"In mathematics, a negative number represents an opposite."
Numbers that are less than zero in value.
Introduction to Negative Numbers: Introduction to the concept of negative numbers and how they are represented in number systems.
Absolute Value: The distance of a number from zero, regardless of its sign.
Number Lines: A visual representation of numbers that helps understand the relationship between positive and negative numbers.
Addition and Subtraction of Negative Numbers: The rules for adding and subtracting negative numbers.
Multiplication and Division of Negative Numbers: The rules for multiplying and dividing negative numbers.
Order of Operations: The rules for performing mathematical operations in a specific order to avoid confusion and errors.
Integers: A class of numbers that includes both positive and negative whole numbers and zero.
Rational Numbers: Numbers that can be expressed as a ratio of two integers.
Fractions with Negative Denominators: The rules for working with fractions with negative denominators.
Coordinate Plane: A visual tool used to represent points and graph functions on a Cartesian system.
Applications of Negative Numbers: Real-world scenarios where negative numbers are used, such as temperature, money, and elevation.
Comparing and Ordering Negative Numbers: The rules for comparing and ordering negative numbers based on their size.
Exponents and Negative Numbers: The rules for working with negative exponents.
Solving Equations with Negative Numbers: The steps for solving equations that involve negative numbers.
Absolute Value Equations: Equations that involve absolute value and how to solve them when they include negative numbers.
Complex Numbers: Numbers that involve both real and imaginary parts, and how to represent them in a number line.
Signed Magnitude: Representing negative numbers using a negative sign and their positive equivalent. For example, -5 can be represented as -5 or -00101 in binary (5 in binary is 00101).
One's Complement: Using the complement of a number to represent negative numbers. For example, the one's complement of 5 in binary is 11110, and so -5 is represented as 11110 in one's complement.
Two's Complement: Using the two's complement of a number to represent negative numbers. For example, the two's complement of 5 in binary is 1011, and so -5 is represented as 1011 in two's complement. This is the most commonly used system in computing as it simplifies arithmetic operations.
Excess-K: This system uses a fixed value (K) as an offset to represent negative numbers. For example, in excess-3, the number 0 is represented as 3, the number 1 is represented as 4, and so on. Therefore, -1 is represented as 2 in excess-3.
Biased (or Offset Binary) Notation: Similar to excess-K, but the offset value can be positive or negative. For example, in biased 7 notation, 0 is represented as 7, 1 is represented as 8, and so on. Therefore, -1 is represented as 6 in biased 7 notation.
"In the real number system, a negative number is a number that is less than zero."
"Negative numbers are often used to represent the magnitude of a loss or deficiency."
"A debt that is owed may be thought of as a negative asset."
"Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature."
"The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic."
"Negative numbers are usually written with a minus sign in front."
"The positivity of a number may be emphasized by placing a plus sign before it, e.g., +3."
"Every real number other than zero is either positive or negative."
"The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...)."
"Amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers."
"It has been proposed that negative numbers were used on the Greek counting table at Salamis, known as the Salamis Tablet, dated to 300 BC."
"Negative numbers were also used in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty (202 BC – AD 220)."
"Liu Hui (c. 3rd century) established rules for adding and subtracting negative numbers."
"Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients."
"Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems 'false'."
"Western mathematicians like Leibniz (1646–1716) held that negative numbers were invalid, but still used them in calculations."