sum and difference without using formulas is to try different pairs of numbers and check:

• Do the two numbers add up to the given sum?
• Is the difference between them (first number $-$ second number) equal to the given difference?

For example, if the sum is $25$ and the difference is $11$, the numbers are $18$ and $7$ because:

• $18 + 7 = 25$
• $18 - 7 = 11$

If the sum is $25$ and the difference is $-11$, swapping the numbers gives the correct pair: $7$ and $18$.

Movement on a line can be modeled using integers:

• Rightward movement is considered positive.
• Leftward movement is considered negative.

If a coin starts at $0$, and the first strike moves it by $a$ units and the second strike moves it by $b$ units, the final position $P$ is given by the formula:

$$P = a + b$$

The magnitude represents the distance traveled, while the sign represents the direction (rightwards or leftwards).

The token model uses colored tokens to represent integers:

• A green token represents $+1$.
• A red token represents $-1$.
• A green and a red token together form a zero pair and cancel each other out.

Subtracting a number is equivalent to adding its additive inverse:

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

For example, $7 - 18 = 7 + (-18) = -11$.

We can interpret multiplication $m \times n$ using a bag of tokens:

• When the multiplier $m$ is positive, we add tokens to the bag $m$ times.
• When the multiplier $m$ is negative, we remove tokens from the bag $|m|$ times.

Examples:

• $4 \times (-2)$: Place $2$ negative (red) tokens into an empty bag $4$ times. Total = $8$ negative tokens ($-8$).
• $(-4) \times 2$: Remove $2$ positive (green) tokens from an empty bag $4$ times. This requires adding zero pairs first. After removing $8$ positive tokens, $8$ negative tokens remain ($-8$).
• $(-4) \times (-2)$: Remove $2$ negative (red) tokens $4$ times. After removing $8$ negative tokens from zero pairs, $8$ positive tokens remain ($+8$).

Observing multiplication sequences reveals the rules of signs:

• When the multiplicand is positive, for every unit decrease in the multiplier, the product decreases by the multiplicand.
• When the multiplicand is negative, for every unit decrease in the multiplier, the product increases by the magnitude of the multiplicand.

This establishes the following rules:

• $\text{Positive} \times \text{Positive} = \text{Positive}$
• $\text{Positive} \times \text{Negative} = \text{Negative}$
• $\text{Negative} \times \text{Positive} = \text{Negative}$
• $\text{Negative} \times \text{Negative} = \text{Positive}$

• Multiplicative Identity: For any integer $a$, $1 \times a = a$.
• Commutative Property: For any two integers $a$ and $b$, multiplication is commutative:

  $$a \times b = b \times a$$

• Brahmagupta's Rules (628 CE): Explicit rules using "fortune" ($dhana$, positive) and "debt" ($ṛṇa$, negative):
  • The product or quotient of two fortunes is a fortune.
  • The product or quotient of two debts is a fortune.
  • The product or quotient of a debt and a fortune is a debt.

For any three integers $a$, $b$, and $c$, the grouping of multiplication does not affect the product:

$$a \times (b \times c) = (a \times b) \times c$$

• If a negative number is multiplied an even number of times, the product is positive.
• If a negative number is multiplied an odd number of times, the product is negative.

Multiplication distributes over addition. For any integers $a$, $b$, and $c$:

$$a \times (b + c) = (a \times b) + (a \times c)$$