Understanding Standard Form in Mathematics
NoteLearning objectives:
- Operations with numbers in the form $a \times 10^k$ where
$1 \leq a < 10$ and $k$ is an integer.
What is Standard Form?
- Standard form, also known as scientific notation, is a way of expressing numbers that are too large or too small to be conveniently written in decimal form.
- It's particularly handy in fields like science and engineering where you deal with extremely large or tiny numbers.
The Structure of Standard Form
In standard form, a number is expressed as:
$$ a \times 10^n $$
Where:
- $a$ is a number, $1 \leq a < 10$.
- $n$ is an integer.
Think of $a$ as the "significant figures" part of the number, and $10^n$ as the "scale" that tells you how big or small the number is.
Converting Numbers to Standard Form
Let's walk through the process of converting a number to standard form with some examples.
ExampleA Large Number
Convert the number 450,000 to standard form.
| Operation | Explanation |
|---|---|
| $450,000 \to 45,000 \to 4,500 \to 450 \to 45 \to 4.5 $ | Identify $a$: Move the decimal point in 450,000 to the left until you have a number between 1 and 10. |
| 5 places | Determine $n$: Count how many places you moved the decimal point. |
| $4.5 \times 10^5$ | Write in standard form. |
A Small Number
Convert the number 0.00032 to standard form.
| Operation | Explanation |
|---|---|
| $0.00032 \to 0.0032 \to 0.032 \to 0.32 \to 3.2$ | Identify $a$: Move the decimal point to the right until you have a number between 1 and 10, which is 3.2. |
| 4 | Determine $n$: Count the places moved, which is 4. Since you moved right, $n$ is negative. |
| $3.2 \times 10^{-4}$ | Write in standard form |
- A common mistake is forgetting to adjust the exponent's sign when moving the decimal point to the right.
- If you're converting a small number to standard form, remember that a negative exponent makes things smaller.
Practice Makes Perfect
Exercise 1: Convert to standard form.
- 3,900
- 0.000967
- 0.021
- 8,230,000
Exercise 2: Convert to general form.
- $3.45 \times 10^3$
- $1.134 \times 10^1$
- $4.65 \times 10^{-3}$
- $7.75 \times 10^{-2}$
Solution
Exercise 1:
- $3.9 \times 10^3$
- $9.67 \times 10^{-4}$
- $2.1 \times 10^{-2}$
- $8.23 \times 10^6$
Exercise 2:
- 3,450
- 11.34
- 0.00465
- 0.0775
Summary
- Standard form (scientific notation) is a way to write very large or very small numbers clearly and efficiently.
- It uses the structure $a \times 10^n$, where $a$ is a number between 1 and 10, and $n$ shows how many times the decimal point moves.
- Positive exponents are used for large numbers, and negative exponents for small ones.
- This form is useful in science, engineering, and mathematics to simplify calculations and make numbers easier to read.
Exercise 1: Convert the following numbers to standard form:
- 0.00000451
- 602,000,000,000,000,000,000,000
- 0.000010000345
- 59,800,090
Exercise 2: Convert the following numbers from standard form to decimal form:
- $4.5×10^5$
- $3.2×10^{−4}$
- $9.81×10^1$
- $11.23×10^{−3}$
- $6.022×10^{8}$
Exercise 3: Which number is larger: $5.6 \times 10^7$ or $6.1 \times 10^6$.
Exercise 4: Write the product of $(3 \times 10^4)\times (2 \times 10^3)$ in standard form.
Exercise 5: Simplify and express in standard form: $\frac{7.5 \times 10^{-2}}{2.5 \times 10^3}$
Answers to self-review exercises.
Solution
Exercise 1:
- 0.00000451 → $4.51×10^{−6}$
- 602,000,000,000,000,000,000,000 → $6.02×10^{23}$
- 0.000010000345 → $1.0000345×10^{−5}$
- 59,800,090 → $5.980009×10^7$
Exercise 2: Convert the following numbers from standard form to decimal form
- $4.5×10^5$ → 450,000
- $3.2×10^{−4}$ → 0.00032
- $9.81×10^1$ → 98.1
- $11.23×10^{−3}$ → 0.01123
- $6.022×10^8$ → 602,200,000
Exercise 3: $5.6×10^7$ is larger.
Exercise 4: $6×10^7$
Exercise 5: $3×10^{−5}$
