Converting Decimals To Scientific Notation
Scientific notation is a compact way to express very large or very small numbers as a product of a decimal coefficient (between 1 and 10) and a power of 10. The number 540,000,000 becomes 5.4 times 10 to the 8th, and to convert back, move the decimal 8 places to the right. This Grade 7 math skill from Saxon Math, Course 2 is essential in science and engineering for expressing measurements that range from subatomic scales (10 to the -15) to astronomical distances (10 to the 26), making calculations and comparisons practical.
Key Concepts
Property To convert a decimal smaller than 1, move the decimal point to the right to create a number between 1 and 10. The new exponent will be negative, counting the places moved. Then combine the powers.
Examples $0.25 \times 10^4 = (2.5 \times 10^{ 1}) \times 10^4 = 2.5 \times 10^{ 1+4} = 2.5 \times 10^3$ $0.16 \times 10^6 = (1.6 \times 10^{ 1}) \times 10^6 = 1.6 \times 10^{ 1+6} = 1.6 \times 10^5$ $0.75 \times 10^{ 8} = (7.5 \times 10^{ 1}) \times 10^{ 8} = 7.5 \times 10^{ 1+( 8)} = 7.5 \times 10^{ 9}$.
Explanation Tiny decimals need a makeover! To fit the 'between 1 and 10' rule, boost the number by moving its decimal point to the right. The cost of this boost is a negative exponent. Then, just add the exponents together to complete the problem.
Common Questions
What is scientific notation?
Scientific notation expresses numbers as c times 10 to the n, where c is a decimal number between 1 and 10, and n is an integer exponent. For example, 81,200,000 = 8.12 times 10 to the 7.
How do I convert a large number to scientific notation?
Move the decimal point left until you have one non-zero digit before it. Count the moves — that is your positive exponent. For 540,000,000: move 8 places left to get 5.4, so it is 5.4 times 10 to the 8.
How do I convert from scientific notation back to standard form?
Move the decimal point right by the number of the exponent. For 8.12 times 10 to the 7, move the decimal 7 places right: 81,200,000.
What is a negative exponent in scientific notation?
A negative exponent means a very small number. Move the decimal to the left. For 5.12 times 10 to the -7, move the decimal 7 places left: 0.000000512.
When do students learn scientific notation?
Scientific notation is introduced in Grade 7-8. Saxon Math, Course 2 covers it in Chapter 9, including both large numbers (positive exponents) and small numbers (negative exponents).
What are common mistakes with scientific notation?
Students sometimes move the decimal in the wrong direction (left instead of right or vice versa). For positive exponents, move right to make the number larger. For negative exponents, move left to make it smaller.
How is scientific notation used in science?
Scientists use scientific notation for measurements: the speed of light is 3.0 times 10 to the 8 m/s, an atom's diameter is about 1 times 10 to the -10 meters, and Earth's mass is 5.97 times 10 to the 24 kg.