Adding and Subtracting: Negative Numbers
Adding and subtracting negative numbers are fundamental mathematical operations that apply to all real numbers, serving important roles in various real-world contexts such as finance, temperature changes, and elevation differences. The concept of negative numbers has a rich historical background, originating as early as the third century BCE for recording debts, and evolving through contributions from mathematicians like Brahmagupta and Samau'al al-Maghribi. Negative numbers were initially viewed with skepticism, but by the nineteenth century, their operations became well-established within mathematical frameworks.
In practical terms, adding negative numbers can signify increases in quantities, such as calculating a temperature rise by combining a negative starting point with a positive increment. Conversely, subtraction of negative numbers can represent decreases, as seen in scenarios like determining elevation changes. Visual aids, like counting chips, are often employed to illustrate these operations, making the abstract concepts more tangible by representing positive and negative values distinctly. Overall, understanding how to add and subtract negative numbers is essential for tackling various real-life situations and mathematical problems.
Adding and Subtracting: Negative Numbers
Addition and subtraction are two closely related operations that can be performed on all real numbers, including negative numbers. Addition and subtraction of negative numbers is particularly useful when modeling decreases and increases among quantities that represent a deficit. For example, changes in monetary debts, temperatures, and elevations are all represented using the addition and subtraction operations of negative numbers.
Overview
Negative numbers originated around the third century b.c.e. when debts and deficits were denoted using a specialized Chinese rod notation. In the seventh century, an Indian mathematician named Brahmagupta introduced the concept of zero from which rules for negative number operations followed. The algebra of real number operations, including addition and subtraction of negative quantities, was formally introduced by Samau’al al-Maghribi in his twelfth-century volume titled The Brilliant in Algebra. The existence of negative numbers and their operations were contentious among mathematicians for many centuries and were used solely as a computational tool. By the nineteenth century, an axiomatic logic of negative number operations was widely accepted in light of the work of the British mathematician Augustus DeMorgan.
Adding and Subtracting Negative Numbers
Increases in quantities can be calculated by adding negative numbers. For example, a 10 degree rise in temperature from an initial temperature of –18 degrees can be modeled by adding 10 to −18 using the expression –18 + 10 = –8. Likewise, debts of $1000 and $2000 on two respective credit cards can be consolidated by adding –1000 and –2000, represented by the expression –1000 + ( –2000) = –3000. Addition of negative numbers is often modeled with counting chips where black chips represent positive quantities, red chips represent negative quantities, and a pair of one red and one black chip represent additive inverses whose sum is 0.
Subtraction of negative numbers can be used to determine decreases in quantities. For instance, the change in temperature from –18 degrees to –24 degrees can be calculated by subtracting –18 from –24, or –24 – (–18) = –6. The change in elevation of a hiker who begins 200 meters above sea level and ends 50 meters below sea level can be found by subtracting –50 from 200, or 200 – (–50) = 250. These results are represented on the number line by the distance between the two quantities.
Bibliography
Flores, Alfinio. "Subtraction of Positive and Negative Numbers: The Difference and Completion Approaches with Chips." Mathematics Teaching in the Middle School 14 (2008): 21-23.
Hodgkin, Luke. A History of Mathematics : From Mesopotamia to Modernity. Oxford UP, 2005.
Martinez, Alberto A. Negative Math: How Mathematical Rules Can Be Positively Bent. Princeton UP, 2006.
Ponce, G. A. "It's All in the Cards: Adding and Subtracting Integers." Mathematics Teaching in the Middle School 13 (2007): 10-17.