Scientific Notation | Mathematics for ACT PDF Download

Introduction

• Understanding and effectively using scientific notation is a fundamental skill throughout your education. It involves breaking down numbers into three components: the simplified number, multiplied by 10 raised to some power. The purpose of scientific notation is to represent large or small numbers in a compact and readable format without altering their value.
• For instance, let's consider the number 15. While it may seem unnecessary to express 15 in scientific notation, it serves as a simple example to illustrate the process. In scientific notation, 15 is written as 1.5x10^1. This means 1.5 times 10 raised to the power of 1. Essentially, it's just a different way to express the same value.
• Scientific notation simplifies the representation of numbers, especially when dealing with large or small quantities. Throughout this chapter, we'll focus on mastering the conversion to and from scientific notation. This standardized approach ensures consistency and accuracy, regardless of the specific number being converted.

Converting from standard to scientific notation

Converting from standard notation to scientific notation involves simplifying a number so that there is one digit to the left of the decimal point. While there can be varying numbers to the right of the decimal, we'll focus on having just one for simplicity.

The process of converting to scientific notation revolves around reducing a large number or growing a small number. To achieve this, we'll count decimal places. Let's consider a more complex example: 12,345,678, a very large number. We aim to express it as 1.2 multiplied by some power of 10. Follow these steps for a clear explanation of the conversion process:

• Begin with the whole number to the left of the decimal: 12,345,678.0
• Move the decimal one place to the left (signifying division by ten): 12,345,67.80
• Repeat this step until only 1 is left to the left of the decimal: 1.23456780
• Count the number of times the decimal was moved (the groups of ten removed): 7 times.
• Round off the number to have one digit on each side of the decimal: 1.2
• Add the number of groups of ten removed to the end of the smaller number in the form of "x10⁷": 1.2x10⁷

This completes the conversion to scientific notation.

Similarly, when dealing with very small numbers, the process is quite similar. For instance, let's examine the number 0.000437:

• Since the decimal is already visible, we can skip the initial step.
• Move the decimal one place to the right (to ensure one non-zero digit to the left): 0.00437
• Continue moving the decimal until there is only one non-zero digit to the left: 0004.37
• Count the number of times the decimal was moved (the groups of ten added): 4 times.
• Round off the rest of the number to have one digit on each side of the decimal: 4.4
• Add the number of groups of ten added to the end of the larger number in the form of "x10^-4": 4.4x10^-4

It's crucial to remember that multiplying by a negative power of 10 indicates moving the decimal back to its original position, in case of a backtrack. Double-checking the sign on the power of 10 ensures accuracy. For larger numbers, the power of ten is positive, while for smaller numbers, it is negative, reflecting the direction of decimal movement.

Converting from scientific to standard notation

Converting from scientific notation to standard notation is a straightforward process, especially if you're familiar with the steps outlined above. Essentially, it involves carrying out the expression given in the scientific notation. Let's break down what this means:

Consider the example 2.6x10³, a simple representation of scientific notation. To convert this to standard notation, we simply follow what the expression dictates:

• 2.6 multiplied by 10^3

That's all there is to it! Using a calculator, we can directly perform this operation to obtain the conversion in standard notation:

• 2.6x10³ = 2600

Without a calculator, we would need to add zeros equal to the power of 10 included in the scientific notation. In this case, since 10^3 equals 1000 (as the exponent indicates the number of zeros to attach to 1), we multiply 2.6 by 1000:

• 2.6x1000 = 2600

Alternatively, you can visualize this as moving decimal places, as we've been doing. If the exponent on the 10 is positive, you move the decimal that many places to the right. Conversely, if the exponent is negative, you move the decimal that many places to the left. Ensure to fill any gaps between the decimal and the numbers with 0s for accuracy.

Multiplication & division in scientific notation

Multiplication and division in scientific notation follow similar principles to those outlined above. We can either convert everything to standard numbers before performing the operation, or we can carry out the operation while keeping the numbers in scientific notation. To do this, we treat the exponent part of scientific notation (the part represented as 10 raised to some power) as the remainder of our primary number (the part without the exponential term). By doing this, we can disregard the exponent and directly perform multiplication and division on our simplified numbers. Let's illustrate this with an example:

Consider the product of the following expressions:

• (2.4∗10⁸)∗(8.9∗10³)

We focus solely on the simplified parts of the numbers, which gives us 2.4∗8.9.

This simpler product equals 21.4 (rounded to the nearest decimal). Note that we have increased the number of digits before the decimal to two (21). To revert the format to proper scientific notation, we move the decimal place one space to the left. What happens to our exponent term when we do this? It increases by one in the exponent!

So, after our simple product, we have 2.1 (rounded to one decimal place), an increase of our exponent by 1, and our original exponents,

• 10^8 * 10³

We multiply these exponents together as well, resulting in 10¹¹ following the rule for multiplying exponents.

Now, putting it all together:

• 2.1∗10¹¹, accounting for the increase in our exponent. Let’s incorporate that as well!
• This yields our final answer of 2.1∗10¹².

Division follows a similar process to multiplication. You perform the same steps, but when your number changes significantly, you'll move your decimal to the right, which subtracts one from your original exponent. Consequently, you'll "carry a negative one" and subtract it from your total exponent instead of adding it.

Key points

• Format: Scientific notation consists of two components: the simple number and the exponent (10 raised to some power).
• Converting standard to scientific: Determine how many places the decimal point needs to move, and that value becomes the exponent of 10 in scientific notation. Moving the decimal to the left results in a positive exponent, while moving it to the right yields a negative exponent.
• Converting scientific to standard: The exponent above 10 indicates the number of decimal places to move the decimal point. A positive exponent means shifting it to the right (since it originally came from the right), while a negative exponent means shifting it to the left (since it originally came from the left).
• Addition/subtraction: Utilize a calculator for the most efficient computation. Alternatively, convert scientific numbers to standard, perform addition or subtraction, then convert them back to scientific notation.
• Multiplication/division: Employ a calculator for the most efficient calculation. Alternatively, separate the scientific numbers into their simple number and 10 exponent parts, perform multiplication or division separately, and then combine them at the end (remembering to handle any remainders appropriately!).