Lesson 3.5: Large Numbers

New Concept

Writing and comparing huge numbers, like the national debt or astronomical distances, is challenging. Scientific notation is a standard method that uses powers of 10 to express these numbers compactly, making them easier to calculate and understand.

What’s next

First, we'll review multiplying by powers of ten. Then, you'll tackle interactive examples and practice cards to master converting large numbers into scientific notation.

Property

One million is a thousand thousands. One billion is 1000 millions, and one trillion is 1000 billions. Each number name comes in groups of three.

• Million: $1,000,000$
• Billion: $1,000,000,000$
• Trillion: $1,000,000,000,000$

Examples

• The number $7,450,000$ is read as 'seven million, four hundred fifty thousand'.
• Fifteen billion is written as the numeral $15,000,000,000$.
• A quadrillion, which is $1,000,000,000,000,000$, is 1000 times larger than a trillion.

Explanation

This naming system groups large numbers by thousands. A new name like 'billion' or 'trillion' means the number is 1000 times bigger than the previous major name ('million' or 'billion'), making huge values easier to read and comprehend.

Property

To multiply a number by a power of ten:

1. Move the decimal point to the right the same number of places as the exponent on ten.
2. Write in zeros to fill any empty places at the end of the new number.

Examples

• To compute $56 \times 10^4$, move the decimal point 4 places to the right from $56.0$, giving $560,000$.
• To compute $4.81 \times 10^6$, move the decimal point 6 places to the right, which results in $4,810,000$.
• To compute $0.092 \times 10^3$, move the decimal point 3 places to the right, giving $92$.

Explanation

This rule is a shortcut based on our base-10 system. Multiplying by $10^n$ means making the number $10^n$ times larger. This is visually represented by shifting the digits $n$ places to the left, which we accomplish by moving the decimal point right.

Property

To write a large number as a product involving a power of ten, we reverse the multiplication process. We can rewrite a number by moving the decimal point to the left and multiplying by a power of 10. The exponent on 10 will be the number of places the decimal point was moved.

Examples

• To write $4,700,000$ as $4.7$ times a power of ten, we move the decimal 6 places left, so $4,700,000 = 4.7 \times 10^6$.
• To write $93,500$ as $93.5$ times a power of ten, we move the decimal 3 places left, so $93,500 = 93.5 \times 10^3$.
• To write $810,000,000,000$ as $8.1$ times a power of ten, we move the decimal 11 places left, so $810,000,000,000 = 8.1 \times 10^{11}$.

Explanation

This technique lets us 'factor out' powers of ten from a large number, making it more compact. It is the opposite of multiplying by powers of ten and is a key step toward writing numbers in scientific notation.

Property

To write a number in scientific notation:

1. Move the decimal point so that there is only one (non-zero) digit to the left of the decimal.
2. Multiply by a power of ten, where the exponent on ten is the number of places you moved the decimal.

Regular numbers are said to be in standard notation.

Examples

• The number $5,900,000$ in scientific notation is $5.9 \times 10^6$, because we move the decimal 6 places.
• The number $451.7$ is written in scientific notation as $4.517 \times 10^2$, because we move the decimal 2 places.
• To write $2.08 \times 10^9$ in standard notation, we move the decimal 9 places right, resulting in $2,080,000,000$.

Explanation

Scientific notation provides a universal, compact way to handle extremely large numbers. By standardizing the format to one digit before the decimal, it makes comparing magnitudes and performing calculations much simpler.