^{*}

^{a}

^{b}

^{a}

Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers’ whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers.

Possessing “number sense” refers to the ability to use numbers flexibly and adaptively (e.g.,

There is broad empirical evidence that the ability to assess magnitudes of whole numbers and fractions is correlated with and even predictive of further mathematical achievement (^{th}- and 8^{th}-grade students from the U.S., China, and Belgium, even after controlling for several other related variables. In a recent meta-analysis,

To summarize, the ability to assess magnitudes of number symbols is fundamental for numerical development from both a theoretical and an empirical point of view. While previous empirical studies have focused exclusively on magnitudes of

Irrational numbers are real numbers that cannot be represented as the quotient of two integers. These numbers differ from rational numbers in several respects, two of which are relevant in the context of the present study. First, while for rational numbers the magnitude information can be decoded from their symbolic notation in a more or less straightforward way, this is not the case for irrational numbers. Assessing the magnitudes of integers requires an understanding of the magnitude information of numerals, an ability to take into account the base-ten system, and, if applicable, an understanding of the minus sign. Assessing the magnitudes of fractions requires an understanding of the magnitude information of the whole number components (the numerator and the denominator) and some reasoning about the relation between these magnitudes. In contrast, the algorithm to determine the magnitude of an irrational number (e.g., ^{ii} For example, the square root of 2 is approximately 1.4142135…. Because of these differences, and because people arguably rarely encounter irrational numbers in their daily lives, it should be much more difficult to assess the magnitudes of irrational numbers than those of integers or fractions. For that reason, we studied people with good mathematical skills, to find out whether it is

To investigate whether people are able to assess magnitudes of number symbols, many studies have used a number comparison task. In this task, participants have to decide which of two numbers is numerically larger. Next to participants’ accuracy when solving number comparison problems, the occurrence of a numerical distance effect across a set of problems is of particular interest. The distance effect means that people tend to be more accurate and faster in comparing the numerical values of two numbers when the numerical difference between the two numbers is large compared to when it is small.

In fraction comparison problems, the central question is whether accuracy and response times depend on the numerical distance between the fraction magnitudes, or between the fraction components (the numerators and the denominators). While the former is evidence that people assess the fraction magnitudes, the latter is evidence that people assess the magnitudes of the components. Recent research has converged in the finding that whether people assess the magnitudes of fractions in a comparison problem depends on the type of fraction pair. When the two fractions have common components (e.g., 3/7 versus 5/7, or 5/9 versus 5/6), people are more likely to rely on the non-equal fraction components rather than on the magnitudes of the whole fractions. When fractions do not have common components (e.g., 5/9 versus 6/11), people are more likely to assess fractions according to their magnitudes rather than their components alone (

There is only a limited amount of research on people’s understanding of irrational numbers (

Notwithstanding most people’s fundamental ability to assess fraction magnitudes in comparison problems, assessing magnitudes of fractions seems to be much more demanding and less automatic than assessing magnitudes of whole numbers. For example, comparing fractions that have common components (so that assessing the magnitudes of the components is sufficient) is much easier than comparing fraction pairs without common components (where assessing the magnitudes of the components is not sufficient) (

Research on fractions provides an important basis for studying the ability to assess magnitudes of irrational numbers that are represented as roots (e.g.,

We addressed the question of whether mathematically highly skilled adults can assess numerical magnitudes of irrational numbers, presented in their (exact) symbolic format^{iii}, to solve number comparison problems. We also aimed to characterize their reasoning process in such problems. We presented numbers in “root notation” (i.e.,

Common Components | No Common Components | |
---|---|---|

Congruent | ||

Incongruent | ||

Neutral | – |

One can solve comparison problems with common components easily by comparing the components while not assessing the number magnitudes. This is also true for the neutral problems with no common components, because the number with the larger radicand and the smaller index is always the larger number. In contrast, congruent and incongruent problems without common components require assessing the magnitudes of the irrational numbers, because componential comparison does not allow for a valid conclusion.^{iv}

Previous studies have shown that educated adults were fairly accurate about comparison problems with whole numbers and comparison problems with fractions. For example, the academic mathematicians in the study by

Furthermore, we expected to find a natural number bias, because participants would heavily rely on component strategies. That is, we expected to find significantly higher accuracy rates and lower response times for congruent as compared to incongruent problems with common components (Hypothesis 2a). We also expected to find higher accuracy rates and lower response times for congruent rather than incongruent problems without common components (Hypothesis 2b).

