Question

Write the following numbers in standard notation. Use a terminal decimal point when needed. (a) \(5.55 \times 10^{3} ;\) (b) \(1.0070 \times 10^{4}\) (c) \(8.85 \times 10^{-7}\) (d) \(3.004 \times 10^{-3}\).

Short answer

5550; 10070; 0.000000885; 0.003004.

Step-by-step solution

Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. When a number is written as a product of a number (between 1 and 10) and a power of 10, it is in scientific notation.

Convert 5.55 × 10^{3} to Standard Notation

To convert the number, move the decimal point 3 places to the right.\(5.55 \times 10^{3} = 5550\)

Convert 1.0070 × 10^{4} to Standard Notation

Move the decimal point 4 places to the right.\(1.0070 \times 10^{4} = 10070\)

Convert 8.85 × 10^{-7} to Standard Notation

Move the decimal point 7 places to the left.\(8.85 \times 10^{-7} = 0.000000885\)

Convert 3.004 × 10^{-3} to Standard Notation

Move the decimal point 3 places to the left.\(3.004 \times 10^{-3} = 0.003004\)

Key concepts

Standard Notation

Standard notation is the familiar way of writing numbers using the digits 0-9. It includes placing the decimal point in the appropriate position and understanding the value of each digit based on its position. For example, the number 345 in standard notation means 3 hundreds, 4 tens, and 5 ones. Standard notation is used to express numbers without the use of exponents or powers of 10.
To convert from scientific notation to standard notation, you just need to move the decimal point to the right for positive exponents or to the left for negative exponents.

• For positive exponents, like in the given example of 5.55 × 10^{3}, you move the decimal point 3 places to the right, resulting in 5550.
• For negative exponents, as shown in 3.004 × 10^{-3}, you move the decimal point 3 places to the left, giving you 0.003004.

Powers of 10

Powers of 10 are critical in understanding and using scientific notation. A 'power of 10' is simply 10 multiplied by itself a certain number of times. The expression for a power of 10 is written as 10 with a small number, called an exponent, to its right. For example, 10^3 means 10 * 10 * 10 = 1000.

• A positive exponent shows how many times to multiply 10 by itself.
• A negative exponent shows how many times to divide 1 by 10. For example, 10^{-3} is equivalent to 1 divided by 10 three times, resulting in 0.001.

Let's apply this in the exercise above. For instance, 8.85 × 10^{-7} involves moving the decimal point 7 places to the left, giving 0.000000885. Understanding powers of 10 helps to quickly determine the magnitude of numbers in scientific notation.

Decimal Point

The decimal point is vital in both standard and scientific notation. It separates the whole number part from the fractional part of a number. In scientific notation, the position of the decimal point is crucial because it determines the value of the number based on the exponent.

• When the exponent is positive, move the decimal point to the right.
• When the exponent is negative, move the decimal point to the left.

For example, in converting 1.0070 × 10^4 to standard notation, move the decimal point 4 places to the right, resulting in 10070. This correct placement indicates the actual value represented by the number. Similarly, in 8.85 × 10^{-7}, moving the decimal point 7 places to the left results in a very small number, 0.000000885. Understanding how to manipulate the decimal point is essential for converting between scientific and standard notation.