Integers and Subtraction Quiz

Integers and Subtraction: A Crash Course in Whole Number Arithmetic

Integers, simply put, are the whole numbers that include positive and negative values. They play a critical role in mathematics as they form the foundation of our number system. In this article, we'll explore integers and dive into the subject of subtracting integers, one of the core operations in this realm of mathematics.

Integers: Positive, Negative, and Zero

An integer is a number that can be represented by the set of whole numbers (…, -3, -2, -1, 0, 1, 2, 3, …). The negative numbers are denoted by a negative sign before the integer, e.g., -2, -10, -75. Zero is a special type of integer that is neither negative nor positive.

Subtraction of Integers

Subtracting integers is a process that relies on a few rules and concepts.

Sign Rules

1. When subtracting integers with the same sign, follow the usual rules of arithmetic: subtract the smaller absolute value from the larger absolute value.

   • For example, -7 - (-3) = -7 + 3 = -4

2. For integers with opposite signs:

   • Subtract the absolute values and then keep the sign of the integer with the larger absolute value.
   • For example, -7 - 3 = -10 (since |-7| > |3|, the sign of -7 is retained)

Place Value and Borrowing

Subtracting integers can be thought of as borrowing from the next place value to the left when subtracting a larger number from a smaller one.

• For example, to calculate 12 - 18:
  • Subtract 10 from 12 to get 2.
  • Since we borrowed 10 from the tens place, add 1 to the ones place to compensate: 2 + 1 = 3.
  • The final result is 3 - 8 = -5.

Steps for Subtracting Integers

1. Check the signs:

   • If the signs are the same, follow the usual rules of subtraction.
   • If the signs are opposite, follow the sign rules mentioned earlier.

2. Perform the subtraction based on the sign rules and the usual rules of arithmetic, paying attention to the place value and borrowing rules when necessary.

Applications of Integers and Subtraction

Integers and subtraction of integers have numerous applications in everyday life and the sciences:

• In finance, subtracting integers can calculate profit, loss, or debt.
• In physics, subtracting integers can find the final position of an object after it is displaced in a given direction.
• In chemistry, subtracting integers can help determine the number of moles of a substance reacting in a chemical reaction.

Practice and Further Study

Practice makes perfect, so try out the following exercises to solidify your understanding of integers and subtraction.

1. Subtract the following integers and verify your answers:

   • -17 - 9
   • -5 - (-12)
   • 13 - 25

2. Solve problems involving integers in various contexts:

   • In a store, a customer purchases an item for $12 and receives a $4 discount. What is the final cost?
   • In a chemical reaction, 8 moles of one substance react to produce 2 moles of another substance. How many moles of the first substance remain unreacted?

Conclusion

Integers are an essential part of mathematics, and subtraction is a fundamental operation in this realm. By understanding the rules and concepts related to integers and subtraction, you'll find that the subject is not as intimidating as it might initially seem. As with any mathematical topic, practice and persistence will help you gain a thorough and intuitive understanding of integers and subtraction.