Question

Express the following ordinary numbers in scientific notation: (a) 1,010,000,000,000,000 (b) 0.000000000000456 (c) 94,500,000,000,000,000 (d) 0.00000000000000001950

Short answer

(a) \( 1.01 \times 10^{15} \), (b) \( 4.56 \times 10^{-13} \), (c) \( 9.45 \times 10^{16} \), (d) \( 1.95 \times 10^{-17} \).

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way to express very large or very small numbers in the form of \( a \times 10^n \), where \( 1 \leq |a| < 10 \) and \( n \) is an integer. This makes it easier to handle and compare numbers.

Express 1,010,000,000,000,000

For the number 1,010,000,000,000,000, move the decimal point to the left until one digit is to the left of the decimal. The number becomes 1.01 and you count the places moved, which is 15.Thus, in scientific notation, the number is \( 1.01 \times 10^{15} \).

Express 0.000000000000456

For the number 0.000000000000456, move the decimal point to the right until one digit is to the left of the decimal. The number becomes 4.56, and you count the places moved, which is 13.Thus, in scientific notation, the number is \( 4.56 \times 10^{-13} \).

Express 94,500,000,000,000,000

For the number 94,500,000,000,000,000, move the decimal point to the left until one digit is to the left of the decimal. This becomes 9.45, with a movement of 16 places.Thus, in scientific notation, the number is \( 9.45 \times 10^{16} \).

Express 0.00000000000000001950

For the number 0.00000000000000001950, move the decimal point to the right until one digit is in front of the decimal. This gives 1.95, with a movement of 17 places.Thus, in scientific notation, the number is \( 1.95 \times 10^{-17} \).

Key concepts

Exponents

Exponents are a crucial part of scientific notation, and they offer a way to express how many times you need to multiply a base number. In scientific notation, the base is always 10. Exponents reflect the number of times you move the decimal point:

• If you move the decimal to the left, the exponent is positive.
• If you move the decimal to the right, the exponent is negative.

This means for a large number, such as 1,010,000,000,000,000, represented in scientific notation as \( 1.01 \times 10^{15} \), you moved the decimal 15 places to the left. Conversely, for a small number like 0.000000000000456, which is \( 4.56 \times 10^{-13} \), you moved the decimal 13 places to the right.

Decimal Place Value

Understanding decimal place value is fundamental to expressing numbers in scientific notation. Every shift in the decimal point changes the value of the number:

• As you move the decimal to the right, you're dividing the number by ten with each shift, which is why small numbers have negative exponents.
• As you move the decimal to the left, you're multiplying the number by ten with each shift, resulting in positive exponents for large numbers.

In the number 94,500,000,000,000,000, when you adjust the decimal to 9.45, you've shifted the decimal place from the original position across 16 zeroes - translating into \( 9.45 \times 10^{16} \). Meanwhile, with 0.00000000000000001950, shifting the decimal to 1.95 involves moving it 17 places to the right, making it \( 1.95 \times 10^{-17} \).

Large and Small Numbers

Large and small numbers often contain many zeros, making them clumsy and hard to read. Scientific notation simplifies these numbers:

• Large numbers shrink into shorter, more manageable forms with positive exponents.
• Small numbers shed their zeroes using negative exponents.

For instance, 1,010,000,000,000,000 is far less wieldy than \( 1.01 \times 10^{15} \). Similarly, \( 4.56 \times 10^{-13} \) is much neater than 0.000000000000456.The beauty of scientific notation is that it allows quick and efficient calculations while maintaining the number's actual value. Especially in scientific work, handling extensive datasets or minute measurements becomes feasible.

Number Representation

Number representation using scientific notation distills complex numerals into an accessible format. This can be particularly useful when dealing with enormous figures in astronomy or tiny atoms in chemistry.The main parts of scientific notation are:

• a coefficient \( (a) \) - must be at least 1 but less than 10.
• a base - always 10 in scientific notation.
• an exponent \( (n) \) - shows the number of decimal place shifts.

For example, 0.00000000000000001950 becomes \( 1.95 \times 10^{-17} \), with `1.95` as the coefficient, `10` as the base, and `-17` as the exponent. This structure offers a universal method for representing numbers concisely and precisely, regardless of their size or scale.