Properties of Real Numbers
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Arithmetic and Fractions
Summary
ACT math uses the word number to mean any real number, so you must consider positives, negatives, zero, fractions, decimals, and irrational values unless the question gives a narrower set. Success depends on distinguishing number sets correctly, knowing basic arithmetic vocabulary, and recognizing the key properties of real numbers that govern operations.
- A real number is any point on the number line, including whole numbers, fractions, decimals, negatives, zero, and values like pi or the square root of seven, so the word number never means only counting numbers.
- Zero is the only real number that is neither positive nor negative, and ACT questions often hinge on keeping positive numbers, negative numbers, and zero conceptually separate.
- You must distinguish among numbers, integers, and positive integers: integers include positive and negative whole numbers, while positive integers are the counting numbers.
- Basic operation vocabulary matters: a sum comes from addition, a difference from subtraction, a product from multiplication, and a quotient from division.
- Addition and multiplication allow order changes and regrouping, while subtraction and division generally do not, though subtraction can sometimes be rewritten as adding a negative.
- Multiplication distributes over both addition and subtraction, making the distributive property especially important for simplifying expressions and solving problems.
- Special rules for 1 and 0 are essential: multiplying or dividing by 1 leaves a number unchanged, multiplying by 0 gives 0, division by 0 is not allowed, a product of 0 means at least one factor is 0, and any nonzero number divided by itself equals 1.
Chapters
00:00
Real Numbers and Common ACT Traps
01:20
Number Sets: Real Numbers, Integers, and Positive Integers
01:55
Essential Arithmetic Vocabulary
02:24
Core Operation Properties
05:22
Special Rules Involving 1 and 0
Q: In the video, the speaker mentions summaries of the video lectures. Where do I find the summaries?
A: The summaries are at the end of the videos. So, they aren't kept somewhere in Magoosh; they are within the videos, at the end.
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