Order of Magnitude: Scale and Comparisons

What Is Order of Magnitude?

To find a number's magnitude in powers of ten, we first write it in scientific notation as $a \times 10^b$, where $1 \leq a < 10$. The order of magnitude is the power of ten that best approximates the number.

Because the powers of ten scale is multiplicative, the halfway point between $10^0$ (which equals $1$) and $10^1$ (which equals $10$) is not $5.5$. Instead, it is exactly the square root of $10$. We can use fractional exponents to write this as $10^{0.5}$, which is approximately $3.16$.

Using this logarithmic midpoint, the standard rounding convention is:

• If $a < 3.16$, the order of magnitude is simply the exponent $b$.
• If $a \geq 3.16$, the value is closer to the next power of ten, so the order of magnitude is $b + 1$.

For example, $2.0 \times 10^4$ has an order of magnitude of $4$ because $2.0$ is less than $3.16$.

However, $8.0 \times 10^4$ has an order of magnitude of $5$ because $8.0$ is greater than $3.16$.

Key Ideas and Vocabulary

When learning about exponents and powers, you will frequently encounter terms related to relative size.

• Base 10: The numbering system foundation used to define orders of magnitude.
• Logarithmic Scale: A scale where each step represents a multiplication by $10$ rather than the addition of a fixed amount.
• Orders of Magnitude Difference: A way to compare orders of magnitude by subtracting their exponents.

When a problem states that one object is "three orders of magnitude larger" than another, it means the object is approximately $10^3$, or $1{,}000$, times larger.

Visual Explanation

On a standard linear number line, the distance between $1$ and $2$ is the same as the distance between $2$ and $3$. However, the powers of ten scale operates logarithmically, where each equal distance on the axis represents multiplying by $10$.

Every jump along this line adds $1$ to the exponent, which mathematically translates to multiplying the actual value by $10$. A movement of three jumps to the right represents an increase of three orders of magnitude, or a multiplication by $10^3 = 1{,}000$.

Worked Examples

Review these order of magnitude examples to see how the mathematical rounding rules apply to various quantities.

Example 1: Finding the order of magnitude for large quantities

Question: Determine the order of magnitude of $72{,}000$.

Method:

1. Write the number in scientific notation. We move the decimal point $4$ places to the left, resulting in $7.2 \times 10^4$.
2. Identify the coefficient $a = 7.2$ and the exponent $b = 4$.
3. Compare the coefficient to the $3.16$ threshold.
4. Since $7.2 \geq 3.16$, we add $1$ to the exponent. The calculation is $4 + 1 = 5$.

Answer: The order of magnitude is $5$.

Check: We can evaluate $10^4 = 10{,}000$ and $10^5 = 100{,}000$. Logarithmically, $72{,}000$ is much closer to $100{,}000$ than it is to $10{,}000$, confirming our result of $5$.

Example 2: Small numbers and negative exponents

Question: What is the order of magnitude of $0.0018$?

Method:

1. Write the number in scientific notation. Move the decimal point $3$ places to the right to get $1.8 \times 10^{-3}$.
2. The coefficient is $a = 1.8$ and the exponent is $b = -3$.
3. Compare $1.8$ to the $3.16$ threshold.
4. Since $1.8 < 3.16$, the exponent remains unchanged.

Answer: The order of magnitude is $-3$.

Check: The number $0.0018$ lies between $10^{-3}$ ($0.001$) and $10^{-2}$ ($0.01$). It is much closer to $0.001$, so the order of magnitude correctly aligns with $-3$.

Example 3: Comparing orders of magnitude

Question: A planet has a mass of $6.0 \times 10^{24}$ kg, and a moon has a mass of $2.0 \times 10^{22}$ kg. How many orders of magnitude larger is the planet's mass?

Method:

1. Determine the order of magnitude for the planet. The coefficient $6.0 \geq 3.16$, so the order is $24 + 1 = 25$.
2. Determine the order of magnitude for the moon. The coefficient $2.0 < 3.16$, so the order is $22$.
3. Subtract the moon's order from the planet's order: $25 - 22$.

Answer: The planet's mass is $3$ orders of magnitude larger.

Check: Divide the masses directly: $\dfrac{6.0 \times 10^{24}}{2.0 \times 10^{22}} = 3.0 \times 10^2$.

The coefficient $3.0 < 3.16$, so the order of magnitude of the ratio is $2$. Wait, estimating orders independently before finding the difference can sometimes yield a slightly different check than dividing first. The difference in their rounded orders is $3$, while the order of their exact ratio is $2$.

When comparing quantities directly, the most mathematically rigorous approach is to find the order of magnitude of their ratio. Using the ratio method, the difference is $2$ orders of magnitude.

Common Mistakes and Non-Examples

A common misconception is thinking that "one order of magnitude larger" means adding $10$. An order of magnitude is strictly multiplicative. A tree that is one order of magnitude taller than a $2$-meter bush is $20$ meters tall, not $12$ meters tall.

Another frequent error involves the rounding threshold. In standard arithmetic, we round up at $5$. However, the logarithmic scale midway point is $\sqrt{10}$. While working with radicals and surds, you will see that many square roots produce long decimals. The rule for simplifying radicals shows that $\sqrt{9} = 3$ and $\sqrt{16} = 4$. The value of $\sqrt{10}$ is an irrational number approximately equal to $3.162$.

Do not round the coefficient using the number $5$. If a scientific notation coefficient is $4.2$, it is greater than $3.16$, so the order of magnitude rounds up to the next power of ten.

Real-World Connections

Orders of magnitude are universally applied in physics and astronomy to classify objects into manageable categories. Rather than saying the observable universe is roughly $8.8 \times 10^{26}$ meters across, a physicist will simply say its size is on the order of $10^{27}$ meters.

By stripping away the precision of the coefficient, researchers can quickly evaluate whether two quantities belong to the same scale. If a new particle is discovered to be three orders of magnitude heavier than an electron, scientists instantly know its mass is approximately $1{,}000$ times greater without performing complex multiplication.