Standard form Revision Notes

Standard form is a way of writing very large or very small numbers in a compact and manageable way. It expresses any number as a product of two parts: a decimal number A, where 1 ≤ A < 10, and a power of ten, 10 raised to an integer n.

The general format is: A × 10^n, where n is an integer (positive, negative, or zero). This notation simplifies working with numbers that would otherwise be cumbersome to write or calculate with directly.

For example, the number 45000 can be written as 4.5 × 10^4 because 4.5 × 10,000 = 45000. Similarly, 0.00032 can be written as 3.2 × 10^-4 because 3.2 × 0.0001 = 0.00032.

• Standard form = A × 10^n
• A is a decimal number with 1 ≤ A < 10
• n is an integer (positive, negative, or zero)
• Used for very large or very small numbers

To convert an ordinary number into standard form, follow these steps:

1. Identify the significant figures of the number and write them as a decimal number A between 1 and 10.

2. Count how many places the decimal point moves to reach this number A from the original number. This count determines the power n of 10: move left for positive n, right for negative n.

• Example 1: Convert 720000 to standard form.
• Write 720000 as 7.2 (decimal between 1 and 10).
• The decimal point moves 5 places left (from after the last zero to after 7).
• So, 720000 = 7.2 × 10^5.
• Example 2: Convert 0.0045 to standard form.
• Write 0.0045 as 4.5 (decimal between 1 and 10).
• The decimal point moves 3 places right (from after 0 to after 4.5).
• So, 0.0045 = 4.5 × 10^-3.

To convert a number from standard form back to ordinary form:

1. Multiply the decimal number A by 10 raised to the power n.

2. Move the decimal point in A to the right if n is positive, or to the left if n is negative, by |n| places. Fill with zeros if necessary.

• Example: Convert 3.6 × 10^4 to ordinary number.
• Move decimal point in 3.6 four places to the right: 36000.
• So, 3.6 × 10^4 = 36000.
• Example: Convert 5.2 × 10^-3 to ordinary number.
• Move decimal point in 5.2 three places to the left: 0.0052.
• So, 5.2 × 10^-3 = 0.0052.

Calculations involving numbers in standard form include multiplication, division, addition, and subtraction. Each operation requires careful handling of the decimal parts and powers of ten.

Multiplication: Multiply the decimal parts (A values) and add the exponents (n values).

Division: Divide the decimal parts and subtract the exponents (numerator exponent minus denominator exponent).

• Multiplication example: (3 × 10^5) × (2 × 10^3)
• Multiply decimals: 3 × 2 = 6
• Add exponents: 5 + 3 = 8
• Result: 6 × 10^8
• Division example: (6 × 10^7) ÷ (2 × 10^3)
• Divide decimals: 6 ÷ 2 = 3
• Subtract exponents: 7 - 3 = 4
• Result: 3 × 10^4

Addition and subtraction require the numbers to have the same power of ten before combining the decimal parts.

Steps: 1. Adjust one or both numbers so their powers of ten match by rewriting one number’s decimal and exponent.

2. Add or subtract the decimal parts accordingly, keeping the common power of ten unchanged. 3. If necessary, convert the result back to standard form by adjusting the decimal part to be between 1 and 10.

• Example: Add 3.2 × 10^4 and 4.5 × 10^3
• Rewrite 4.5 × 10^3 as 0.45 × 10^4 (move decimal one place right, decrease exponent by 1)
• Add decimals: 3.2 + 0.45 = 3.65
• Result: 3.65 × 10^4
• Example: Subtract 5.0 × 10^-2 from 7.2 × 10^-2
• Powers of ten are the same (10^-2), subtract decimals: 7.2 - 5.0 = 2.2
• Result: 2.2 × 10^-2