As in previous studies on fractions, we analyzed the distance effect as an indicator of comparison strategies. We distinguished between a distance effect for the magnitudes of the irrational numbers (hereafter: holistic distance effect) and a distance effect for the whole number components (i.e., the indices and the radicands; hereafter: component distance effect). We expected to find a holistic distance effect for problems of types No-CC-CO and No-CC-IC because component strategies are not successful in these cases (Hypothesis 3a). However, we did not expect to find a holistic distance effect for problems of the other types (Hypothesis 3b) because one can easily compare their magnitudes using component strategies. Theoretically, we would expect to find a component distance effect for problems of types CC-CO, CC-IC, and No-CC-N because we expected participants to rely on component comparison strategies rather than holistic strategies in these cases. However, since the distance effect for whole numbers is known to decrease with age and level of expertise (

The participants in this study were 45 mathematically skilled adults (mean age: 34.6 years; 17 female, 28 male). They were recruited at the Mathematical Department of a university in Germany. Twenty of the participants had a Bachelor’s degree in mathematics (15) or physics (5) and were graduate students majoring in mathematics or physics, respectively. Another 20 participants had a Master’s degree in mathematics and were research assistants at the Mathematical Department of the university. Another five participants were professors of mathematics. Although the participants in our study certainly do not engage in assessing magnitudes of irrational numbers on a regular basis, we were confident that they had a sound concept of irrational numbers and were perfectly able to understand the meaning of the symbolic notation of irrational numbers.

We constructed 70 number comparison problems with irrational numbers that were represented as the

The participants worked on the problems individually in a quiet room at the university. The problems were presented on a Laptop, using E-Prime software (

We used SPSS 23 to analyze the data. In line with data analysis procedures in previous studies on number comparison, we excluded incorrectly solved problems (13.8%, see below) and problems for which the response time deviated more than two standard deviations from the sample mean of the respective problem type (another 3.7% of all problems) for analyzing response times. To compare mean response times between problem types, we used paired samples

Type | Accuracy (%) |
Response Time (ms) |
|||
---|---|---|---|---|---|

CC | 97.22 | 3.87 | 3218 | 747 | |

CC-CO | 96.19 | 6.91 | 2986 | 788 | |

CC-IC | 98.25 | 3.78 | 3451 | 870 | |

No-CC | 78.94 | 8.07 | 5620 | 2085 | |

No-CC-CO^{a} |
51.59 | 28.73 | 7502 | 3749 | |

No-CC-IC | 89.84 | 9.81 | 5012 | 1758 | |

No-CC-N | 95.40 | 7.17 | 4568 | 1584 | |

All | 86.25 | 5.40 | 4659 | 1485 |

^{a}Due to our exclusion criteria (described in the first paragraph of the Results section), the sample size is reduced to 43 for No-CC-CO problems (i.e. for two persons all No-CC-CO were excluded).

Overall, accuracy rate was 86%, which is fairly high but somewhat lower than the academic mathematicians’ accuracy rate on fraction comparison problems (97%) reported by

Among those problems that had common components, there was no significant difference between congruent and incongruent problems in terms of the accuracy rates,

Among those problems that had no common components, the difference between the accuracies of congruent and incongruent problems was large and highly significant,

These results suggest that participants not only struggled more with congruent than with incongruent comparison problems, but were actually unable to find successful strategies to solve the congruent comparison problems while being well able to solve the incongruent comparison problems. The analyses of the distance effects will reveal whether the participants relied on assessing the magnitudes of the irrational numbers, or on component strategies alone (which, however, should have resulted in better performance on congruent than incongruent problems).

To analyze whether the participants engaged in processing the magnitudes of the irrational numbers or of their components, we ran multiple regression analyses, including the distance between the indices, the distance between the radicands, and the distance between the irrational numbers as predictors. For No-CC-CO and No-CC-IC problems, we ran these analyses with both accuracy and—in a separate analysis—response times as the dependent variables. However, for problems of types CC-CO, CC-IC, and No-CC-N, we ran only these analyses which had response times as the dependent variable. The reason was that for the latter problem types, accuracies were extremely high (> 94%), so that linear regression analyses would not yield reliable results.

For modeling accuracy data of No-CC-CO and No-CC-IC problems, we ran the regression analyses twice. First, we ran a linear regression analysis on the sample level, using the sample mean accuracy as the data point for each numerical distance value.

