Division is Multiplication by the Reciprocal

The Concept

Dividing by a number and multiplying by its reciprocal are the same operation. A fraction can always be read two ways.

$\frac{a}{b}=\frac{1}{b}\times a$

These are not two different things. They are two ways of writing the same thing.

With Simple Numbers

$\frac{10}{2}=5$

Read as division: $10÷2=5$

Read as multiplication: $\frac{1}{2}\times 10=0.5\times 10=5$

Same answer. Dividing by 2 and multiplying by one-half are the same operation.

Fraction	As division	As multiplication	Result
$\frac{10}{2}$	$10÷2$	$\frac{1}{2}\times 10$	5
$\frac{12}{4}$	$12÷4$	$\frac{1}{4}\times 12=0.25\times 12$	3
$\frac{10}{0.55}$	$10÷0.55$	$\frac{1}{0.55}\times 10\approx 1.82\times 10$	$\approx 18.18$

Why Economists Write It This Way

In the multiplier formula, $\frac{\Delta A}{1-z}$ and $\frac{1}{1-z}\times \Delta A$ are mathematically identical. Economists prefer the second form because it separates two distinct ideas:

$\underset{\text{the multiplier (structural)}}{\underbrace{\frac{1}{1-z}}}\times \underset{\text{the change (what you plug in)}}{\underbrace{\Delta A}}$

The multiplier $\frac{1}{1-z}$ is a property of the economy — it depends on $b$, $t$, and $m$, which don’t change when you change spending. The $\Delta A$ is the specific policy or shock you’re analyzing. Writing them separately makes it clear: the same multiplier applies to any autonomous change.

The Reciprocal

The reciprocal of a number is 1 divided by that number. It’s the number that, when multiplied by the original, gives 1.

Number	Reciprocal	Check
2	$\frac{1}{2}=0.5$	$2\times 0.5=1$ ✓
4	$\frac{1}{4}=0.25$	$4\times 0.25=1$ ✓
0.55	$\frac{1}{0.55}\approx 1.82$	$0.55\times 1.82\approx 1$ ✓
$(1-z)$	$\frac{1}{1-z}$	$(1-z)\times \frac{1}{1-z}=1$ ✓

Where This Shows Up in Economics

Every time you see a formula written as $\frac{1}{\text{something}}\times \text{change}$, it’s just a fraction rewritten as multiplication:

Formula	As a fraction	As multiplication
$\Delta Y=\frac{\Delta A}{1-z}$	$\Delta A÷(1-z)$	$\frac{1}{1-z}\times \Delta A$
$\Delta C=\frac{\Delta a}{1-b}$ (Ch6)	$\Delta a÷(1-b)$	$\frac{1}{1-b}\times \Delta a$

---

North: Where this comes from

• Basic fraction arithmetic — the definition of division
• The concept of a reciprocal (multiplicative inverse)

East: What’s the opposite?

• Multiplying by a number vs dividing by it — opposite operations that undo each other ($\times 2$ then $÷2$ returns to start)

South: Where this leads

• Only the Slope Survives a Change in a Linear Function — the change formula uses this rewriting to separate the multiplier from the change
• ECON-1221 Chapter 7 - Notes from the Textbook_backup — the simple multiplier $\frac{1}{1-z}\times \Delta A$
• Solving any equation by “dividing both sides” is really “multiplying both sides by the reciprocal”

West: What’s similar?

• Subtraction is addition of a negative ($a-b=a+(-b)$) — same pattern of rewriting one operation as another
• Percentage calculations: “50% of X” is $\frac{1}{2}\times X$ — a fraction acting as a multiplier