Question

Write this number in scientific notation: one hundred fifty-three million.

Short answer

Question: Write the number one hundred fifty-three million in scientific notation. Answer: \(1.53 \times 10^8\)

Step-by-step solution

Convert the number into a value between 1 and 10

Divide the given number (153,000,000) by 100,000,000 to get a value between 1 and 10: \(153,000,000 \div 100,000,000 = 1.53\).

Count the number of decimal places the decimal has moved

When we divided the number by 100,000,000, the decimal point moved 8 places to the left.

Write the number in scientific notation

Multiply the value in Step 1 (1.53) by 10 raised to the power of the number of decimal places moved in Step 2 (8): \(1.53 \times 10^8\). The number is now in scientific notation: \(1.53 \times 10^8\).

Key concepts

Significant Figures

When working with scientific data, precision is key, and significant figures play a crucial role in maintaining this precision. Significant figures are the digits in a number that carry meaning contributing to its measurement resolution. This includes all the nonzero digits, zeroes between nonzero digits, and trailing zeroes in the decimal portion.

For example, in the number 153,000,000, which is written in scientific notation as \(1.53 \times 10^8\), the significant figures are the digits 1, 5, and 3. The zeroes are not considered significant because they serve merely as placeholders. This concept ensures that when numbers are converted into scientific notation, just like in our exercise example, the significant figures remain clear and indicate the true precision of the original number.

Exponential Notation

Exponential notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. In exponential notation, a number is written as a product of a number between 1 and 10 and a power of 10. This is synonymous with scientific notation and is useful for simplifying calculations and comparisons of very large or very small numbers.

In the exercise, the large number 153,000,000 is written in exponential notation as \(1.53 \times 10^8\). The '8' in \(10^8\) tells us by how many decimal places the original number can be multiplied to arrive back at the standard form. It simplifies to read and write the numbers and maintains the accuracy without compromising on the number's scale.

Powers of Ten

The powers of ten are the foundation of the decimal system and are fundamental in scientific notation. This system allows us to express very large or very small numbers compactly. The power of ten represents how many times to multiply the number 10 by itself. For example, \(10^2 = 10 \times 10 = 100\), whereas \(10^{-2} = 1/(10 \times 10) = 0.01\).

By using the powers of ten, numbers are scaled up or down, expressing how many places the decimal point moves. In the exercise solution, the decimal moves 8 places, which we note as the exponent in \(10^8\). It's crucial to understand this concept to perform multiplication or division involving numbers in scientific notation correctly.

Decimal Places

Decimal places refer to the position of digits to the right of the decimal point in a number. They are important in determining precision and accuracy during calculations. For instance, the number 0.012 has three decimal places. The concept becomes especially pertinent when rounding numbers to a certain number of significant figures or converting to scientific notation.

In the given exercise, the process of dividing by 100,000,000 effectively moved the decimal point 8 places to the left, condensing the number 153,000,000 to 1.53, a number with two decimal places. Understanding decimal places help us keep track of the value's precision throughout this conversion to scientific notation.