Type | Predictor | SE |
β | ^{2} |
||
---|---|---|---|---|---|---|

No-CC-CO | .67 | |||||

Dist_Indices | -0.05 | 0.02 | -.65 | .024 | ||

Dist_Radicands | 0.00 | 0.00 | .26 | .352 | ||

Dist_Numbers | 0.44 | 0.19 | .51 | .037 | ||

No-CC-IC | .54 | |||||

Dist_Indices | 0.01 | 0.01 | .24 | .309 | ||

Dist_Radicands | -0.00 | 0.00 | -.67 | .018 | ||

Dist_Numbers | -0.03 | 0.10 | -.08 | .737 |

In a second analysis, we ran a binary logistic regression, using 45 data points (one per participant) for each numerical distance value.

Type | Predictor | SE |
Nagelkerkes ^{2} |
||
---|---|---|---|---|---|

No-CC-CO | .10 | ||||

Dist_Indices | -0.22 | 0.06 | .000 | ||

Dist_Radicands | 0.01 | 0.01 | .121 | ||

Dist_Numbers | 1.91 | 0.53 | .000 | ||

No-CC-IC | .10 | ||||

Dist_Indices | 0.10 | 0.07 | .138 | ||

Dist_Radicands | -0.04 | 0.01 | .001 | ||

Dist_Numbers | -0.20 | 0.92 | .833 |

^{2} is an estimate of the variance explained by the logistic regression model. Dist_Indices = distance between indices, Dist_Radicands = distance between radicands, Dist_Numbers = distance between the numbers, No-CC = no common components, CO = congruent, IC = incongruent.

Although the first regression analysis explained much more variance in the accuracy data than the second one, both types of analyses yielded the same overall pattern of results: for the No-CC-CO problems, the distance between the irrational numbers as well as the distance between the indices were significant predictors of accuracies. This result suggests that for these problems, the participants at least tried to assess the irrational number magnitudes to some extent. However, considering their low accuracies, they were not very successful in doing so. On the contrary, they may have made systematic mistakes on the most difficult problems featuring small numerical distances (accuracies far below 50%). Interestingly, while the distance between the irrational numbers was positively related to accuracies, the distance between indices was negatively related to accuracies. We will discuss an interpretation of this finding in the discussion section below.

For the No-CC-IC problems, the distance between the radicands, but no other distance, was a significant predictor of accuracies. This result suggests that the participants relied on comparing the radicands rather than the magnitudes of the irrational numbers. However, we should interpret this result with caution because accuracies were relatively high for the No-CC-IC problems (90%), which, as mentioned earlier, limits the reliability of the linear regression analyses.

For the response times as the dependent variable, we ran the regression analyses separately for all types of comparison problems. Again, we ran each analysis twice. First, we used the sample mean response times for each numerical distance value.

Type | Predictor | SE |
β | ^{2} |
||
---|---|---|---|---|---|---|

CC-CO | .11 | |||||

Dist_Radicands | 3.85 | 3.70 | .36 | .321 | ||

Dist_Numbers | -88.33 | 446.72 | -.07 | .847 | ||

CC-IC | .20 | |||||

Dist_Indices | 48.38 | 36.33 | .36 | .210 | ||

Dist_Numbers | 249.33 | 311.15 | .22 | .440 | ||

No-CC-CO | .02 | |||||

Dist_Indices | -89.59 | 199.54 | -.19 | .663 | ||

Dist_Radicands | 4.20 | 19.04 | .10 | .830 | ||

Dist_Numbers | -163.47 | 1966.97 | -.03 | .935 | ||

No-CC-IC | .57 | |||||

Dist_Indices | 44.08 | 53.68 | .18 | .431 | ||

Dist_Radicands | 34.14 | 12.31 | .64 | .020 | ||

Dist_Numbers | -950.43 | 959.64 | -.23 | .345 | ||

No-CC-N | .70 | |||||

Dist_Indices | -39.74 | 48.07 | -.15 | .428 | ||

Dist_Radicands | 20.85 | 5.65 | .68 | .004 | ||

Dist_Numbers | -847.14 | 528.02 | -.29 | .140 |

In a second analysis, we used all 45 response times (one per participant) for each numerical distance value.

