Properties of Operations and Integer Expressions

In a multiplication, the numbers you multiply are factors and the result is the product.

With whole numbers, $3 \times 4$ and $4 \times 3$ count the same $12$ objects, arranged as $3$ rows of $4$ or as $4$ rows of $3$. The order of the factors does not matter. The rules for multiplying negative numbers were built so that facts like this stay true for every integer.

That gives you three safe moves in any multiplication of integers:

• Swap the order of the factors.
• Regroup the factors, which means choosing which two to multiply first.
• Split a factor that is written as a sum, and multiply each part. For example, $3$ rows of $14$ are $3$ rows of $10$ together with $3$ rows of $4$.

The table names the property behind each move and shows what the special factors $1$, $-1$ and $0$ do. In the table, $a$, $b$ and $c$ stand for any integers. The opposite of a number, also called its additive inverse, is the number you add to it to get $0$: the opposite of $8$ is $-8$, and the opposite of $-8$ is $8$.

The properties also explain the sign of a longer product. A negative factor is $-1$ times a positive number: $-5 = (-1) \times 5$. Because you may swap and regroup factors, you can gather every $-1$ together:

$(-5) \times 7 \times (-2) = (-1) \times (-1) \times 5 \times 7 \times 2 = 1 \times 70 = 70$

Each pair of $-1$ factors multiplies to $1$, and multiplying by $1$ changes nothing. The product is negative only when one $-1$ is left without a partner. That gives a rule with two cases:

• No factor is zero: the product is negative exactly when the number of negative factors is odd. Otherwise it is positive.
• Some factor is zero: the product is $0$.

A division follows the same count. The dividend is the number being divided, the divisor is the number you divide by, and the result is the quotient. When the dividend and the divisor are both nonzero (not $0$), the quotient is negative exactly when just one of the two is negative, since one is an odd count. Otherwise it is positive. Zero divided by a nonzero number is $0$, as in $0 \div (-5) = 0$. Division by $0$ is undefined, so a divisor is never $0$.

The distributive property is a strategy too. When one factor is close to a number that is easy to multiply, such as $100$, write that factor as a sum and multiply each part.

Splitting produces a sum of integers, so recall how a sum works. To add two integers $p$ and $q$, start at $p$ and move $|q|$ units in the direction of $q$: the positive direction when $q$ is positive, the negative direction when $q$ is negative. Here $|q|$, the absolute value of $q$, is the distance of $q$ from $0$.

To find $(-7) \times 98$, write $98$ as $100 + (-2)$, because starting at $100$ and moving $2$ units in the negative direction lands on $98$. Then:

• $(-7) \times 100 = -700$. Neither factor is zero and one is negative, an odd count, so the product is negative.
• $(-7) \times (-2) = 14$. Neither factor is zero and two are negative, an even count, so the product is positive.
• $-700 + 14 = -686$. Start at $-700$ and move $14$ units in the positive direction.

The whole calculation fits on one line:

Division does not share every property of multiplication, so check before you rewrite a division.

• Swapping can change the quotient. $(-12) \div 4 = -3$, but $4 \div (-12) = -\frac{1}{3}$.
• Regrouping can change the quotient. $(48 \div 8) \div 2 = 6 \div 2 = 3$, but $48 \div (8 \div 2) = 48 \div 4 = 12$.
• Splitting the dividend keeps the quotient. For any integers $a$ and $b$ and any nonzero divisor $c$, $(a + b) \div c = a \div c + b \div c$.

Why is the split safe? Dividing by $8$ undoes multiplying by $8$. By the distributive property, $8 \times (-10 + (-2)) = 8 \times (-10) + 8 \times (-2) = -80 + (-16)$. Read that line backward: dividing $-80 + (-16)$ by $8$ gives $-10 + (-2)$, which is each part divided by $8$.

Split only the dividend. No property lets you split the divisor: $60 \div (10 + 2) = 60 \div 12 = 5$, but $60 \div 10 + 60 \div 2 = 6 + 30 = 36$.

A numerical expression is a string of numbers and operation signs, such as $-6 + 4 \times (-3)$. To evaluate an expression is to find its single value.

For expressions built from the four operations and grouping symbols, follow this order:

1. Work inside grouping symbols, such as parentheses, first.
2. Then multiply and divide, moving from left to right.
3. Then add and subtract, moving from left to right.

Parentheses around one negative number, as in $(-3)$, only hold its sign. They contain nothing to compute.

The order is an agreement that gives every expression one meaning. In $-6 + 4 \times (-3)$, it says the $4$ belongs to the product $4 \times (-3) = -12$, so the expression means $-6 + (-12)$. Start at $-6$, move $12$ units in the negative direction, and you reach $-18$.

Two cautions for step 3:

• Subtracting a number means adding its opposite: $5 - 9 = 5 + (-9)$ and $12 - (-16) = 12 + 16$.
• The odd-or-even count of negative numbers works for products and quotients of nonzero numbers. It does not tell you the sign of a sum. The sum $-6 + (-12)$ has two negative numbers, yet it is negative.

A property of operations changes how a calculation looks, never what it equals. Swap, regroup or split factors when you multiply. Split only the dividend when you divide, and never divide by $0$. Read the sign of a product or quotient of nonzero numbers from its count of negative numbers. To evaluate an expression, follow the order of operations, and rewrite only where a property allows it.

Next, you will interpret products and quotients of integers in real-world contexts.