Question

Convert each number from scientific notation to decimal notation. $$3.667 \times 10^{-3}$$

Short answer

0.003667

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a shorthand way to write very large or very small numbers. A number in scientific notation has two parts: a decimal part called the coefficient (here, 3.667) and an exponent of 10 (here, -3). The exponent indicates how many places to move the decimal point.

Convert to Decimal Notation

Since the exponent is -3, move the decimal point in the coefficient 3 places to the left. If there are not enough digits, add zeros to the left of the number.

Writing the Decimal Result

After moving the decimal point three places to the left, the number in decimal notation is 0.003667.

Key concepts

Scientific Notation

Scientific notation is a succinct method to express numbers that are exceptionally large or minuscule. It's composed of a coefficient and a base of 10 raised to an exponent. The coefficient is normally a number between 1 and 10, while the exponent signifies the number of places the decimal point must be shifted. For instance, in the scientific notation \(3.667 \times 10^{-3}\), \(3.667\) is the coefficient and \(-3\) is the exponent. This system is extremely useful in fields like physics and engineering where extreme values are common.

Decimal Notation

Decimal notation is the standard form most people use for writing numbers. It's based on the value of each digit depending on its position, or place value. Converting from scientific to decimal notation involves adjusting the decimal point in a systematic way that reflects the exponent's instruction. When converting the example \(3.667 \times 10^{-3}\) to decimal form, the decimal point moves three places to the left, yielding \(0.003667\). This notation is more familiar in everyday life and foundational in teaching basic mathematics.

Exponents

Exponents are a fundamental mathematical concept that denote the repeated multiplication of a number by itself. The number of times the multiplication is performed is signified by the exponent. When the exponent is positive, it indicates the number of times the base (10 in the case of scientific notation) is multiplied. A negative exponent, such as \(-3\) in the example, implies that the base is divided rather than multiplied, or alternatively, that we're dealing with the reciprocal. So, \(10^{-3}\) equals \(1/10^3\) or \(1/1000\).

Mathematical Concepts

Mathematical concepts provide the building blocks for understanding and working with numbers in diverse ways. Scientific notation and decimal notation exemplify how the same number can be represented differently based on context or utility. Exponents are another layer, allowing concise expression of large numbers or miniscule fractions. It's essential to understand that these concepts aren't isolated but interlinked, forming a cohesive system that mathematicians, scientists, and engineers use to solve real-world problems in a precise and efficient manner.