Type | Predictor | SE |
β | ^{2} |
||
---|---|---|---|---|---|---|

CC-CO | .00 | |||||

Dist_Radicands | 3.77 | 3.92 | .05 | .337 | ||

Dist_Numbers | -79.85 | 471.24 | -.01 | .866 | ||

CC-IC | .01 | |||||

Dist_Indices | 48.78 | 33.82 | .06 | .150 | ||

Dist_Numbers | 244.85 | 294.04 | .03 | .405 | ||

No-CC-CO | .00 | |||||

Dist_Indices | 38.60 | 175.69 | .02 | .826 | ||

Dist_Radicands | 3.20 | 14.50 | .02 | .825 | ||

Dist_Numbers | -154.78 | 1396.57 | -.01 | .912 | ||

No-CC-IC | .03 | |||||

Dist_Indices | 44.49 | 45.54 | .04 | .329 | ||

Dist_Radicands | 32.70 | 11.18 | .14 | .004 | ||

Dist_Numbers | -960.80 | 829.14 | -.06 | .247 | ||

No-CC-N | .02 | |||||

Dist_Indices | -40.51 | 71.86 | -.02 | .573 | ||

Dist_Radicands | 21.08 | 8.44 | .11 | .013 | ||

Dist_Numbers | -863.63 | 775.87 | -.05 | .266 |

While the first type of analysis could explain a substantial portion of the variance in response times, the second type of analysis had poor explanatory power. As with the accuracy data, however, both types of analyses yielded the same pattern of results. The distance between the irrational numbers was not a significant predictor of response time for the problems of any type. This result suggests that the participants did not rely on the irrational number magnitudes to solve the comparison problems of any of the problem types.

For incongruent and neutral problems without common components, the distance between the radicands was a significant predictor of response times, suggesting that the participants relied predominantly on comparing the radicands to solve the comparison problems of these types.

In sum, the analyses of distance effects are largely not in line with Hypothesis 3a, which predicted holistic distance effects for comparison problems without common components. The only holistic distance effect we could detect regarded accuracies on congruent problems without common components. Unexpectedly, we found a small distance effect for the components (indices in case of congruent problems, radicands in case of incongruent and neutral problems) for problems without common components. In line with Hypothesis 3b, we did not find holistic distance effects for those problems for which component strategies are valid. However, we did not find component distance effects for these problems either. We will discuss these results in the next section.

We investigated whether mathematically skilled adults are at all able to readily assess magnitudes of irrational numbers presented in symbolic notation. Our study extends previous research that has amply shown that educated adults are able to assess magnitudes of number symbols for integers and fractions.

Participants in our study were able to correctly solve almost all problems for which component strategies were applicable (i.e., problems of type CC-CO, CC-IC, and No-CC-N). This finding suggests that mathematically skilled adults are able to choose very efficient component strategies—and avoid holistic strategies—to solve problems for which these component strategies are valid. In line with that interpretation, the participants were less accurate for comparison problems for which component strategies were not valid (i.e., problems of type No-CC-CO and No-CC-IC). Although we would theoretically expect to find a component distance effect if participants relied on component strategies, we found this effect neither for the radicands nor for the indices for any of the comparison problems with common components. As mentioned earlier, it is likely that although participants actually used component strategies, the component distance effects were too small to be detectable in our sample of mathematically skilled adults. We should also note that due to the restrictions we set for constructing the items, there was not much variation among the indices between comparison problems within each problem type, which makes it less likely to detect a distance effect for these indices.

Surprisingly, while accuracy rates were rather high for incongruent problems without common components, they were at random chance level for congruent problems without common components. The low accuracies for congruent problems suggest that the participants were not able to assess the magnitudes of the irrational numbers. The question is how the participants were able to solve most of the incongruent problems correctly, since there is no straightforward alternative strategy (such as a component strategy) for these problems either. To find an answer to that question, we looked for systematic differences between the congruent and incongruent problems other than congruency. The differences in the numerical distances between the components might explain the results: while the mean numerical distance between indices was similar between congruent (3.00) and incongruent (4.07) problems, the mean numerical distance between radicands was much higher for congruent (61.57) than for incongruent (16.29) problems. This is necessarily the case, given the constraint that the number ranges of the irrational number components as well as the irrational number magnitudes were comparable between problem types. For an irrational number with a larger index to be the larger number (a defining feature of congruent problems), it is necessary to greatly increase the radicand. The participants in our study might have used an approximation strategy: if the distance between the radicands is relatively small, one can assume the radicands are equal and choose the number with the smaller index as the larger number. As even small differences in the indices strongly affect the size of the irrational numbers, comparing the indices yields reliable information in these cases. In fact, choosing the number with the smaller index was (by definition) a successful strategy for all the incongruent problems. The “reverse” distance effects for the radicands of the No-CC-IC problems support our interpretation: The distance between radicands was positively related to response times and slightly negatively related to accuracies (see

Following this interpretation, we would also expect a distance effect for the indices of the No-CC-IC problems, because after assuming that the radicands are equal, one has to compare the values of the indices. We did not find such a distance effect for the indices, which might, however, be due to the reasons discussed in the Introduction and earlier in this section.

