# Multiplication Using a Base - Part 2 (Speed Math)

We can extend base method to multiply two numbers when both of them are close to a multiple of power of ten. (Note: speed math calculator can be used to practise problems using this method.)

### Example: Calculate 397 × 392

Select $100$ as base and $400$ as working base. Subtract working base from both numbers. That is, $397 - 400 = -3$ and $392 - 400 = -8$

Find any diagonal sum, and multiply result by $4$ (because working base is $4$ times the base) to obtain left side of answer. Note that both diagonal sums will be same.

$4(392 - 3) = 1556$

Multiply differences to obtain right side of answer.

$-3 × -8 = 24$

Combine left and right sides, that is, $1556$ and $24$

Answer is $155624$

### Example: Calculate 1996 × 2016

Select $1000$ as base and $2000$ as working base. Subtract working base from both numbers. That is, $1996 - 2000 = -4$ and $2016 - 2000 = 16$

Find any diagonal sum, and multiply result by $2$ (because working base is $2$ times the base) to obtain left side of answer. Note that both diagonal sums will be same.

$2(2016 - 4) = 4024$

Multiply differences to obtain right side of answer.

$-4 × 16 = -64$

Make right side positive by borrowing $1$ from left side. This borrowed $1$ becomes $1000$ when coming to right side (because base is $1000$ and $1 × 1000 = 1000$) and therefore right side becomes $1000 - 64 = 936.$ Since we borrowed $1,$ left side becomes $4024 - 1 = 4023$

Combine left and right sides, that is, $4023$ and $936$

Answer is $4023936$

### Example: Calculate 40003 × 40015

Select $10000$ as base and $40000$ as working base. Subtract working base from both numbers. That is, $40003 - 40000 = 3$ and $40015 - 40000 = 15$

Find any diagonal sum, and multiply result by $4$ (because working base is $4$ times the base) to obtain left side of answer. Note that both diagonal sums will be same.

$4(40015 + 3) = 160072$

Multiply differences to obtain right side of answer.

$3 × 15 = 45$

Since base $10000$ has four zeros, write $45$ as $0045$

Combine left and right sides, that is, $160072$ and $0045$

Answer is $1600720045$