Problem 39 - Algebra 1: Concepts and Skills

Question

Compare using \(<,>\) or \(=\) $$ 15.7 ? 15.700 $$

Step-by-Step Solution

Answer

The comparison operator between \(15.7\) and \(15.700\) is \(=\)

Step 1: Understand Decimal Numbers

Decimal numbers consist of the whole number part, a decimal point, and the fractional part. The decimal places count from the decimal point towards the right: the first number after the decimal point counts the tenths, the second counts the hundredths, and so on. So, \$15.700 means 15 plus 7 hundredths plus zero thousandths plus zero ten thousandths.

Step 2: Comparing Decimal Numbers

Comparing the given numbers, \(15.7\) and \(15.700\), from left to right, all the digits and the number of decimal places are the same except that \(15.700\) has two more zeros at the end. However, adding zeros at the end of a decimal number doesn't change its value. Thus, they are equal.

Step 3: Write The Answer

So, for the comparison \(15.7 ? 15.700\), the correct symbol to use is \(=\). The full expression would be \(15.7 = 15.700\).

Key Concepts

Decimal Places

When we talk about decimal places, we're focusing on the digits that appear to the right of the decimal point. Decimal places are essential because they tell us how precise a number is. For example, in the number 15.700, each digit plays a role in increasing the precision:

• The first digit after the decimal is the tenths place (7 in 15.7 and 15.700).
• The second digit (a zero in this example) represents the hundredths place.
• The third digit is the thousandths place, which also happens to be a zero here.

Understanding the position of each digit after the decimal point helps in comparing and understanding the value represented by decimals. Even if more zeros are added after the decimal places, they do not change the value of the number. So, 15.7 and 15.700 are equivalent in value despite being different in form.

Equality of Decimals

Equality of decimals means that two decimal numbers represent the same value, even if they have different numbers of digits after the decimal point. It's crucial to recognize that adding trailing zeros does not affect the value of the decimal.
The decimals 15.7 and 15.700 might look different at first glance due to the differing numbers of digits, but they are indeed equal. This is because zeros that appear at the end of a decimal (after some non-zero digits) are simply there to indicate the precision, not to change the number's value.
To identify equality of decimals, you can:

• Compare the main digits without considering trailing zeros.
• Recognize that additional zeros do not affect the numerical value of a decimal.

Recognizing decimal equality is fundamental, especially in scientific and mathematical calculations, where precision and notation matter.

Mathematical Symbols

In mathematics, symbols are used to express various relationships between numbers concisely. When comparing numbers like decimals, the symbols we commonly use are less than (\(<\)), greater than (\(>\)), and equal to (\(=\)).
The process of comparing decimals involves looking at the values without getting distracted by the number of decimal places. For example:

• If 15.7 and 15.700 are compared, the appropriate symbol to denote their relationship is "equal to," or\(=\).
• If the numbers were something like 15.7 and 15.8, then the symbol \(<\) would show that 15.7 is less than 15.8.

Using these symbols helps accurately describe the relationships between values, ensuring clear and precise mathematical communication.