In contrast to the incongruent problems without common components, the supposed approximation strategy (assuming the radicands are equal and choosing the number with the smaller index) was not applicable for the congruent problems without common components, because the distances between the radicands were too great. In fact, in these types of problems, the distances between the radicands were always so great that the number with the smaller index always had the smaller numerical value. The participants in our study might have overestimated the effect of the index on the magnitude of the irrational number, which would explain the difference in accuracies between congruent and incongruent problems without common components. Together, the data suggest that the participants in our study did not find valid strategies other than component strategies (including approximation regarding these components) for comparing the magnitudes of irrational numbers.

The analyses of distance effects support our conclusion that the participants did not engage in processing the magnitudes of the irrational numbers as a whole. We found no holistic distance effects for any type of comparison problems, except for accuracy in the congruent problems without common components. This means that only for this type of problems did the participants engage in assessing the magnitudes of the irrational numbers as a whole to some extent, but often failed to do so successfully. In fact, mean accuracies for many of the congruent problems without common components were far below chance level, suggesting that the participants made systematic errors on some problems. The finding that the distance between indices was negatively related to accuracies in problems of type No-CC-CO might support our interpretation that the participants overestimated the influence of the index on the numerical value of the irrational numbers. While for the No-CC-IC problems, a larger index always meant a smaller number, the reverse was true for the No-CC-CO problems.

The methodology of this study does not allow final conclusions concerning our participants’ strategies, because accuracies and response times are an only indirect measure of strategy use. Furthermore, we analyzed data on the level of the whole sample but not on an individual level. Further studies could analyze individual solution strategies to provide more insights into individual participants’ reasoning processes. Individual interviews with retrospective verbal reports or think aloud protocols (as described, e.g., in

We found a natural number bias for problems with common components in terms of response times but not accuracies. This finding is in line with previous research that has detected a natural number bias in comparison problems with fractions (

We did not find a natural number bias for problems without common components. On the contrary, the participants were particularly inaccurate and slow in solving congruent problems without common components, but highly accurate and much faster in solving incongruent problems. As discussed above, these differences might be due to other task features, so that we cannot conclude that participants were actually not biased.

This study suggests that even mathematically skilled adults struggle with assessing the magnitudes of irrational numbers. The study thus challenges the idea that quickly assessing numerical magnitudes is an essential part of numerical abilities beyond rational numbers. If quick assessment of magnitudes was essential for being competent with real numbers in general, then people with high mathematical skills should be able to assess magnitudes also of irrational numbers, even if doing so might be more demanding and time-consuming than for rational numbers. Although the participants spent most time on the congruent problems without common components, they performed only at random chance level. It seems that assessing magnitudes for irrational numbers is extremely demanding and cannot be achieved with straightforward strategies.

In view of evidence showing that activating magnitudes of natural numbers is much easier than activating magnitudes of fractions, some researchers have argued that the human cognitive architecture is privileged for processing natural numbers (

In view of its implication for education, this study suggests that assessing magnitudes can be extremely difficult even for mathematically skilled adults. While quickly assessing the magnitudes of number symbols is a feasible goal for specific cases of irrational numbers (e.g.,

Common Components |
Common Components |
No Common Components Congruent |
No Common Components Incongruent |
No Common Components Neutral |
---|---|---|---|---|

The authors would like to thank the mathematically skilled participants of this study for their willingness to work on irrational number comparison problems.

The authors have declared that no competing interests exist.

Note that in academic mathematics, other numbers exist (such as complex numbers), which do not have magnitudes in this sense. That is, there is no order relation for these numbers, so it is not possible to decide which of two numbers is “larger”.

Note that there are also rational numbers that have an infinite number of digits after the decimal point (those with recurrent decimals such as 1/3 = 0.333…). However, for these numbers it is possible to notate all recurrent digits (because their number is finite), whereas this is not the case for irrational numbers. This means that irrational numbers cannot be represented in decimal notation.

Note that in the reminder of this article, we use the term “irrational numbers in their symbolic notation” or simply “symbolic notation” to refer to the exact value of irrational numbers presented with the root sign. We do not refer to the decimal notation of numbers, which would only allow approximate representations of irrational number values.

Of course, if one is informed about the category of the comparison problem (congruent or incongruent), one can make valid decisions without assessing the number magnitudes. However, participants are typically

The authors have no funding to report.