What is Standard Form?
Standard form is a way to write very large or very small numbers in a compact and manageable way. It expresses a number as the product of two parts: a decimal number A and a power of ten, written as 10 raised to an integer n.
The number A must be at least 1 but less than 10 (1 ≤ A < 10). The integer n can be positive, negative, or zero, indicating how many places the decimal point has moved.
For example, 3 500 can be written as 3.5 × 10^3 because 3.5 × 10 × 10 × 10 = 3 500. Similarly, 0.0042 can be written as 4.2 × 10^-3 because moving the decimal point three places to the right gives 4.2.
- Standard form = A × 10^n, where 1 ≤ A < 10 and n is an integer.
- A is called the coefficient or mantissa.
- n is the exponent or power of ten.
- Useful for very large or very small numbers.
Converting Ordinary Numbers to Standard Form
To convert a number to standard form, follow these steps:
1. Identify the position of the decimal point in the original number.
2. Move the decimal point to create a new number A between 1 and 10 (including 1 but less than 10). Count how many places you move the decimal point; this count determines the exponent n of 10. Moving the decimal point to the left gives a positive n, moving it to the right gives a negative n.
- Example: Convert 47 200 to standard form.
- Step 1: Original number is 47 200 (decimal point after 0).
- Step 2: Move decimal point 4 places left to get 4.72 (which is between 1 and 10).
- Step 3: Since decimal moved left 4 places, n = 4.
- Answer: 4.72 × 10^4.
Converting Standard Form to Ordinary Numbers
To convert a number from standard form back to ordinary decimal notation:
1. Look at the exponent n on 10^n. If n is positive, move the decimal point in A to the right by n places.
2. If n is negative, move the decimal point in A to the left by |n| places (the absolute value of n). Add zeros if necessary to fill empty places.
- Example: Convert 6.3 × 10^-3 to ordinary number.
- Step 1: n = -3, so move decimal point 3 places left.
- Step 2: 6.3 becomes 0.0063 after adding zeros.
- Answer: 0.0063.
Calculations Using Standard Form
When multiplying or dividing numbers in standard form, use the properties of indices to simplify calculations:
Multiplication: Multiply the coefficients (A values) and add the exponents (n values).
Division: Divide the coefficients and subtract the exponents (denominator exponent from numerator exponent).
- Example 1: Multiply (3 × 10^4) by (2 × 10^3).
- Multiply coefficients: 3 × 2 = 6.
- Add exponents: 4 + 3 = 7.
- Answer: 6 × 10^7.
- Example 2: Divide (5 × 10^6) by (2 × 10^2).
- Divide coefficients: 5 ÷ 2 = 2.5.
- Subtract exponents: 6 - 2 = 4.
- Answer: 2.5 × 10^4.
Checking and Writing Final Answers
After performing calculations, ensure the coefficient A is between 1 and 10. If not, adjust the coefficient and exponent accordingly.
For example, if the result is 12 × 10^5, rewrite as 1.2 × 10^6 by moving the decimal point one place left and increasing the exponent by 1.
Always write answers in the correct standard form to gain full marks in exams.
- Adjust coefficient to satisfy 1 ≤ A < 10 after calculations.
- Rewrite answers if necessary.
- Check calculations by converting back to ordinary numbers if unsure.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Convert to standard form | Move decimal point to get A between 1 and 10; count moves for n | 47 200 → 4.72 × 10^4 | Show clear decimal moves and exponent sign |
| Convert from standard form | Move decimal point right (n positive) or left (n negative) | 6.3 × 10^-3 → 0.0063 | Write zeros if needed; check decimal placement |
| Multiply in standard form | Multiply coefficients; add exponents | (3 × 10^4)(2 × 10^3) = 6 × 10^7 | Show coefficient and exponent steps separately |
| Divide in standard form | Divide coefficients; subtract exponents | (5 × 10^6) ÷ (2 × 10^2) = 2.5 × 10^4 | Keep track of exponent subtraction order |
| Adjust final answer | Ensure 1 ≤ A < 10; adjust exponent accordingly | 12 × 10^5 → 1.2 × 10^6 | Write final answer in correct standard form |
Active recall
Quick check
1. Express 0.00056 in standard form.
5.6 × 10^-4 because the decimal point moves 4 places to the right to get 5.6.
2. Convert 7.3 × 10^5 to an ordinary number.
730 000 because the decimal point moves 5 places to the right.
3. Calculate (4 × 10^3) × (3 × 10^2) and write your answer in standard form.
(4 × 3) × 10^(3+2) = 12 × 10^5 = 1.2 × 10^6 after adjusting the coefficient.
4. Calculate (6 × 10^7) ÷ (2 × 10^3) and write your answer in standard form.
(6 ÷ 2) × 10^(7-3) = 3 × 10^4.