Integers and Signed-Number Operations
Integers extend whole-number arithmetic in both positive and negative directions and include zero. They model gains and losses, rises and falls, positive and negative offsets, and many other quantities measured relative to a reference point. The core challenge is not magnitude alone but the interaction between magnitude and sign.
Learning objectives
- Define integers, positive numbers, negative numbers and zero
- Compare signed numbers on a number line
- Add and subtract integers using consistent sign logic
- Multiply and divide integers using same-sign/different-sign rules
- Use parity of negative factors when multiplying several signed values
Source scope
Lesson 1, pp. 13-20
The article paraphrases and restructures the supplied source. Source-branded names, personal names, promotional material and original test questions are not reproduced.
Core concepts and decision rules
Integer set and order
Integers are the positive whole numbers, their negative opposites and zero. Moving right on a number line increases value; moving left decreases value. Therefore -3 is greater than -8 even though 8 has the larger absolute magnitude.
Adding like signs
When two addends have the same sign, add their absolute magnitudes and keep the common sign. This combines quantities acting in the same direction.
Adding unlike signs
When signs differ, subtract the smaller absolute magnitude from the larger and keep the sign of the quantity with the larger absolute magnitude. This is a net-effect calculation.
Subtraction as adding the opposite
Rewrite a - b as a + (-b). This single transformation reduces subtraction to the addition rules and prevents many sign mistakes.
Products and quotients
For multiplication or division, equal signs produce a positive result and different signs produce a negative result. With several factors, an even number of negative factors gives a positive product; an odd number gives a negative product.
Step-by-step method
Worked examples
Problem: Evaluate -14 + 9.
- The signs differ.
- Compare absolute values: 14 is larger than 9.
- Subtract magnitudes: 14 - 9 = 5.
- Keep the sign of -14.
Problem: Evaluate 6 - (-11).
- Rewrite subtraction as addition of the opposite.
- The opposite of -11 is +11.
- Compute 6 + 11.
Problem: Evaluate (-2)(5)(-3)(-4).
- There are three negative factors, an odd count, so the product is negative.
- Multiply magnitudes: 2 × 5 × 3 × 4 = 120.
How to reason through integers and signed-number operations
1. Identify the mathematical structure
Before calculating, classify what you are looking at. Decide whether the expression is a sum, product, quotient, power, equation, inequality, graph or system. Then identify the terms, signs, grouping symbols and variables that control the next legal move. This classification step prevents a common failure mode in algebra: applying a familiar rule to the wrong structure.
Use notation as information. A sign attached to a term belongs to that term; parentheses define a unit of work; an exponent applies to its stated base; and an equals or inequality symbol separates two related expressions. Read the structure before manipulating it.
2. Preserve equivalence or implication
Algebra is not a sequence of arbitrary rearrangements. Each line should follow from the previous line by a named rule. When simplifying an expression, preserve its value for every admissible input. When solving an equation, preserve the equality unless you knowingly use an operation such as squaring that can introduce extra candidates and therefore requires a final check.
A useful discipline is to ask: What operation did I apply, and to what complete object did I apply it? This question catches incomplete distribution, partial denominator clearing, lost signs and unbalanced equation operations.
3. Separate exact work from approximation
Keep fractions, powers and radicals exact while the algebra is still being transformed. Approximate decimals are best introduced only when a problem requires a numerical result to a stated precision. Exact intermediate forms are easier to verify and avoid cumulative rounding drift.
When an application does require rounding, retain enough guard digits during the calculation and round only the reported result. This is an illustrative good-calculation practice rather than a numerical requirement from the source.
4. Build an independent check
Use a check that is different from the step that produced the answer. Substitute a solved variable into the original equation, expand proposed factors, square a simplified radical, test a point on a graph, or evaluate both original and simplified expressions at a convenient value. Independent checks are more valuable than rereading the same arithmetic because they test the relationship from another direction.
If the check fails, work backwards through the written transformations until the first inconsistent line appears. Correct that line, not merely the final number.
Quick-reference table
| Rule or idea | How to use it |
|---|---|
| Integer set and order | Integers are the positive whole numbers, their negative opposites and zero. Moving right on a number line increases value; moving left decreases value. Therefore -3 is greater than -8 even though 8 has the larger absolute magnitude. |
| Adding like signs | When two addends have the same sign, add their absolute magnitudes and keep the common sign. This combines quantities acting in the same direction. |
| Adding unlike signs | When signs differ, subtract the smaller absolute magnitude from the larger and keep the sign of the quantity with the larger absolute magnitude. This is a net-effect calculation. |
| Subtraction as adding the opposite | Rewrite a - b as a + (-b). This single transformation reduces subtraction to the addition rules and prevents many sign mistakes. |
Common mistakes and controls
- Thinking that the number with the larger written digit is always larger; for example, -9 is less than -2
- Treating subtraction and a negative sign as the same symbol without considering context
- Using the multiplication sign rule for addition
- Forgetting that zero is neither positive nor negative, even though it belongs to the integers
- Counting negative factors incorrectly in a long product
Applications
Signed deviations
An illustrative process deviation of -0.4 mm can represent a value below a nominal reference, while +0.2 mm represents a value above it. The signs describe direction; the magnitudes describe size.
Classification: Illustrative application unless directly stated as a source concept.
Net change
If a level falls by 8 units, rises by 5, then falls by 1, model the changes as -8 + 5 - 1 to obtain a net change of -4.
Classification: Illustrative application unless directly stated as a source concept.
Practice and self-check
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Related KEVOS knowledge
Source fidelity note: Topic selection and instructional sequence are grounded in the supplied algebra source. Mathematical explanations have been paraphrased and reorganised into a web-handbook format. No external standards, company-specific requirements or numerical engineering limits are asserted.