Powers of \(10\)
Our standard numbering system is decimal (base-\(10\)), meaning each place value shift to the left represents a multiplication by \(10\).
We can capture these large place values concisely using exponents:
\(\begin{align*} 1 & = 10^0 \\ 10 & = 10^1 \\ 100 & = 10 \times 10 = 10^2 \\ 1,000 & = 10 \times 10 \times 10 = 10^3 \\ 10,000 & = 10 \times 10 \times 10 \times 10 = 10^4 \\ 1,000,000 \text{ (1 Million)} & = 10^6 \\ 1,000,000,000 \text{ (1 Billion)} & = 10^9 \end{align*}\)
Representing large quantities using powers of \(10\)
The power of \(10\) can be used to represent larger values in expanded form, and decimal values can be expressed using powers of \(10\).
Example:
1. \(27381\)
\(=(2 \times 10000) + (7 \times 1000) + (3 \times 100) + (8 \times 10) + (1 \times 1)\)
\(=(2 \times 10^{4}) + (7 \times 10^{3}) + (3 \times 10^{2}) + (8 \times 10^{1}) + (1 \times 10^{0})\)
2. \(742.901\)
\(=(7 \times 100) + (4 \times 10) +(2 \times 1) + (9 \times \frac{1}{10}) + (0 \times \frac{1}{100}) + (1 \times \frac{1}{1000})\)
\(=(7 \times 10^{2}) + (4 \times 10^{1}) +(2 \times 10^{0}) + (9 \times 10^{-1}) + (0 \times 10^{-2}) + (1 \times 10^{-3})\)
Converting large numbers into standard exponential form
Any large number can be changed into a decimal factor multiplied by a power of \(10\). This is known as standard exponential form(or scientific notation).
A number is successfully written in standard form when it is expressed as:
\(m \times 10^n\)
Where:
1. \(m\) is a decimal factor greater than or equal to \(1\), but strictly less than \(10\) (\(1 \le m < 10\)).
2. \(n\) is an integer representing the power of \(10\).
Step-by-Step Conversion Algorithm
Step 1: Locate the decimal point. If the number is a whole number, the decimal point is understood to be at the end of the number.
Step 2: Shift the decimal point to the left until it sits directly behind the first non-zero digit. This creates your value for \(m\).
Step 3: Count the exact number of places you skipped over. This count becomes your exponent \(n\).
Step 4: If the decimal point is moved to the left, the exponent is positive. If it is moved to the right, the exponent is negative.
Example:
1. Convert \(100\text{ million}\) into standard form.
Solution:
\(100,000,000 = 1 \times 10^8\)
2. Convert \(0.00006737\) into standard form.
Solution:
\(0.00006737 = 6.737 \times 10^{-5}\)
Large quantities
In the real world, we frequently encounter quantities that are too large to read or comprehend easily at a glance.
Example A (Global Population):
\(8,200,000,000\) (\(8\) billion, \(200\) million)
Example B (Sheep Population):
\(1,000,000,000\) (\(1\) billion)
When numbers have many trailing zeros, they become difficult to count, prone to writing errors, and hard to compare.
To make very large numbers easier to write, read, compare, and calculate, we express them using powers of \(10\) in standard form.
Comparing large quantities using powers of \(10\)
When numbers are written in standard form, we no longer need to count zeros to see which number is larger.
We use a structured two-step comparison:
Check the Exponents (\(n\)):
Compare the powers of \(10\) first.
The number with the higher exponent is always the larger quantity, regardless of its front decimal value.
Check the Decimals (\(m\)):
If the exponents are identical, compare the decimal factors \(m\).
The number with the larger decimal factor is the greater quantity.
Example:
Compare the mass of Planet A (\(4.3 \times 10^{24}\text{ kg}\)) and Planet B (\(8.9 \times 10^{22}\text{ kg}\)). Which is heavier?
Solution:
Look at the powers of \(10\).
\(10^{24} > 10^{22}\)
Since \(24 > 22\), \(10^{24}\) is \(100\) times larger than \(10^{22}\).
Conclusion:
Even though \(8.9\) looks bigger than \(4.3\), Planet A is significantly heavier because it is multiplied by a much higher power of \(10\).
\(4.3 \times 10^{24} > 8.9 \times 10^{22}\)