Question

Express each of the following ordinary numbers as a power of 10: (a) 1,000,000,000 (b) 0.00000001

Short answer

(a) 10^9, (b) 10^{-8}

Step-by-step solution

Analyzing the Structure of 1,000,000,000

The number 1,000,000,000 consists of 1 followed by 9 zeroes. Each zero represents a position in the decimal system. Therefore, the number can be considered as 10 raised to the power of 9.

Expressing 1,000,000,000 as a Power of 10

Since 1,000,000,000 equals 10 raised to the power of 9, we can write this as:$${10}^9$$

Analyzing the Structure of 0.00000001

The number 0.00000001 represents a small decimal number which is 1 divided by 100,000,000. This can also be written as 10 raised to the negative 8 because moving the decimal point 8 places to the right converts it into the number 1.

Expressing 0.00000001 as a Power of 10

Since 0.00000001 is equivalent to 10 raised to the power of -8, we write:$${10}^{-8}$$

Key concepts

Scientific notation

Scientific notation is a way of expressing very large or very small numbers in a concise format. It's like shorthand for numbers. This system leverages "powers of ten" to simplify numbers that would otherwise be cumbersome to write and read.

In scientific notation, a number is transformed into a product of two factors: a number usually between 1 and 10, and an exponential part that tells you how many places to move the decimal point. For example, the number "1,000,000,000" can be written as $1\times {10}^9$. Similarly, "0.00000001" can be expressed as $1\times {10}^{-8}$.

This method is particularly useful in fields like science and engineering, where we often deal with extremely large or small numbers. It aids in making calculations much simpler, by reducing them down to operations on the powers of ten.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It’s written in the form $b^n$. Here, $b$ is the base and $n$ is the exponent or power.

When you see ${10}^9$, for instance, it means multiply the base, which is 10, by itself 9 times. So, ${10}^9$ equals 10 multiplied by itself 9 times, resulting in 1,000,000,000.

On the other side, ${10}^{-8}$ indicates 10 raised to the power of -8. Negative exponents mean division instead of multiplication. So, ${10}^{-8}$ translates to 1 divided by ${10}^8$, i.e., 1/100,000,000.

Exponents are a powerful tool because they allow us to express large-scale multiplications or divisions in a simple and easy-to-understand way, drastically reducing the size and complexity of the numbers.

Decimal system

The decimal system, also called the base-10 system, is the standard system for denoting integer and non-integer numbers. It is based on 10 distinct digits from 0 to 9.

In this system, each position of a number represents a power of 10. For instance, in the number 1,000,000,000 the rightmost zero represents ${10}^0$, the next zero represents ${10}^1$, and so on up to ${10}^9$.

• Every step to the left increases the power of 10 by one.
• Inversely, every step to the right for decimals decreases the power of 10 by one.

For the decimal 0.00000001, the first digit "1" is at the ${10}^{-8}$ place, which essentially represents 1 divided by 100,000,000.

This framework of the decimal system simplifies our understanding and handling of both whole numbers and fractions, translating them into a standard, easily comparable form.