Question

Convert the following numbers from scientific notation to standard notation: (a) \(5.28 \times 10^{3}\) (b) \(8.205 \times 10^{-2}\) (c) \(1.84 \times 10^{-5}\) (d) \(6.37 \times 10^{4}\)

Short answer

(a) 5280, (b) 0.08205, (c) 0.0000184, (d) 63700.

Step-by-step solution

Understanding Scientific Notation

Scientific notation expresses numbers as a product of a number (usually between 1 and 10) and a power of ten. To convert this to standard notation, we must multiply the base number by 10 raised to the given exponent.

Converting (a) to Standard Notation

For (a) \(5.28 \times 10^{3}\), move the decimal point 3 places to the right because the exponent is 3: \(5.28 \rightarrow 5280\).

Converting (b) to Standard Notation

For (b) \(8.205 \times 10^{-2}\), move the decimal point 2 places to the left because the exponent is -2: \(8.205 \rightarrow 0.08205\).

Converting (c) to Standard Notation

For (c) \(1.84 \times 10^{-5}\), move the decimal point 5 places to the left because the exponent is -5: \(1.84 \rightarrow 0.0000184\).

Converting (d) to Standard Notation

For (d) \(6.37 \times 10^{4}\), move the decimal point 4 places to the right because the exponent is 4: \(6.37 \rightarrow 63700\).

Key concepts

Standard Notation

Standard notation is the regular way of writing numbers without using exponents. It's how we usually see and use numbers in everyday life. For example, writing 5000 instead of as a product with powers of ten is an example of standard notation.
In standard notation:

• Numbers are written as whole numbers or fractions/decimals.
• There are no powers of ten involved.
• It's the most common and practical form for direct calculations.

When converting from scientific to standard notation, the goal is to eliminate the exponent by appropriately moving the decimal point.

Mathematics Education

Understanding concepts like scientific and standard notation is crucial in mathematics education. It helps students grasp the size and scale of numbers efficiently. These skills extend to many areas in science and engineering, where large or tiny values are frequent.
By learning to convert between notations:

• Students improve their number sense.
• They gain an appreciation for the magnitude of numbers.
• It builds a foundation for more advanced mathematical operations and concepts.
• It lays groundwork for scientific understanding and literacy.

This concept connects classroom learning to real-world applications.

Powers of Ten

The concept of powers of ten is fundamental when working with large and small numbers. It involves expressing numbers as a base of ten raised to an exponent. The exponent indicates how many times the number ten is used as a factor or fraction.
In powers of ten:

• A positive exponent moves the decimal to the right, indicating a multiplication.
• A negative exponent moves the decimal to the left, indicating a division.
• The base number, between 1 and 10, is scaled by these powers.

Powers of ten simplify numerical calculations and expressions, especially for extremely large or small values, such as in scientific measurements.

Decimal Point Movement

Decimal point movement is a key process when converting between scientific and standard notations. The direction and distance the decimal moves are defined by the exponent in the scientific notation.

• If the exponent is positive, the decimal point shifts right by that number of places.
• If the exponent is negative, the decimal point shifts left.
• Zeroes may be added to fill empty places left by the decimal move.

For example, in the number \(5.28 \times 10^{3}\), a positive exponent '3' indicates moving the decimal three places to the right, turning it into 5280. This movement translates the abstract, shorthand of scientific notation into a regular number form we recognize.