Question

Express these numbers in standard notation. a) \(2.87 \times 10^{-8}\) b) \(1.78 \times 10^{11}\) c) \(1.381 \times 10^{-23}\)

Short answer

a) 0.0000000287 b) 178,000,000,000 c) 0.00000000000000000000001381

Step-by-step solution

Understanding Scientific Notation

Scientific notation expresses numbers as a product of two factors: a decimal part between 1 and 10, and a power of 10. For example, in the expression \(2.87 \times 10^{-8}\), 2.87 is the decimal part and \(10^{-8}\) is the power of 10. To convert this to standard notation, we need to move the decimal point of the decimal part in relation to the exponent of 10.

Convert \(2.87 \times 10^{-8}\) to Standard Notation

The exponent \(-8\) indicates we need to move the decimal point in 2.87 eight places to the left. Starting from 2.87, moving the decimal eight places yields 0.0000000287.

Convert \(1.78 \times 10^{11}\) to Standard Notation

The exponent \(11\) indicates we should move the decimal point in 1.78 eleven places to the right. Begin with 1.78 and add eleven zeros after it, resulting in 178,000,000,000.

Convert \(1.381 \times 10^{-23}\) to Standard Notation

The exponent \(-23\) requires moving the decimal point in 1.381 twenty-three places to the left. This will result in a very small number: 0.00000000000000000000001381.

Key concepts

Exponents

Exponents play a crucial role in mathematical expressions, particularly in scientific notation. An exponent tells you how many times a number, known as the base, is multiplied by itself. For example, in the expression \(10^3\), the exponent is 3, which tells us to multiply 10 by itself three times: \(10 \times 10 \times 10 = 1000\).

In scientific notation, exponents are primarily used with base 10. They simplify how we express very large or very small numbers. For instance:

• A positive exponent, like in \(10^4\), allows us to express larger numbers, as it shifts the decimal point to the right.
• A negative exponent, such as \(10^{-3}\), represents very small numbers by moving the decimal point to the left.

Understanding how to manipulate exponents helps you accurately convert between scientific and standard notation.

Decimal Point

The decimal point is a critical marker in any decimal number, separating the whole number part from the fractional part. In the context of scientific notation, the position of the decimal point is adjusted based on the exponent of 10. In order to convert from scientific notation to standard notation, you either move the decimal point to the right for positive exponents or to the left for negative exponents.

For example:

• In \(2.87 \times 10^{-8}\), the exponent \(-8\) means you move the decimal point 8 places left, resulting in 0.0000000287.
• With \(1.78 \times 10^{11}\), the exponent \(11\) means the decimal point moves 11 places right, yielding 178,000,000,000.

This movement is crucial to accurately express numbers in standard notation.

Standard Notation

Standard notation is the more common way to express numbers without exponents. It's used in everyday mathematics to write and interpret numbers. When expressing numbers in standard notation, there is no power of 10 visible in the number. Instead, all digits are shown directly.

For example:

• In scientific notation, \(1.381 \times 10^{-23}\) becomes a standard notation number: 0.00000000000000000000001381.
• Similarly, \(1.78 \times 10^{11}\) is expressed in standard notation as 178,000,000,000.

The transition from scientific notation to standard notation involves moving the decimal point, providing the exact decimal representation.

Powers of Ten

Powers of ten are a foundational concept, especially for scientific notation. They make dealing with large and small numbers manageable. The base here is always 10, and the exponent determines how many times 10 is multiplied by itself. This fundamental principle is used in scientific notation to indicate shifts in the decimal point's position, making it easier to read and write very large or very small numbers.

For instance:

• In \(10^{11}\), the exponent \(11\) indicates moving the decimal point 11 places to the right, leading to a significantly larger number.
• Contrarily, \(10^{-8}\) signifies moving the decimal point 8 places to the left, resulting in a much smaller number.

Ultimately, working with powers of ten helps simplify and streamline complex calculations in mathematics and science.