Basic Math · 6 min read
Basic Math · Numbers
Basic Math is the number sense behind every GED Mathematical Reasoning question. It covers classifying and ordering numbers; computing with integers, fractions, decimals, powers, and roots; factors and multiples; and the skills the test sets most often in real-world contexts: fraction–decimal–percent conversions, discounts, tax, tips, commission, simple interest, unit rates, scale, and proportions. The first few questions of the test must be answered without a calculator, so accurate mental and paper computation matters as much as calculator skill.
Numbers are sorted into sets, and each set builds on the one before it.
| Set | Defi nition | Examples |
|---|---|---|
| Naturalnumbers | Thecountingnumbers startingat1;nozero fractions or , , , | 1 2 3 4 5 , , , , , … |
| negatives | ||
| Wholenumbers | Thenaturalnumberspluszero;nevernegative neverfractional , | 0 1 2 3 , , , , … |
| Integers | Wholenumbersandtheiropposites;nofractionsordecimals | −3 −2 −1 0 1 2 3 … , , , , , , … |
| Rationalnumbers | Anynumberthatcanbewrittenasa/bwithintegersaandb b , | 3/4 −2/5 0 .75 0 .333 , , , … |
| ≠0;includeseveryinteger(5=5/1) everyfraction andevery , , | ||
| terminatingorrepeatingdecimal | ||
| Irrationalnumbers | Cannotbewrittenasafraction;thedecimalneverterminates | π≈314159 √2≈141421 . …, . … |
| andneverrepeats | ||
| Realnumbers | Allrationalandirrationalnumbers;everypointonthenumber | −4 0 2/3 √5 π , , , , |
| line |
Prime and Composite Numbers
A prime number is a natural number greater than 1 with exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … The number 2 is the only even prime. A composite number is a natural number greater than 1 with more than two factors; 12 is composite because its factors are 1, 2, 3, 4, 6, and 12. Every composite number can be written as a product of primes. The numbers 0 and 1 are neither prime nor composite, and negative numbers and fractions are never prime.
Which of −7, 0, 2/3, √9, √10, 0.444… are integers? rational?
√9 = 3 a perfect square root is an integer
Integers: −7, 0, √9
Rational: −7, 0, 2/3, √9, 0.444… (0.444… = 4/9)
Irrational: √10 10 is not a perfect square
Simplify first, then classify. A square root is rational only when the number under the radical is a perfect square.
Place Value
In the base-10 system each position is worth 10 times the position to its right. Moving left from the decimal point the places are ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions; moving right they are tenths, hundredths, thousandths, ten-thousandths, each one-tenth of the place to its left.
Placevaluesin4,527.386
| Digit | 4 | 5 | 2 | 7 | . | 3 | 8 | 6 |
|---|---|---|---|---|---|---|---|---|
| Place | thousands | hundreds | tens | ones | tenths | hundredths | thousandths | |
| Value | 4000 , | 500 | 20 | 7 | 03 . | 008 . | 0006 . |
Rounding
To round, find the digit in the target place and look at the digit immediately to its right. If that digit is 5 or greater, increase the target digit by 1; if it is less than 5, leave the target digit unchanged. Replace the digits to the right with zeros (whole- number places) or drop them (decimal places). To round 3,847 to the nearest hundred, look at the 4 after the 8: 4 < 5, so the result is 3,800. To round 6.7851 to the nearest hundredth, look at the 5 after the 8: round up to 6.79.
Gas costs $3.459 per gallon. What does 12.6 gallons cost?
3.459 × 12.6 = 43.5834 multiply first
Hundredths digit is 8; next digit is 3 (less than 5)
Cost = $43.58 round only the final answer
Money answers are rounded to the nearest cent. Rounding in the middle of a problem can shift the final answer by a cent or more.
Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction, so it is never negative. It is written with vertical bars: |7| = 7, |−7| = 7, |0| = 0, and |−15| = |15| = 15. Treat the bars as grouping symbols and simplify inside them first: |3 − 10| = |−7| = 7.
Prime numbers have exactly two factors; 2 is the only even
To classify a number, check its sign, whether it is whole or
prime.
fractional, and whether its decimal terminates, repeats, or does neither.
0 and 1 are neither prime nor composite; negatives and fractions are never prime.
Round only the final answer, and round money to the nearest cent.
The Number Line
The number line shows every real number in order from least to greatest. Zero sits at the center, positive numbers increase to the right, and negative numbers decrease (become more negative) to the left. Consecutive integers are always the same distance apart, and the line extends without end in both directions.
Comparing and Ordering
Of any two numbers, the one farther to the right is greater. This holds for all real numbers: −3 > −7 because −3 lies to the right of −7, and 0.5 > 0.25 because 0.5 lies to the right of 0.25. Among negative numbers, the one closer to zero is greater, so from least to greatest −8, −3, −5, −1 are ordered −8, −5, −3, −1. The number −8 is farthest from zero in the negative direction, which makes it the smallest.
To order fractions and decimals, convert them to one form, either decimals or fractions with a common denominator. For 3/4, 2/3, and 5/6: 3/4 = 0.75, 2/3 ≈ 0.667, and 5/6 ≈ 0.833, so the order from least to greatest is 2/3, 3/4, 5/6.
least greatest −4 −2½ −¾ ½ −5 −4 −3 −2 −1 distance = |2 − (−4)| = 6 units
Values increase from left to right, so −4 < −2½ < −¾ < ½ < 2. The distance between two points is the absolute value of their difference.
Distance Between Two Points
The distance between two numbers on the number line equals the absolute value of their difference, |a − b|. The order of subtraction does not matter because absolute value removes the sign. The distance between 4 and 9 is |9 − 4| = 5; between −6 and 2 it is |2 − (−6)| = |2 + 6| = 8; and between −3 and −8 it is |−3 − (−8)| = |−3 + 8| = 5.
At 6 a.m. the temperature is −8°F. By noon it is 15°F.
How many degrees did it rise?
|15 − (−8)| = |15 + 8| subtracting a negative = adding
= 23°F
A change that crosses zero is the sum of the distances on each side of zero: 8 degrees up to 0, then 15 more.
Order from least to greatest: −0.6, −2/3, 1/8, −1/2
−2/3 ≈ −0.667 −1/2 = −0.5 1/8 = 0.125 convert to decimals
−0.667 < −0.6 < −0.5 < 0.125 farther left is smaller
Order: −2/3, −0.6, −1/2, 1/8
With negatives, the number with the larger absolute value is the smaller number: |−2/3| > |−0.6|, so −2/3 < −0.6.
Right is greater, left is smaller, for every real number.
Convert fractions and decimals to one form before ordering them.
For negatives, the smaller absolute value is the greater number: −3 > −7.
Distance between a and b is |a − b|; distance is never negative.
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