Multiplication

Multiplication is repeated addition of the same number. The notation $a \cdot b = c$ reads "a times b equals c" and means adding the number $b$ to itself $a$ times. The numbers $a$ and $b$ are the factors, and $c$ is the product.

For example $4 \cdot 3 = 3 + 3 + 3 + 3 = 12$ — four threes.

The names in multiplication

• factor — each of the numbers you multiply ($a$ and $b$),
• product — the result ($c$).

Laws of multiplication

Commutativity

The order of the factors does not change the result:

$a \cdot b = b \cdot a$

So $6 \cdot 7$ and $7 \cdot 6$ give the same value — $42$. That is half the work when learning the times tables.

Associativity

When multiplying three or more factors, how you group them does not change the result:

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

Distributivity over addition

Multiplication "spreads out" over a sum:

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

This is the basis of mental arithmetic: $7 \cdot 12 = 7 \cdot (10 + 2) = 70 + 14 = 84$.

Identity element and zero

• multiplying by one leaves the number unchanged: $a \cdot 1 = a$,
• a product with zero is always zero: $a \cdot 0 = 0$.

Column multiplication

Multi-digit numbers are multiplied in columns: multiply each digit of one factor by the other factor, then add the results (shifted by the right number of places). Take $23 \cdot 4$: $4 \cdot 3 = 12$ — write $2$, carry $1$; $4 \cdot 2 = 8$, plus the carried $1$ makes $9$. The result is $92$.

Common mistakes

• Forgetting the carry in column multiplication — just like in addition, the tens travel to the next column.
• "Times zero gives the number back" — no: $a \cdot 0 = 0$, while $a \cdot 1 = a$.
• Confusing the product with the sum — $3 \cdot 4 = 12$, not $7$.