Problem 62 - Prealgebra

Question

Write each number in standard form. $$2 \times 10^{3}$$

Step-by-Step Solution

Answer

The standard form is 2000.

Step 1: Understand the Notation

The expression given is written in scientific notation with a coefficient (2) and a power of ten (10^3). Scientific notation is used to express very large or very small numbers in a compact form.

Step 2: Apply the Power of Ten

The power of ten, which is 10 raised to the power of 3 (10^3), means the number 10 is multiplied by itself three times: \[10^3 = 10 \times 10 \times 10 = 1000\]

Step 3: Multiply the Coefficient

Multiply the coefficient by the result of the power of ten. Thus, 2 multiplied by 1000 gives:\[2 \times 1000 = 2000\]

Step 4: Write in Standard Form

Combine the results to write the number in standard form, which is 2000.

Key Concepts

standard form

The standard form is a way of writing numbers that is commonly used in mathematics to make complex and detailed calculations simpler and more manageable. In essence, standard form is the method of expressing a number as a product of its coefficient and powers of ten. This form, sometimes referred to as scientific notation, is particularly useful for handling extremely large or small numbers.

When converting to standard form, follow these steps:

• Identify the coefficient. This is usually a number between 1 and 10.
• Use a power of ten to scale the coefficient to its actual size.
• Express the final result as a product of the coefficient and that power of ten.

This process makes reading and computing with large numbers, such as distances in astronomy or particle sizes in quantum physics, a much more straightforward task. In our example, the number 2000 can be expressed as \( 2 \times 10^3 \). This indicates the coefficient 2 and a scale factor represented by the power of ten.

powers of ten

Powers of ten are the backbone of the scientific notation and standard form. They are based on the number 10 raised to an integer power. This concept allows us to represent large quantities in a compressed format.

For example:

• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^3 = 1000\)

The exponent indicates how many times 10 is multiplied by itself. For instance, \(10^3\) means multiplying 10 three times: \(10 \times 10 \times 10\).

Using powers of ten is immensely useful in mathematics as it simplifies calculations by allowing complex numbers to be depicted as easily manipulatable and interpretable terms. This makes processes like multiplication and division much simpler, as they can often be reduced to simple arithmetic operations on the exponents themselves.

mathematical notation

Mathematical notation is the system of symbols and signs used to represent numbers and operations in mathematics. It functions as the universal "language" of math, allowing for clear communication of numerical information and relationships.

Scientific notation, of which the standard form is a part, is a prime example. It uses specific symbols such as numbers, coefficients, powers, and operations like multiplication to convey complex numerical ideas succinctly.

Consider another example, \(5 \times 10^4\). Here:

• "5" is the coefficient.
• "\(\times 10^4\)" indicates scaling by a factor of 10, raised to the fourth power, signifying resizing of 5.

Through mathematical notation, what could be cumbersome to read and write (50,000 in this instance) becomes concise and easy to understand. It also extends beyond just numbers, encompassing a wide range of operations and concepts such as summation \(\Sigma\), derivatives \(\frac{d}{dx}\), and integrals \(\int\). Mathematical notation helps ensure that calculations, theories, and proofs are accurately and efficiently communicated.