Question

Convert each number from enginecring notation to decimal notation. $$385 \times 10^{3}$$

Short answer

385 * 10^{3} in decimal notation is 385000.

Step-by-step solution

Identify the Parts of Engineering Notation

Recognize that the number is presented in engineering notation, which means it is expressed as a coefficient multiplied by 10 raised to an exponent. In this case, the coefficient is 385 and the exponent is 3.

Convert to Decimal Notation

To convert the number to decimal notation, simply multiply the coefficient by 10 raised to the power of the exponent. In decimal notation, this means shifting the decimal point of the coefficient to the right as many times as the value of the exponent.

Complete the Conversion

After shifting the decimal point three places to the right, the number becomes 385000. The final decimal notation is therefore 385000.

Key concepts

Mathematical Notation

Mathematical notation is a system of symbols and numbers used to represent quantities, operations, relations, and other concepts within mathematics. It provides a standard way for mathematicians and scientists to express their ideas clearly and concisely. Different branches of mathematics and engineering use various notations to represent large or small numbers efficiently, one of those being 'engineering notation'.

In engineering notation, quantities are often expressed in a format that is akin to scientific notation. This format includes a coefficient (a number typically between 1 and 1000) multiplied by a power of 10, but the exponents are restricted to multiples of three, aligning with the prefixes used in the metric system such as kilo (k), mega (M), and micro (µ). For example, instead of writing 15000, an engineer might write it as '15 x 10^3' to highlight that this represents 15 kilo-units, making it easier to grasp the scale of the number.

Exponents and Powers

Exponents and powers are core tools in mathematical notation that allow us to concisely represent repeated multiplication. When we write an exponent, we use a base number followed by a superscript that indicates how many times to multiply the base by itself. The expression 'base^exponent' directs you to multiply the base number by itself as many times as indicated by the exponent. For instance, in the expression '10^3', 10 is the base, and it should be multiplied by itself a total of 3 times: 10 x 10 x 10, which equals 1000.

When converting from engineering notation to decimal notation, understanding exponents is crucial. The exponent tells us how many positions to move the decimal point in the coefficient. A positive exponent means we move the decimal point to the right, making the number larger, whereas a negative exponent means moving the decimal point to the left, making the number smaller. The act of adjusting the decimal point can transform numbers into a format that is often more readable and relatable to the situation at hand.

Decimal Notation

Decimal notation is the standard form of representing numbers that most people are familiar with. It is used to represent both whole numbers and fractions and is based on the base-10 number system. Each position to the left or right of the decimal point signifies a power of 10, with the immediate right-hand side representing tenths, hundredths, thousandths, and so on.

Converting engineering notation to decimal notation involves expanding the number so that it's written without the use of exponents, allowing us to see and interpret the actual magnitude directly. This is particularly useful when precision matters, as well as in everyday banking, commerce, and measuring activities. The process usually requires shifting the decimal point of the coefficient to the right for positive exponents (as seen in the provided example), or to the left for negative exponents, based on the exponent's value, which will result in an expanded form of the number where the power of 10 is accounted for in this new position of the decimal point.