1. The problem is to find the product of the complex numbers $1+i$ and $1-i$. 2. Recall the formula for multiplying complex numbers: if $z_1 = a+bi$ and $z_2 = c+di$, then $$z_1 \times z_2 = (ac - bd) + (ad + bc)i$$ 3. Here, $z_1 = 1+i$ and $z_2 = 1-i$, so $a=1$, $b=1$, $c=1$, and $d=-1$. 4. Substitute these values into the formula: $$ (1)(1) - (1)(-1) + ((1)(-1) + (1)(1))i = 1 + 1 + ( -1 + 1 )i = 2 + 0i $$ 5. Simplify the expression: $$ 2 + 0i = 2 $$ 6. Therefore, the product of $1+i$ and $1-i$ is $2$.