Scientific Notation: The Mathematical Equivalent of Shorthand

According to astronomers, the observable universe contains at least 120 sextillion stars, a number that is undeniably astounding. To put it into perspective, one sextillion is represented as a "1" trailed by 21 zeros. When written out fully, 120 sextillion appears as 120,000,000,000,000,000,000,000.

The concept of one sextillion is mind-boggling and difficult to grasp. Fortunately, scientific notation offers a practical method for representing such enormous or tiny numbers in a clear and standardized manner. By eliminating the need for lengthy strings of zeros, it simplifies complex calculations and comparisons, making them more approachable.

Let’s dive into the details of scientific notation and examine some practical examples to see how it works in real-world scenarios.

What Is Scientific Notation?

Scientific notation is a streamlined method for representing very large or very small numbers in a clear and consistent format. It consists of two key parts: a coefficient (usually a number between 1 and 10) and an exponent (a power of 10), where the coefficient’s absolute value indicates the scale of the number.

Imagine scientific notation as a recipe card. The title and image on the card (like the coefficient) give you a quick sense of the dish’s size, but they don’t include the step-by-step details. The cooking time (akin to the exponent) specifies exactly how long to prepare the dish, ensuring the perfect result.

In this analogy, the coefficient’s absolute value is like glancing at the recipe card’s title and image to gauge the dish’s scale, while the exponent acts as the cooking time, providing precise guidance for the desired outcome. Now, let’s explore coefficients in more detail.

Coefficients

In basic terms, a coefficient is a numerical value that multiplies another number or variable in a mathematical equation. It acts as a scaling factor, indicating how many times the associated variable or number should be multiplied.

Exponents

An exponent is a small number that specifies how many times a larger number (known as the base) should be multiplied by itself. For instance, in the expression 2³, the base is 2, and the exponent is 3. This means you multiply 2 by itself three times: 2 x 2 x 2, resulting in 8.

Exponents provide a compact way to represent repeated multiplication, simplifying the handling of extremely small or large numbers in mathematical calculations.

Additional Examples of Scientific Notation

As anyone familiar with basic math knows, 100 is the same as 10 multiplied by 10. Instead of writing "10 x 10," we can save space and express it as 10². But what does that tiny "2" beside the 10 signify?

That small number is known as an exponent, and the larger number next to it (in this case, 10) is called the base. The exponent indicates how many times the base should be multiplied by itself. For example, 10² is shorthand for 10 x 10, while 10³ stands for 10 x 10 x 10, which equals 1,000.

(Incidentally, when working on math problems using a computer or scientific calculator, the caret symbol — ^ — is often used to represent exponents. So, 10² can also be written as 10^2, but we’ll discuss that topic another time.)

Scientific notation is built on the use of exponents. Take the number 2,000 as an example. To express this in scientific notation, you’d write it as 2.0 x 10³. Essentially, you’re taking a number (2.0) and multiplying it by a power of 10 (10^3).

The exponent (3, in this case) indicates that the coefficient (2.0) is multiplied by 10 raised to the power of 3. This shifts the decimal point three places to the right, yielding the original number: 2,000.

Scientific Notation vs. Decimal Notation

Decimal notation and scientific notation are two distinct methods for representing numbers. Decimal notation uses a standard sequence of digits and a decimal point, making it practical for everyday applications. For example, 3,500 stands for 3,500 units, and 22.5 represents 22 and a half units.

On the other hand, scientific notation is tailored for efficiently expressing extremely large or small numbers. It involves a coefficient between 1 and 10 multiplied by a power of 10. For instance, x 10³ is the same as 3,500.

The primary distinction is in their approach to scale: Decimal notation is more straightforward for everyday numbers, whereas scientific notation excels at simplifying calculations involving numbers of vastly different magnitudes.

Applications of Scientific Notations

Scientific notation is extensively employed to represent extremely large or tiny numbers in a concise and uniform manner. Below are some practical examples:

Other Types of Notation

Notation refers to a system of symbols, signs, or characters used to represent or communicate information, typically in a structured and standardized format. Here are some widely used types:

A Sextillion by Another Name

Now, let’s have some fun. Using the steps we’ve discussed, we can express 4,000 in scientific notation as 4.0 x 10³. Similarly, 27,000 becomes 2.7 x 10⁴, and 525,000,000 transforms into 5.25 x 10⁸.

But how do we tackle 120 sextillion, that massive, cumbersome number from earlier? Start by examining 120,000,000,000,000,000,000,000 closely. There are 23 digits following the "1." (Feel free to count them; we’ll wait.)

Therefore, in scientific notation, 120,000,000,000,000,000,000,000 is written as 1.2 x 10²³.

But let’s be honest, the latter is far more visually appealing. Plus, the exponent instantly conveys the sheer enormity of the number, something counting zeros could never achieve. This is the elegance and simplicity of scientific notation.

Going Negative

You’ll be pleased to learn that this method also works for negative numbers or values less than one.

Imagine you only have one-tenth of an apple. Mathematically, that’s 0.10 apples. Similarly, if you’re left with one-millionth of an apple, you’re stuck with a mere 0.000001 apples. Not ideal.

Fortunately, you can express these quantities using scientific notation — and it’s remarkably similar to the approach we’ve been using.

In this case, we’ll move the decimal point to the right of the first non-zero digit in the number. This will leave us with a simple "1." For mathematical precision, we’ll represent this as "1.0."

To express 0.000001, we’ll multiply our 1.0 by an exponent of 10. However, there’s a catch: The exponent will be a negative number.

Take another look at 0.000001. Notice the six digits after the decimal point? This means we must multiply our 1.0 by 10⁻⁶. In short, 1.0 x 10⁻⁶ is how we write one-millionth, or 0.000001, in scientific notation.

Similarly, 6.0 x 10⁻³ represents 0.006. Likewise, 0.00086 would be expressed as 8.6 x 10⁻⁴. And so on. Enjoy your calculations!