Problem 69 - College Algebra

Question

Write each number in decimal notation. $$ 7.86 \times 10^{-4} $$

Step-by-Step Solution

Answer

The number \(7.86 \times 10^{-4}\) in decimal notation is 0.000786.

Step 1: Understand the Scientific Notation

The given form of the number is \(7.86 \times 10^{-4}\). In this case, -4 is the exponent to the base 10 and 7.86 is the mantissa.

Step 2: Move the Decimal Point

Since the exponent is negative, the decimal point in 7.86 should be moved 4 places to the left. After moving, zeros should be inserted as placeholders for the moved places.

Step 3: Write Number in Decimal Notation

After moving the decimal point 4 places to the left and inserting zeros, the number in decimal notation becomes 0.000786.

Key Concepts

Decimal Notation

To understand any mathematical concept, clarity on its basic terminology is important. Decimal notation, commonly referred to just as 'decimals', is a way of expressing numbers that include a fraction represented by digits following a decimal point. For instance, when we write 0.75, we're using decimal notation to show a number that is three quarters of a whole.

Most of us are familiar with this notation from everyday use, such as in prices or measurements. The beauty of decimal notation is in its simplicity — each digit to the right of the decimal point represents a fractional part of a power of 10. The first digit after the decimal point is in the tenths place, the second in the hundredths, and so on. Therefore, 0.75 is the same as 75/100 or seventy-five hundredths.

Exponent

An exponent, in its basic form, is a shorthand notation to express repeated multiplication of the same number. It's written as a small number positioned slightly above and to the right of the base number. For example, in the expression \(2^3\), 2 is the base, and 3 is the exponent, indicating that 2 should be multiplied by itself three times: \(2 \times 2 \times 2 = 8\).

The concept of an exponent is not limited to whole numbers or positive values. When the exponent is negative, such as in the exercise \(7.86 \times 10^{-4}\), it implies division, more precisely, the inverse of the base raised to the absolute of that exponent. In this case, it indicates dividing 1 by 10, four times, which is the same as moving the decimal point four places to the left. This understanding is crucial when converting from scientific notation to decimal notation.

Mantissa

Now, let's dive into the term 'mantissa'. When numbers are written in scientific notation, the mantissa is the part that contains the significant digits of the number, without the exponent part. It's usually a value between 1 and 10. In the given example \(7.86 \times 10^{-4}\), 7.86 is the mantissa.

Understanding the role of the mantissa is essential, as it carries the significant figures of the number. It's the 'meat' of the number, so to speak, while the exponent indicates the size or scale. When converting from scientific to decimal notation, the mantissa must be adjusted by the value indicated by the exponent, as demonstrated in the textbook exercise. This adjustment changes its position but not its significant digits, which are fundamental to the accuracy and precision of the number.