Scientific Notation

By the end of this lesson, you will be able to write any number in scientific notation, convert back, multiply and divide without a calculator, and fix an answer whose first part has drifted out of range.

One Digit, Then a Power of Ten

$$a \times 10^n \qquad \text{where } 1 \leq a < 10$$

The whole point is that huge and tiny numbers become readable and comparable.

$$4{,}500{,}000 = 4.5 \times 10^6$$

$$0.00032 = 3.2 \times 10^{-4}$$

The range on $a$ is what makes it a standard form. Exactly one nonzero digit before the decimal point, so every number has exactly one representation and two of them can be compared at a glance. $45 \times 10^5$ is the same value but is not in scientific notation, and $0.45 \times 10^7$ is not either.

The exponent counts places moved, and its sign says which direction.

$$\text{big number} \Rightarrow \text{positive exponent}$$

$$\text{small number} \Rightarrow \text{negative exponent}$$

A negative exponent does not mean a negative number. $3.2 \times 10^{-4}$ is positive and small; $-3.2 \times 10^4$ is negative and large. The sign of the NUMBER lives on the $a$, and the sign of the EXPONENT describes size.

Sort It: Positive or Negative Exponent

Example 1: Converting Both Ways

Here is every step, worked all the way through. $$0.00047$$

• Which digit has to sit in front of the decimal point? 4
• Write the value of a, the whole part before the times sign 4.7
• How many places does the point move to get from 0.00047 to 4.7? 4
• The number is small, so is the exponent positive or negative? Type one word negative
• What is the exponent? -4

Multiplying and Dividing

This is where scientific notation earns its keep, because the exponent laws do the hard part.

$$(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$$

Multiply the fronts, add the exponents. That is the product rule from two lessons ago, applied to base $10$.

$$\frac{8 \times 10^7}{2 \times 10^3} = 4 \times 10^4$$

Divide the fronts, subtract the exponents.

Doing that with $30{,}000 \times 200{,}000$ written out means counting nine zeros correctly. In scientific notation it is $3 \times 2$ and $4 + 5$, which is arithmetic anyone can do in their head. That is the actual reason scientists use it, not the neatness.

Worked Example 1: Multiplying

$$(2.5 \times 10^6)(3 \times 10^{-2})$$

When the Front Drifts Out of Range

$$(5 \times 10^3)(4 \times 10^6) = 20 \times 10^9$$

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