Everything covered in this lesson, in one place - useful for revision or printing.
\[ (+6)(+4) \quad (+6)(-4) \quad (-6)(-4) \]
Before calculating, predict the sign. Same signs give positive products. Different signs give negative products. This pattern holds for all integer multiplication.
\[ \text{Same signs} \rightarrow + \quad \text{Different} \rightarrow - \]
Multiply the numbers normally, then apply the sign rule. (-14) × 5 = -70 because signs differ. (-19) × (-2) = 38 because signs are same.
\[ (-1)(3)=-3 \quad (-1)(0)=0 \quad (-1)(-1)=1 \]
The pattern increases by 1 as the second factor decreases by 1. This proves why negative times negative equals positive.
\[ (-24) \div 6 = -4 \]
Division uses the same sign rules as multiplication because it is the inverse. Estimate the sign first, then calculate. Check by multiplying back.
\[ -7626 \div 31 \approx -250 \]
Estimate: -7500 ÷ 30 ≈ -250. The actual answer is -246, which is close to our estimate. Always check reasonableness.
\[ a+b=b+a \quad a \times b=b \times a \]
Order does not matter for addition or multiplication. 8 + (-3) = (-3) + 8. But subtraction and division are NOT commutative.
\[ (a+b)+c=a+(b+c) \]
Grouping does not matter for addition or multiplication. [(-6)+4]+(-1) = (-6)+[4+(-1)] = -3. Subtraction and division are NOT associative.
\[ -4(5+6) = -20 + (-24) = -44 \]
Multiplication distributes over addition/subtraction. Multiply the outside term by EACH inside term. Same result as simplifying brackets first.
\[ a+0=a \quad a \times 1=a \]
Zero is the additive identity. One is the multiplicative identity. Adding zero or multiplying by one leaves the number unchanged.
\[ \text{Brackets} \rightarrow \text{Orders} \rightarrow \text{DM} \rightarrow \text{AS} \]
For mixed operations: Brackets first, then Orders (powers/roots), then Division/Multiplication left-to-right, then Addition/Subtraction left-to-right.
\[ 4 - \{2 \times 30 \div 5 + 3(-3)\} = 1 \]
Inner bracket first: -12÷4=-3. Then multiply: 2×30=60, 60÷5=12, 3×(-3)=-9. Bracket: 12+(-9)=3. Finally: 4-3=1.
\[ (-5)^2=25 \quad (-4)^3=-64 \]
Negative base with even exponent = positive. Odd exponent = negative. The exponent tells how many times to multiply the base by itself.
\[ (-4)^2=16 \quad -4^2=-16 \]
Brackets include the negative sign in the power. Without brackets, the exponent applies only to the number, then the negative is applied after.
\[ \sqrt{25}=5 \quad \sqrt[3]{-27}=-3 \]
Square roots undo squaring. Cube roots undo cubing. Negative integers have cube roots but NO real square roots at this level.
\[ \text{Rules make calculations reliable} \]
You can now multiply/divide with sign rules, use properties to rearrange work, follow BODMAS strictly, and handle powers/roots correctly. You are ready for integrated problems.